multiwavelets for second kind integral equations we consider a galerkin method for an elliptic pseudodifferential operator of order zero on a two dimensional manifold we use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis using this basis we can compress the stiffness matrix from n2 to o n log n nonzero entries and still obtain up to log n terms the same convergence rates as for the exact galerkin method

wavelet solvers for hp fem discretizations in 3d using hexahedral elements in this paper we investigate the discretization of an elliptic boundary value problem in 3d by means of the hp version of the finite element method using a mesh of hexahedrons the corresponding linear system is solved by a preconditioned conjugate gradient method the construction of the preconditioner is based on an inexact additive overlapping schwarz method the remaining subproblems are treated by a tensor product based preconditioner this preconditioner uses a basis transformation into a basis which is stable in l 2 and h 1 several numerical examples show the efficiency of the proposed method

an optimal adaptive wavelet method for first order system least squares in this paper it is shown that any well posed 2nd order pde can be reformulated as a well posed first order least squares system this system will be solved by an adaptive wavelet solver in optimal computational complexity the applications that are considered are second order elliptic pdes with general inhomogeneous boundary conditions and the stationary navier stokes equations

appending boundary conditions by lagrange multipliers analysis of the lbb condition this paper is concerned with the analysis of discretization schemes for second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of lagrange multipliers specifically we show how the validity of the ladyskaja babusska brezzi lbb condition for the corresponding saddle point problems depends on the various ingredients of the involved discretizations the main result states that the lbb condition is satisfied whenever the discretization step length on the boundary h gamma sim 2 ell is somewhat bigger than the one on the domain h omega sim 2 j this is quantified through constants stemming from the trace theorem norm equivalences for the multiplier spaces on the boundary and direct and inverse inequalities in order to better understand the interplay of these constants we then specialize the setting to wavelet discretizations in this case the stability criteria can be stated solely in terms of spectral properties of wavelet representations of the trace operator we conclude by illustrating our theoretical findings by some numerical experiments we stress that the results presented here apply to any spatial dimension and to a wide selection of lagrange multiplier spaces which in particular need not be traces of the trial spaces however we do always assume that a hierarchy of nested trial spaces is given

construction of aharonov berry s superoscillations a simple method is described for constructing functions that superoscillate at an arbitrarily chosen wavelength scale our method is based on the technique of oversampled signal reconstruction this allows us to explicitly demonstrate that the observed fragility of superoscillating wavefunctions is indeed mathematically closely connected to what in the communication theory community is known as the instability of oversampled signal reconstruction confirming a previous conjecture this is of potential interest for example concerning the understanding of the practical difficulties in experimentally producing superoscillatory wavefunctions

superoscillations with optimum energy concentration oscillations of a bandlimited signal at a rate faster than the bandlimit are called superoscillations and have applications e g in superresolution and superdirectivity the synthesis of superoscillating signals is a numerically difficult problem minimum energy superoscillatory signals seem attractive for applications because i the minimum energy solution is unique ii it has the smallest energy cost iii it may yield a signal of the smallest possible amplitude on the negative side superoscillating functions of minimum energy depend heavily on cancellation and give rise to expressions that have very large coefficients furthermore these coefficients have to be found by solving equations that are very ill conditioned surprisingly we show that by dropping the minimum energy requirement practicality can be gained rather than lost we give a method of constructing superoscillating signals that leads to coefficients and condition numbers that are smaller by several orders of magnitude than the minimum energy solution yet yields energies close to the minimum in contrast with the minimum energy method which builds superoscillations by linearly combining functions with an ill conditioned gram matrix our method combines orthonormal functions the gram matrix of which is obviously the identity another feature of the method is that it yields the superoscillatory signal that maximises the energy concentration in a given set which may or may not include the superoscillatory segment

sparse finite elements for elliptic problems with stochastic loading we formulate elliptic boundary value problems with stochastic loading in a bounded domain d d we show well posedness of the problem in stochastic sobolev spaces and we derive a deterministic elliptic pde in d d for the spatial correlation of the random solution we show well posedness and regularity results for this pde in a scale of weighted sobolev spaces with mixed highest order derivatives discretization with sparse tensor products of any hierarchic finite element fe spaces in d yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data multilevel preconditioning in d d allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation

superoscillations of prescribed amplitude and derivative superoscillations occur when a bandlimited signal oscillates at a rate higher than its maximum frequency we show that it is possible to construct superoscillations by constraining not only the value of the signal but also that of its derivative this allows a better control of the shape of the superoscillations we find that for any given bandwidth no matter how small there exists a unique signal of minimum energy that satisfies any combination of amplitude and derivative constraints on a sampling grid as fine as desired we determine the energy of the signal for any grid regular or irregular when the set of derivative constraints is empty the results reduce to minimum energy interpolation in the absence of amplitude constraints we obtain pure derivative constrained extremals the flexibility gained by having two different types of constraints makes it possible to design superoscillations based only on amplitudes based only on derivatives or based on both in the last case the amplitude and derivative sampling grids can be interleaved or aligned we explore this flexibility to build superoscillations that cost less energy illustrating examples are given

adaptive solution of operator equations using wavelet frames in adaptive wavelet methods ii beyond the elliptic case of cohen dahmen and devore found comput math 2 2002 pp 203 245 an adaptive method has been developed for solving general operator equations using a riesz basis of wavelet type for the energy space the operator equation is transformed into an equivalent matrix vector system this system is solved iteratively where the application of the infinite stiffness matrix is replaced by an adaptive approximation assuming that the stiffness matrix is sufficiently compressible i e that it can be sufficiently well approximated by sparse matrices it was proved that the adaptive method has optimal computational complexity in the sense that it converges with the same rate as the best n term approximation for the solution assuming that the latter would be explicitly available the condition concerning compressibility requires that dependent on their order the wavelets have sufficiently many vanishing moments and that they be sufficiently smooth however except on tensor product domains wavelets that satisfy this smoothness requirement are not easy to construct in this paper we write the domain or manifold on which the operator equation is posed as an overlapping union of subdomains each of them being the image under a smooth parametrization of the hypercube by lifting wavelets on the hypercube to the subdomains we obtain a em frame for the energy space with this frame the operator equation is transformed into a matrix vector system after which this system is solved iteratively by an adaptive method similar to the one from the work of cohen dahmen and devore with this approach frame elements that have sufficiently many vanishing moments and are sufficiently smooth something needed for the compressibility are easily constructed by handling additional difficulties due to the fact that a frame gives rise to an underdetermined matrix vector system we prove that this adaptive method has optimal computational complexity

some mathematical properties of superoscillations in this paper we give a possible mathematical setting for superoscillations we define the set of superoscillation in terms of the uniform convergence of functions on such a set and study the problem of the approximation of a function by superoscillating functions

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

the stiffness of self similar fractals abstract a method to derive the stiffness of self similar elastic fractals is presented based on structural mechanics principles and a physically motivated similarity criterion which is assumed as a postulate using this method the stiffnesses of both the von koch curve and the sierpinski gasket in the small deformation regime are derived for these fractal structures it is shown that the stiffness matrix is completely determined by a single elastic constant the procedure to tile a planar domain with sierpinski gaskets is explored and shown to require the consideration of hexagonal shaped combinations of gaskets joined continuously along their edges this continuity leads to a phenomenon of geometrically induced inextensibility along the common edges after deriving the stiffness matrix for the basic hexagon the analog of the boussinesq flamant problem for a tiled half plane is solved numerically to demonstrate the potential of the method in modeling of solid mechanics applications

wavelet galerkin schemes for boundary integral equations implementation and quadrature in the present paper we consider the fully discrete wavelet galerkin scheme for the fast solution of boundary integral equations in three dimensions it produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that stays proportional to the number of unknowns we focus on algorithmical details of the scheme in particular on numerical integration of relevant matrix coefficients we illustrate the proposed algorithm by numerical results

continuous limit of discrete systems with long range interaction discrete systems with long range interactions are considered continuous medium models as continuous limit of discrete chain system are defined long range interactions of chain elements that give the fractional equations for the medium model are discussed the chain equations of motion with long range interaction are mapped into the continuum equation with the riesz fractional derivative we formulate the consistent definition of continuous limit for the systems with long range interactions in this paper we consider a wide class of long range interactions that give fractional medium equations in the continuous limit the power law interaction is a special case of this class

finite element wavelets with improved quantitative properties in w dahmen r stevenson element by element construction of wavelets satisfying stability and moment conditions siam j numer anal 37 1 1999 319 352 electronic finite element wavelets were constructed on polygonal domains or lipschitz manifolds that are piecewise parametrized by mappings with constant jacobian determinants the wavelets could be arranged to have any desired order of cancellation properties and they generated stable bases for the sobolev spaces h s for s 32 or s 1 on manifolds unfortunately it appears that the quantitative properties of these wavelets are rather disappointing in this paper we modify the construction from the above mentioned work to obtain finite element wavelets which are much better conditioned

computation of differential operators in wavelet coordinates in found comput math 2 2002 pp 203 245 cohen dahmen and devore proposed an adaptive wavelet algorithm for solving general operator equations assuming that the operator defines a boundedly invertible mapping between a hilbert space and its dual and that a riesz basis of wavelet type for this hilbert space is available the operator equation is transformed into an equivalent well posed infinite matrix vector system this system is solved by an iterative method where each application of the infinite stiffness matrix is replaced by an adaptive approximation it was shown that if the errors of the best linear combinations from the wavelet basis with n terms are o n s for some s 0 which is determined by the besov regularity of the solution and the order of the wavelet basis then approximations yielded by the adaptive method with n terms also have errors of o n s moreover their computation takes only 0 n operations provided s s assuming that each entry of the stiffness matrix is exactly available at unit cost generally these entries have to be approximated using numerical quadrature in this paper restricting ourselves to differential operators we develop a numerical integration scheme that computes these entries giving an additional error that is consistent with the approximation error whereas in each column the average computational cost per entry is 0 1 as a consequence we can conclude that the adaptive wavelet algorithm has optimal computational complexity

adaptive boundary element methods with convergence rates this paper presents adaptive boundary element methods for positive negative as well as zero order operator equations together with proofs that they converge at certain rates the convergence rates are quasi optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods in particular no saturation type assumption is used the main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods and an inverse type inequality involving boundary integral operators on locally refined finite element spaces

space time approximation with sparse grids in this article we introduce approximation spaces especially suited for the approximation of solutions of parabolic problems which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time proper truncation then leads to so called space time sparse grid spaces for a uniform discretization of the spatial space of dimension d with o nd degrees of freedom these spaces involve for d 1 also only o nd degrees of freedom for the discretization of the whole space time problem but they provide the same approximation rate as classical space time finite element spaces which need o nd 1 degrees of freedoms this makes these approximation spaces well suited for conventional parabolic and time dependent optimization problems we analyze the approximation properties and the dimension of these sparse grid space time spaces for general stable multiscale bases we then restrict ourselves to an interpolatory multiscale basis i e a hierarchical basis here to be able to handle also complicated spatial domains omega we construct the hierarchical basis from a given spatial finite element basis as follows first we determine coarse grid points recursively over the levels by the coarsening step of the algebraic multigrid method then we derive interpolatory prolongation operators between the respective coarse and fine grid points by a least squares approach this way we obtain an algebraic hierarchical basis for the spatial domain which we then use in our space time sparse grid approach we give numerical results on the convergence rate of the interpolation error of these spaces for various space time problems with two spatial dimensions implementational issues data structures and questions of adaptivity also are addressed to some extent

a sparse laplacian in tensor product wavelet coordinates we construct a wavelet basis on the unit interval with respect to which both the infinite mass and stiffness matrix corresponding to the one dimensional laplacian are truly sparse and boundedly invertible as a consequence the infinite stiffness matrix corresponding to the laplacian on the n dimensional unit box with respect to the n fold tensor product wavelet basis is also sparse and boundedly invertible this greatly simplifies the implementation and improves the quantitative properties of an adaptive wavelet scheme to solve the multi dimensional poisson equation the results extend to any second order partial differential operator with constant coefficients that defines a boundedly invertible operator

exact nonparaxial transmission of subwavelength detail using superoscillations any object described by a target function that need not be band limited can be sampled at any chosen set of points and then propagated without evanescent waves using a function which is band limited so as to be imaged exactly i e nonparaxially at multiples of a given repetition distance if the samples span a sub wavelength region the repeated images are superoscillatory the number n of samples is equal to the repetition distance measured in wavelengths and also to the number of plane waves in the propagating field if n 1 the waves form a quasi continuum and asymptotics enables an almost explicit description of the superoscillations but the matrix involved is ill conditioned many of its eigenvalues are very small so this method of sub wavelength imaging would be pathologically sensitive to noise and the depth of focus is exponentially small

representing fractals by superoscillations fractals provide an extreme test of representing fine detail in terms of band limited functions i e by superoscillations we show that this is possible using the example of the weierstrass nondifferentiable fractal if this is truncated at an arbitrarily fine scale it can be expressed to any desired accuracy with a simple superoscillatory function in illustrative simulations fractals truncated with fastest frequency 216 are easily represented by superoscillations with fastest fourier frequency 1

wavelets on manifolds an optimized construction a key ingredient of the construction of biorthogonal wavelet bases for sobolev spaces on manifolds which is based on topological isomorphisms is the hestenes extension operator here we firstly investigate whether this particular extension operator can be replaced by another extension operator our main theoretical result states that an important class of extension operators based on interpolating boundary values cannot be used in the construction setting required by dahmen and schneider in the second part of this paper we investigate and optimize the hestenes extension operator the results of the optimization process allow us to implement the construction of biorthogonal wavelets from dahmen and schneider as an example we illustrate a wavelet basis on the 2 sphere

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

multiresolution approximate inverse preconditioners we introduce a new preconditioner for elliptic pdes on unstructured meshes using a wavelet inspired basis we compress the inverse of the matrix allowing an effective sparse approximate inverse by solving the sparsity vs accuracy conflict the key issue in this compression is to use second generation wavelets which can be adapted to the unstructured mesh the true boundary conditions and even the pde coefficients we also show how this gives a new perspective on multiresolution algorithms such as multigrid interpreting the new preconditioner as a variation on node nested multigrid in particular we hope the new preconditioner will combine the best of both worlds fast convergence when multilevel methods can succeed but with robust performance for more difficult problems the rest of the paper discusses the core issues for the preconditioner ordering and construction of a factored approximate inverse in the multiresolution basis robust interpolation on unstructured meshes automatic mesh coarsening and purely algebraic alternatives some exploratory numerical experiments suggest the superiority of the new basis over the standard basis for several tough problems including discontinuous anisotropic coefficients strong convection and indefinite reaction problems on unstructured meshes with scalability like hierarchical basis methods achieved

compression techniques for boundary integral equations asymptotically optimal complexity estimates matrix compression techniques in the context of wavelet galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense the fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that is proven to stay proportional to the number of unknowns key issues are the second compression which reduces the near field complexity significantly and an additional a posteriori compression the latter is based on a general result concerning an optimal work balance that applies in particular to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time

a continuous wavelet galerkin method for the linear wave equation we consider the continuous space time galerkin method for the linear second order wave equation proposed by french and peterson in 1996 a bottleneck for this approach is how to solve the discrete problems effectively in this paper we tackle this bottleneck by essentially employing wavelet bases in space we show how to decouple the corresponding linear system and we prove that the resulting subsystems can be uniformly preconditioned by simple diagonal preconditioners leading to efficient iterative solutions

on interpolatory divergence free wavelets we construct interpolating divergence free multiwavelets based on cubic hermite splines we give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields we also show that the standard interpolatory wavelets based on the deslauriers dubuc interpolatory scheme or on interpolatory splines cannot be used to construct compactly supported divergence free interpolatory wavelets

a note on differential operators of infinite order a sufficient condition for entire functions f and g to be such that the series m 0 f m 0 g m m represents an entire function is established and in that case the growth of the resulting function is described

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

incomplete sampling series and the recovery of missing samples from oversampled band limited signals it is well known that a band limited oversampled signal is completely determined even if an arbitrary finite number of samples is lost it is shown that an alternative simple proof of this fact carries over to generalized sampling expansions more precisely it is shown that any finite number of missing samples can be recovered from the remaining ones in the case of generalized kramer sampling expansions if an appropriate oversampling constraint is satisfied the recovery can be accomplished either iteratively or noniteratively

composite wavelet bases for operator equations this paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n cube these bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains this covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo differential equations with special emphasis on boundary integral equations in this case it is natural to model the boundary surface as a union of parametric images of the unit cube it will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch here the relevant properties are the validity of norm equivalences in certain ranges of sobolev scales as well as appropriate moment conditions

superoscillation phenomena in so 3 we prove a new asymptotic formula for the wigner d functions as a consequence of the discovery of superoscillatory behaviour in so 3

higher order mortar finite element methods in 3d with dual lagrange multiplier bases mortar methods with dual lagrange multiplier bases provide a flexible efficient and optimal way to couple different discretization schemes or nonmatching triangulations here we generalize the concept of dual lagrange multiplier bases by relaxing the condition that the trace space of the approximation space at the slave side with zero boundary condition on the interface and the lagrange multiplier space have the same dimension we provide a new theoretical framework within this relaxed setting which opens a new and simpler way to construct dual lagrange multiplier bases for higher order finite element spaces as examples we consider quadratic and cubic tetrahedral elements and quadratic serendipity hexahedral elements numerical results illustrate the performance of our approach

pre and post selection weak values and contextuality by analysing the concept of contextuality bell kochen specker in terms of pre and post selection it is possible to assign definite values to observables in a new and surprising way physical reasons are presented for restrictions on these assignments when measurements are performed which do not disturb the pre and post selection i e weak measurements then novel experimental aspects of contextuality can be demonstrated we also prove that every pps paradox with definite predictions directly implies quantum contextuality which is introduced as the analogue of contextuality at the level of quantum mechanics rather than at the level of hidden variable theories finally we argue that certain results of these measurements e g eccentric weak values outside the eigenvalue spectrum cannot be explained by a classical like hidden variable theory

adaptive piecewise tensor product wavelets scheme for laplace interface problems abstract a laplace type boundary value problem is considered with a generally discontinuous diffusion coefficient a domain decomposition technique is used to construct a piecewise tensor product wavelet basis that when normalized w r t the energy norm has riesz constants that are bounded uniformly in the jumps an adaptive wavelet galerkin method is applied to solve the boundary value problem with the best nonlinear approximation rate from the basis in linear computational complexity although the solutions are far from smooth numerical experiments in two dimensions show rates as for a one dimensional smooth solution the latter being possible because of the tensor product construction

yield optimized superoscillations superoscillating signals are band limited signals that oscillate in some region faster their largest fourier component while such signals have many scientific and technological applications their actual use is hampered by the fact that an overwhelming proportion of the energy goes into that part of the signal which is not superoscillating in the present paper we consider the problem of optimization of such signals the optimization that we describe here is that of the superoscillation yield the ratio of the energy in the superoscillations to the total energy of the signal given the range and frequency of the superoscillations the constrained optimization leads to a generalized eigenvalue problem which is solved numerically it is noteworthy that it is possible to increase further the superoscillation yield at the cost of slightly deforming the oscillatory part of the signal while keeping the average frequency we show how this can be done gradually which enables a tradeoff between the distortion and the yield we show how to apply this approach to nontrivial domains and explain how to generalize this to higher dimensions

direct construction of superoscillations oscillations of a bandlimited signal at a rate faster than its maximum frequency are called superoscillations and have been found useful e g in connection with superresolution and superdirectivity we consider signals of fixed bandwidth and with a finite or infinite number of samples at the nyquist rate which are regarded as the adjustable signal parameters we show that this class of signals can be made to superoscillate by prescribing its values on an arbitrarily fine and possibly nonuniform grid the superoscillations can be made to occur at a large distance from the nonzero samples of the signal we give necessary and sufficient conditions for the problem to have a solution in terms of the nature of the two sets involved in the problem since the number of constraints can in general be different from the number of signal parameters the problem can be exactly determined underdetermined or overdetermined we describe the solutions in each of these situations the connection with oversampling and variational formulations is also discussed

biorthogonal bases with local support and approximation properties we construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension in contrast to earlier approaches these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p 1 working with gaus lobatto nodes we find an interesting connection between biorthogonality and quadrature formulas one important application of these newly constructed biorthogonal basis functions are two dimensional mortar finite elements the weak continuity condition of the constrained mortar space is realized in terms of our new dual bases as a result local static condensation can be applied which is very attractive from the numerical point of view numerical results are presented for cubic mortar finite elements

wavelet based schemes for linear advection dispersion equation abstract in this paper two wavelet based adaptive solvers are developed for linear advection dispersion equation the localization properties and multilevel structure of the wavelets in the physical space are used for adaptive computational methods for solution of equation which exhibit both smooth and shock like behaviour the first framework is based on wavelet galerkin and the second is based on multiscale decomposition of finite element method coiflet wavelet filter is incorporated in both the methods the main advantage of both the adaptive methods is the elimination of spurious oscillations at very high peclet number

element by element construction of wavelets satisfying stability and moment conditions in this paper we construct a class of locally supported wavelet bases for c0 lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds the wavelet bases are stable in the sobolev spaces mathcal h s for s frac 3 2 s leq 1 on lipschitz manifolds and the wavelets can in principle be arranged to have any desired order of vanishing moments as a consequence these bases can be used e g for constructing an optimal solver of discretized mathcal h s elliptic problems for s in above ranges the construction of the wavelets consists of two parts an implicit part involves some computations on a reference element which for each type of finite element space have to be performed only once in addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh the only condition we need for this construction to work is that the refinements of initial elements are uniform we will show that the wavelet bases can be implemented efficiently

adaptive wavelet algorithms for elliptic pde s on product domains with standard isotropic approximation by piecewise polynomials of fixed order in a domain d subset of r d the convergence rate in terms of the number n of degrees of freedom is inversely proportional to the space dimension d this so called curse of dimensionality can be circumvented by applying sparse tensor product approximation when certain high order mixed derivatives of the approximated function happen to be bounded in l 2 it was shown by nitsche 2006 that this regularity constraint can be dramatically reduced by considering best n term approximation from tensor product wavelet bases when the function is the solution of some well posed operator equation dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible applying piecewise smooth wavelets we verify this compressibility for general non separable elliptic pdes in tensor domains applications of the general theory developed include adaptive galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust i e independent of the scale parameters resp of the size of the anisotropy

extrapolating a band limited function from its samples taken in a finite interval a band limited signal of finite energy can be reconstructed from its samples taken at the nyquist rate moreover the reconstruction is stable a feature crucial for implementation a small error in the sample values generates only a correspondingly small error in the resulting signal the nyquist sample values are mutually independent so that knowledge of them in a given interval t leq t leq 0 provides hardly any information about the behavior of the signal outside the interval however when the samples are taken at a greater rate a case referred to as over sampling they are interrelated and this redundancy can be exploited in various ways to improve the behavior of the reconstruction procedure a natural question is whether it can also be used to form accurate estimates of the signal outside the interval of observation this problem is relevant as well to prediction theory with oversampling when t infty so that the samples are known on the entire half line t they determine the signal everywhere although the reconstruction is now no longer stable here we examine the case of finite t of course a finite amount of data can yield only limited accuracy we prove that the samples can be used to form an approximation to the signal outside the sampling interval with an error which asymptotically as t rightarrow infty decreases exponentially in t over a range which grows linearly with t however as in the limiting case these approximations are not useful in practice since they require the sample values to be known exactly in the presence of measurement error the nature of the results changes good approximations are now available for only a bounded distance outside the interval of observation regardless of its length and their accuracy and range of validity can be increased only by improving the precision of sample reading since physical measurements are never perfect it is this conclusion which counts for applications the same results hold for the extrapolation of bounded band limited signals

suboscillations with arbitrary shape we report a method for constructing bandpass functions that approximate a given analytic function with arbitrary accuracy over a finite interval a corollary is that bandpass functions can be obtained that oscillate arbitrarily slower than their minimum frequency component a counter intuitive phenomenon known as suboscillations

wavelet galerkin algorithms for boundary integral equations the implementation of a fast wavelet based galerkin discretization of second kind integral equations on piecewise smooth surfaces gamma subset ir 3 is described it allows meshes consisting of triangles as well as quadrilaterals the algorithm generates a sparse approximate stiffness matrix with cn o n log n 2 nonvanishing entries in o n log n 4 operations where n is the number of degrees of freedom on the boundary while essentially retaining the asymptotic convergence rate of the full galerkin scheme a new proof of the matrix compression estimates is given based on derivative free kernel estimates the condition number of the sparse stiffness matrices is bounded independently of the meshwidth the data structure containing the compressed stiffness matrix is described in detail it requires o cn memory and can be set up in o cn operations numerical experiments show that the asymptotic performance estimates apply for moderate n problems with n 106 degrees of freedom were computed in core on a workstation the impact of various parameters in the compression scheme on the performance and the accuracy of the algorithm is studied

static kinematic duality and the principle of virtual work in the mechanics of fractal media abstract the framework for the mechanics of solids deformable over fractal subsets is outlined while displacements and total energy maintain their canonical physical dimensions renormalization group theory permits to define anomalous mechanical quantities with fractal dimensions i e the fractal stress and the fractal strain e a fundamental relation among the dimensions of these quantities and the hausdorff dimension of the deformable subset is obtained new mathematical operators are introduced to handle these quantities in particular classical fractional calculus fails to this purpose whereas the recently proposed local fractional operators appear particularly suitable the static and kinematic equations for fractal bodies are obtained and the duality principle is shown to hold finally an extension of the gauss green theorem to fractional operators is proposed which permits to demonstrate the principle of virtual work for fractal media

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

superoscillations with arbitrary polynomial shape we present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval these functions are obtained as the product of the polynomial with a sufficiently flat bandlimited envelope function whose fourier transform has at least continuous derivatives and an nth derivative of bounded variation n being the order of the polynomial polynomials of arbitrarily high order can be approximated if the fourier transform of the envelope is smooth i e a bump function

evolution of superoscillations for schrodinger equation in a uniform magnetic field aharonov berry superoscillations are band limited functions that oscillate faster than their fastest fourier component superoscillations appear in several fields of science and technology such as aharonov s weak measurement in quantum mechanics in optics and in signal processing an important issue is the study of the evolution of superoscillations using the schrodinger equation when the initial datum is a weak value some superoscillatory functions are not square integrable but they are real analytic functions that can be extended to entire holomorphic functions this fact leads to the study of the continuity of a class of convolution operators acting on suitable spaces of entire functions with growth conditions in this paper we study the evolution of a superoscillatory initial datum in a uniform magnetic field moreover we collect some results on convolution operators that appear in the theory of superoscillatory functions using a direct approach that allows the convolution operators to have non

typical weak and superweak values weak values resulting from the action of an operator on a preselected state when measured after postselection by a different state can lie outside the spectrum of eigenvalues of the operator they can be superweak this phenomenon can be quantified by averaging over an ensemble of the two states and calculating the probability distribution of the weak values if there are many eigenvalues distributed within a finite range this distribution takes a simple universal generalized lorentzian form and the superweak probablility of weak values outside the spectrum can be as large as 1 1 0 293 by contrast the familiar expectation values always lie within the spectral range and their distribution although approximately gaussian for many eigenvalues is not universal

space time adaptive wavelet methods for optimal control problems constrained by parabolic evolution equations an adaptive algorithm based on wavelets is proposed for the efficient numerical solution of a control problem governed by a linear parabolic evolution equation first the constraints are represented by means of a full weak space time formulation as a linear system in ell 2 in wavelet coordinates following a recent approach by schwab and stevenson second a quadratic cost functional involving a tracking type term for the state and a regularization term for the distributed control is also formulated in terms of ell 2 sequence norms of wavelet coordinates this functional serves as a representer for a functional involving different sobolev norms with possibly nonintegral smoothness parameter standard techniques from optimization are then used to derive the first order necessary conditions as a coupled system in ell 2 coordinates for this purpose an adaptive method is proposed which can be interpreted as an inexact gradient method for the control in each iteration step the primal and adjoint systems are solved up to a prescribed accuracy by the adaptive algorithm it is shown that the adaptive algorithm converges moreover the algorithm is proved to be asymptotically optimal the convergence rate achieved for computing each of the components of the solution state adjoint state and control up to a desired target tolerance is asymptotically the same as the wavelet best n term approximation of each solution component and the total computational work is proportional to the number of computational unknowns

wavelet adaptive method for second order elliptic problems boundary conditions and domain decomposition summary wavelet methods allow to combine high order accuracy multilevel preconditioning techniques and adaptive approximation in order to solve efficiently elliptic operator equations one of the main difficulty in this context is the efficient treatment of non homogeneous boundary conditions in this paper we propose a strategy that allows to append such conditions in the setting of space refinement i e adaptive discretizations of second order problems our method is based on the use of compatible multiscale decompositions for both the domain and its boundary and on the possibility of characterizing various function spaces from the numerical properties of these decompositions in particular this allows the construction of a lifting operator which is stable for a certain range of smoothness classes and preserves the compression of the solution in the wavelet basis an explicit construction of the wavelet bases and the lifting is proposed on fairly general domains based on c 0 conforming domain decomposition techniques

adaptive wavelet schemes for parabolic problems sparse matrices and numerical results a simultaneous space time variational formulation of a parabolic evolution problem is solved with an adaptive wavelet method this method is shown to converge with the best possible rate in linear complexity thanks to the use of tensor product bases there is no penalty in complexity due to the additional time dimension special wavelets are designed such that the bi infinite system matrix is sparse this sparsity largely simplifies the implementation and improves the quantitative properties of the adaptive wavelet method numerical results for an ode and the heat equation are presented

covariant information density cutoff in curved space time in information theory the link between continuous information and discrete information is established through well known sampling theorems sampling theory explains for example how frequency filtered music signals are reconstructible perfectly from discrete samples in this letter sampling theory is generalized to pseudo riemannian manifolds this provides a new set of mathematical tools for the study of space time at the planck scale theories formulated on a differentiable space time manifold can be equivalent to lattice theories there is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity

piecewise linear prewavelets on arbitrary triangulations this paper studies locally supported piecewise linear prewavelets on bounded triangulations of arbitrary topology it is shown that a concrete choice of prewavelets form a basis of the wavelet space when the degree of the vertices in the triangulation is not too high

quantum fractals in boxes a quantum wave with probability density confined by dirichlet boundary conditions in a d dimensional box of arbitrary shape and finite surface area evolves from the uniform state for almost all positions the graph of the evolution of p is a fractal curve with dimension for almost all times t the graph of the spatial probability density p is a fractal hypersurface with dimension when d 1 there are in addition to these generic time and space fractals infinitely many special quantum revival times when p is piecewise constant and infinitely many special spacetime slices for which the dimension of p is 5 4 if the surface of the box is a fractal with dimension simple arguments suggest that the dimension of the time fractal is and that of the space fractal is

a simple model of aharonov berry s superoscillations a simple method of constructing superoscillatory functions is given it is based on a zero shifting principle of functions in paley wiener space

evolution of superoscillatory data weak measurements and the theory of weak values have a very deep meaning in quantum mechanics and new phenomena associated with them has recently been observed experimentally this theory has also directly led to the notion of superoscillating sequences of functions in this paper we consider cauchy problems with superoscillatory initial conditions in particular the cauchy problem for the schrodinger equation and some of its variations and we give conditions under which the superoscillations persist in time our work is based on results from the theory of formal solutions to cauchy problems and on the study of the specific growth of superoscillatory sequences when regarded as functions of a complex variable there are two main aims of this paper one is to explain the mathematical tools that are necessary to study superoscillations also repeating a few results that we have already proved in other papers in order to clarify the strategy the second aim is to show that our technique applies to a large class of problems showing under which conditions the superoscillatory phenomenon persists finally we point out that our strategy can be applied also to non constant coefficients differential equations as the quantum harmonic oscillator

c 1 hierarchical riesz bases of lagrange type on powell sabin triangulations in this paper we construct c1 continuous piecewise quadratic hierarchical bases on powell sabin triangulations of arbitrary polygonal domains in r2 our bases are of lagrange type instead of the usual hermite type and under some weak regularity assumptions on the underlying triangulations we prove that they form strongly stable riesz bases for the sobolev spaces hs with s 1 5 2 especially the case s 2 is of interest because we can use the corresponding hierarchical basis for preconditioning fourth order elliptic equations leading to uniformly well conditioned stiffness matrices compared to the hierarchical riesz bases by davydov and stevenson hierarchical riesz bases for hs 1 s 5 2 constructive approximation to appear our construction is simpler

a stabilized lifting construction of wavelets on irregular meshes on the interval we propose a stabilization of the two step lifting construction of sweldens for wavelet bases on irregular meshes on the interval the method yields biorthogonal bases in which the wavelets on both the primal and the dual side have a chosen number of vanishing moments and have local support we combine it with a constrained local semiorthogonalization to obtain a stabilized variant numerical results show that the wavelet bases are well conditioned having approximately the same condition numbers as wavelet bases obtained by global semiorthogonalization while the average support is not much larger than in the unstabilized construction

analysis of superoscillatory wave functions surprisingly differentiable functions are able to oscillate arbitrarily faster than their highest fourier component would suggest the phenomenon is called superoscillation recently a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter intuitive physical effects following up on this work we here present more general methods which allow the calculation of superoscillatory wave functions with custom designed physical properties we give concrete examples and we prove results about the limits to superoscillatory behavior we also give a simple and intuitive new explanation for the exponential computational cost of superoscillations

wavelet discretizations of parabolic integrodifferential equations we consider parabolic problems dot u au f in 0 t times omega t infty where omega subset mathbb r d is a bounded domain and a is a strongly elliptic classical pseudodifferential operator of order rho in 0 2 in tilde h rho 2 omega we use a theta scheme for time discretization and a galerkin method with n degrees of freedom for space discretization the full galerkin matrix for a can be replaced with a sparse matrix using a wavelet basis and the linear systems for each time step are solved approximatively with gmres we prove that the total cost of the algorithm for m time steps is bounded by o mn log n beta operations and o n log n beta memory we show that the algorithm gives optimal convergence rates up to logarithmic terms for the computed solution with respect to l2 in time and the energy norm in space

an adaptive meshfree diffusion wavelet method for partial differential equations on the sphere abstract an adaptive meshfree diffusion wavelet method for solving partial differential equations pdes on the sphere is developed approximation formulae for laplacian beltrami 2 and gradient operators are derived using radial basis functions rbfs and the convergence of these approximations to 2 and is verified for two test functions the matrix approximating the laplace beltrami operator is used for the construction of the diffusion wavelet the diffusion wavelet is used for the adaptation of node arrangement as well as for the fast computation of dyadic powers of the matrices involved in the numerical solution of the pde the efficiency of the method is that the same operator is used for the construction of the diffusion wavelet and for the approximation of the differential operators as a part of the wavelet method the behaviour of the compression error with respect to different parameters involved in the construction of the diffusion wavelet is tested the cpu time taken by the proposed method is compared with the cpu time taken by the rbf based collocation method and it is observed that the proposed method performs better the method is tested on three test problems namely spherical diffusion equation linear problem of computing a moving steep front nonlinear and problem of turing patterns system of nonlinear reaction diffusion equations

adaptive optimization of convex functionals in banach spaces this paper is concerned with optimization or minimization problems that are governed by operator equations such as partial differential or integral equations and thus are naturally formulated in an infinite dimensional function space v we first construct a prototype algorithm of steepest descent type in v and prove its convergence by using a riesz basis in v we can transform the minimization problem into an equivalent one posed in a sequence space of type ell p we convert the prototype algorithm into an adaptive method in ell p this algorithm is shown to be convergent under mild conditions on the parameters that appear in the algorithm under more restrictive assumptions we are also able to establish the rate of convergence of our algorithm and prove that the work accuracy balance is asymptotically optimal finally we give two particular examples

the self similar field and its application to a diffusion problem we introduce a continuum approach which accounts for self similarity as a symmetry property of an infinite medium a self similar laplacian operator is introduced which is the source of self similar continuous fields in this way self similar symmetry appears in an analogous manner as transverse isotropy or cubic symmetry of a medium as a consequence of the self similarity the laplacian is a non local fractional operator obtained as the continuum limit of the discrete self similar laplacian introduced recently by michelitsch et al 2009 phys rev e 80 011135 the dispersion relation of the laplacian and its green s function is deduced in closed forms as a physical application of the approach we analyze a self similar diffusion problem the statistical distributions which constitute the solutions of this problem turn out to be levi stable distributions with infinite variances characterizing the statistics of one dimensional levi flights the self similar continuum approach introduced in this paper has the potential to be applied on a variety of scale invariant and fractal problems in physics such as in continuum mechanics electrodynamics and in other fields

fractional derivatives of weierstrass type functions for a special type of fractional differentiation formulas for the weierstrass and the weierstrass mandelbrot functions are shown for integer valued parameters a well conditioned numerical procedure for computing the derivatives in the mean of these functions is derived and used to compute some values

natural superoscillations in monochromatic waves in d dimensions for monochromatic waves satisfying the helmholtz equation with wavenumber k0 superoscillations correspond to local wavenumbers magnitude of phase gradient greater than k0 large values of local wavenumber are associated with phase singularities for isotropic random waves superpositions of many nonevanescent waves in d dimensions we show that the probability that a point in the field is superoscillatory increases from 0 293 to 0 394 as d increases from 1 to infinity the peculiar case d 1 is examined in detail

fast iterative solution of elliptic control problems in wavelet discretization we investigate wavelet methods for the efficient numerical solution of a class of control problems constrained by a linear elliptic boundary value problem where the cost functional may contain fractional sobolev norms of the control and the state starting point is the formulation of the infinite dimensional control problem in terms of boundary adapted biorthogonal spline wavelets involving only l2 norms of wavelet expansion coefficients where different norms are realized by a diagonal scaling together with a riesz map and constraints in form of an l2 isomorphism the coupled system of equations resulting from optimization is solved by an inexact conjugate gradient cg method for the control which involves the approximate inversion of the primal and the adjoint operator using again cg iterations starting from a coarse discretization level we use nested iteration to solve the coupled system on successively finer uniform discretizations up to discretization error accuracy on each level the resulting inexact cg scheme is a fast solver it is of asymptotic optimal complexity in the sense that the overall computational effort to compute the solution up to discretization error on the finest grid is proportional to the number of unknowns on that grid a consequence of grid independent condition numbers of the linear operators in wavelet coordinates in the numerical examples we study the choice of different norms and the regularization parameter in the cost functional and their effect on the solution moreover for different situations the performance of the fully iterative inexact cg scheme is investigated confirming the theoretical results

superoscillations faster than the nyquist rate it is commonly assumed that a signal bandlimited to mu 2 hz cannot oscillate at frequencies higher than mu hz in fact however for any fixed bandwidth there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals these localized fast transients called superoscillations can only occur in signals that possess amplitudes of widely different scales this paper investigates the required dynamical range and energy squared l2 norm as a function of the superoscillation s frequency number and maximum derivative it briefly discusses some of the implications of superoscillating signals in reference to information theory and time frequency analysis for example it also shows among other things that the required energy grows exponentially with the number of superoscillations and polynomially with the reciprocal of the bandwidth or the reciprocal of the superoscillations period

black holes bandwidths and beethoven it is usually believed that a function t whose fourier spectrum is bounded can vary at most as fast as its highest frequency component max this is in fact not the case as aharonov berry and others drastically demonstrated with explicit counterexamples so called superoscillations it has been claimed that even the recording of an entire beethoven symphony can occur as part of a signal with a 1 hz bandwidth bandlimited functions also occur as ultraviolet regularized fields their superoscillations have been suggested for example to resolve the trans planckian frequencies problem of black hole radiation here we give an exact proof for generic superoscillations namely we show that for every fixed bandwidth there exist functions that pass through any finite number of arbitrarily prespecified points further we show that in spite of the presence of superoscillations the behavior of bandlimited functions can be characterized reliably namely through an uncertainty relation the standard devi

suppression of superoscillations by noise bandlimited functions can vary faster than their highest fourier component such superoscillations result from near perfect destructive interference among the fourier components and correspond to large values of the phase gradient local wavenumber superoscillations that are strong and extend over a large interval occur where functions are exponentially small the associated interference is vulnerable to noise in particular random phases averaging over the phases modelled as independent gaussian variables with a specified rms value enables the suppression of superoscillations to be described quantitatively very weak phase noise suffices strong noise generates functions that are essentially random and the remaining well understood superoscillations are localised in small intervals the theory is illustrated by computations with an explicit superoscillatory function

superoscillations and supershifts in phase space wigner and husimi function interpretations superoscillations namely regions where a band limited function f x varies faster than the fastest of its fourier components k generate the illusion that the fourier content is supershifted so as to lie outside the spectrum of the function the relation between supershifts and superoscillations central to the quantum weak measurements scheme is explored in terms of two different representations of the local fourier transform in the phase space x k the wigner function w x k regarded as a function of k for fixed x inherits the band limited property of f x neverthless its local k average can lie outside the spectrum because w although real posesses negative values the local wigner average of k equals the local wavenumber at x local weak value of momentum defined as the phase variation k loc x x arg f x by contrast the husimi function h x k i e the windowed fourier transform with window width l corresponding to squeezing of the coherent state associated with x k and representing the pointer wavefunction after a weak measurement is positive definite but it is not band limited and the local husimi average of k equals k loc if l is small enough these properties are illustrated numerically with two superoscillatory functions

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization

weak value distributions for spin 1 2 the simplest weak measurement is of a component of spin 1 2 for this observable the probability distributions of the real and imaginary parts of the weak value and their joint probability distribution are calculated exactly for pre and postselected states uniformly distributed over the surface of the poincare bloch sphere the superweak probability that the real part of the weak value lies outside the spectral range is 1 3 this case with just two eigenvalues complements our previous calculation berry and shukla 2010 j phys a math theor 43 354024 of the universal form of the weak value probability distribution for an operator with many eigenvalues

wavelet least squares methods for boundary value problems this paper is concerned with least squares methods for the numerical solution of operator equations our primary focus is the discussion of the following conceptual issues the selection of appropriate least squares functionals their numerical evaluation in the special light of recent developments of wavelet methods and a natural way of preconditioning the resulting systems of linear equations we describe first a general format of variational problems that are well posed in a certain natural topology in order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems their formulation as a first order system transmission problems the system of stokes equations or more general saddle point problems particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions we propose a unified treatment based on wavelet expansions in particular we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context this provides access to difficult norms efficient preconditioners and in the case of first order systems optimal l2 error estimates

composite wavelet bases with extended stability and cancellation properties the efficient solution of operator equations using wavelets requires that they generate a riesz basis for the underlying sobolev space and that they have cancellation properties of a sufficiently high order suitable biorthogonal wavelets were constructed on reference domains as the n cube via a domain decomposition approach these bases have been used as building blocks to construct biorthogonal wavelets on general domains or manifolds where in order to end up with local wavelets biorthogonality was realized with respect to a modified l 2 scalar product the use of this modified scalar product restricts the application of these so called composite wavelets to problems of orders strictly larger than 1 moreover those wavelets with supports that extend to more than one patch generally have no cancellation properties in this paper we construct local composite wavelets that are close to being biorthogonal with respect to the standard l 2 scalar product as a consequence they generate riesz bases for the sobolev spaces h s for the full range of s allowed by the continuous gluing of functions over the patch interfaces the properties of the primal and dual approximation spaces on the reference domain and in the manifold case by the regularity of the manifold moreover all these wavelets have cancellation properties of the full order induced by the approximation properties of the dual spaces on the reference domain we illustrate our findings by a concrete realization of wavelets on a perturbed sphere