radial basis function approximation from gridded centres to scattered centres abstract the paper studies l infinity ir d norm approximations from a space spanned by a discrete set of translates of a basis function theta attention here is restricted to functions theta whose fourier transform is smooth on ird 0 and has a singularity at the origin examples of such basis functions are the thin plate splines and the multiquadrics as well as other types of radial basis functions that are employed in approximation theory the above approximation problem is well understood in case the set of points used for translating theta forms a lattice in ir d and many optimal and quasi optimal approximation schemes can already be found in the literature in contrast only few mostly specific results are known for a set of scattered points the main objective of this paper is to provide a general tool for extending approximation schemes that use integer translates of a basis function to the non uniform case we introduce a single relatively simple conversion method that preserves the approximation orders provided by a large number of schemes presently in the literature more precisely to almost all stationary schemes in anticipation of future introduction of new schemes for uniform grids an effort is made to impose only a few mild conditions on the function theta which still allow for a unified error analysis to hold in the course of the discussion here the recent results of budl on scattered center approximation are reproduced and improved upon

approximation using scattered shifts of a multivariate function the approximation of a general d variate function f by the shifts r d of a fixed functionoccurs in many applications such as data fit ting neural networks and learning theory when hz d is a dilate of the integer lattice there is a rather complete understanding of the approximation problem 6 18 using fourier techniques however in most applications the center set is either given or can be chosen with complete freedom in both of these cases the shift invariant setting is too restrictive this paper studies the approximation problem in the caseis arbitrary it establishes approximation theorems whose error bounds reflect the local density of the points in two different settings are analyzed the first is when the setis prescribed in advance in this case the theorems of this paper show that in analogy with the classical univariate spline ap proximation improved approximation occurs in regions where the density is high the second setting corresponds to the problem of non linear approximation in that setting the setcan be chosen using information about the target function f we discuss how to best make these choices and give estimates for the approximation error

quasi interpolation in shift invariant spaces abstract let s 1 be an integer r s r be a compactly supported function and s denote the linear span of k k z s we consider the problem of approximating a continuous function f r s r on compact subsets of r s from the classes s h h 0 based on samples of the function at scattered sites in r s we demonstrate how classical polynomial inequalities lead to the construction of local quasi interpolatory operators for this purpose

regular multivariate sampling and approximation in lp shift invariant spaces abstract in this paper some results about regular multivariate generalized sampling in the l p setting 1 p are proven thus stable multivariate regular sampling formulas are derived for l p shift invariant spaces v p i e the l p closure of the linear span of the shifts of a finite set of generators these sampling formulas include regular samples of the function its derivatives and or some filtered versions of the function itself taken at a lattice of r d approximation schemes using these generalized sampling formulas are also included

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

exponential bases paley wiener spaces and applications abstract we investigate the connection between translation bases for paley wiener spaces and exponential fourier bases for a domain we apply these results to the characterization of vector valued time frequency translations of a paley wiener window signal

the regularity theory for the parabolic double obstacle problem in this paper we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator the result in this paper are generalizations of the theory for the elliptic problem in lee et al calc var partial differ equ 58 3 104 2019 and lee and park the regularity theory for the double obstacle problem for fully nonlinear operator 2018 to parabolic case and also the theory for the parabolic single obstacle problem in caffarelli et al j am math soc 17 4 827 869 2004 to double obstacle case new difficulties in the theory which are generated by the characteristic of parabolic pdes and the existence of the upper obstacle are discussed in detail furthermore the thickness assumptions to have the regularity of the free boundary are carefully considered

on the structure and interpolation properties of quasi shift invariant spaces abstract the structure of certain types of quasi shift invariant spaces which take the form v x span l 2 x j j z for an infinite discrete set x x j r is investigated additionally the relation is explored between pairs x and y such that interpolation of functions in v x via interpolants in v y solely from the samples of the original function is possible and stable some conditions are given for which the sampling problem is stable and for which recovery of functions from their interpolants from a family of spaces v y is possible

uniform regularity close to cross singularities in an unstable free boundary problem we introduce a new method for the analysis of singularities in the unstable problem delta u chi u 0 which arises in solid combustion as well as in the composite membrane problem our study is confined to points of supercharacteristic growth of the solution i e points at which the solution grows faster than the characteristic invariant scaling of the equation would suggest at such points the classical theory is doomed to fail due to incompatibility of the invariant scaling of the equation and the scaling of the solution in the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of u are unbounded the level set u 0 consists of two c 1 curves meeting at right angles it is important that our result is not confined to the minimal solution of the equation but holds for all solutions

sampling moments and reconstructing signals of finite rate of innovation shannon meets strang fix consider the problem of sampling signals which are not bandlimited but still have a finite number of degrees of freedom per unit of time such as for example nonuniform splines or piecewise polynomials and call the number of degrees of freedom per unit of time the rate of innovation classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited recently it was shown that by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation it is possible to reconstruct such signals uniquely these sampling schemes however use kernels with infinite support and this leads to complex and potentially unstable reconstruction algorithms in this paper we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm the class of kernels that we can use is very rich and includes functions satisfying strang fix conditions exponential splines and functions with rational fourier transform this last class of kernels is quite general and includes for instance any linear electric circuit we thus show with an example how to estimate a signal of finite rate of innovation at the output of an rc circuit the case of noisy measurements is also analyzed and we present a novel algorithm that reduces the effect of noise by oversampling

support and approximation properties of hermite splines abstract in this paper we formally investigate two mathematical aspects of hermite splines that are relevant to practical applications we first demonstrate that hermite splines are maximally localized in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties then we precisely quantify the approximation power of hermite splines for the reconstruction of functions and their derivatives it is known that the hermite and b spline approximation schemes have the same approximation order more precisely their approximation error vanishes as o t 4 when the step size t goes to zero in this work we show that they actually have the same asymptotic approximation error constants too therefore they have identical asymptotic approximation properties hermite splines combine optimal localization and excellent approximation power while retaining interpolation properties and closed form expression in contrast to existing similar functions these findings shed a new light on the convenience of hermite splines in the context of computer graphics and geometrical design

the structure of shift invariant subspaces of l2 rn using the range function approach to shift invariant spaces in l2 rn we give a simple characterization of frames and riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2 zn this in turn gives a simplified approach to the analysis of frames and riesz families done by gramians and dual gramians we prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator as an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function

approximation by translates of a single function of functions in space induced by the convolution with a given function abstract we study approximation by arbitrary linear combinations of n translates of a single function of periodic functions we construct some linear methods of this approximation for functions in a class induced by the convolution with a given function and prove upper bounds of the lp approximation convergence rate by these methods when n for 1 p 2

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

the limits of refinable functions a function q is refinable q c s if it is in the closed span of 5 2x k this set s is not closed in l2 ir and we characterize its closure a necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask the fourier transform of every f c s s vanishes on a set of positive measure as an example we show that all functions with fourier transform supported in 4 r 47t are the limits of refinable functions the relation between a refinable function and its mask is studied and nonuniqueness is proved for inhomogeneous refinement equations we determine when a solution is refinable this result is used to investigate refinable components of multiple refinable functions finally we investigate fully refinable functions for which all translates by any real number are refinable

subspaces of l2 g invariant under translation by an abelian subgroup abstract for a second countable locally compact group g and a closed abelian subgroup h we give a range function classification of closed subspaces in l 2 g invariant under left translation by h for a family a l 2 g this classification ties with a set of conditions under which the translations of a by h form a continuous frame or a riesz sequence when g is abelian our work relies on a fiberization map for the more general case we introduce an analogue of the zak transform both transformations intertwine translation with modulation and both rely on a new group theoretic tool for a closed subgroup g we produce a measure on the space g of right cosets that gives a measure space isomorphism g g outside of the group setting we consider a more general problem for a measure space x and a hilbert space h we investigate conditions under which a family of functions in l 2 x h multiplies with a basis like system in l 2 x to produce a continuous frame or a riesz sequence in l 2 x h finally we explore connections with dual integrable representations of lca groups as introduced by hernandez et al in 25

obstacle problem for a non convex fully nonlinear operator abstract in this paper we study a priori c 2 estimate up to the boundary for f d 2 u 0 and the regularity of the free boundary of the obstacle problem for fully nonlinear operator under specific conditions for the operator and level sets of the operator the conditions are variations of conditions for the zero set of the operator in 7

mra frame wavelets with certain regularities associated with the refinable generators of shift invariant spaces in this paper we will start the discussion with the refinable generators of the shift invariant si spaces in l2 r that possess the largest possible regularities and required vanishing moments for the pseudo scaling generators the corresponding mra frame wavelets with certain regularities are constructed in addition the stability of the refinable si spaces and the corresponding complementary spaces biorthogonality of the si spaces and the approximation property of the spaces are also discussed

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

extra invariance of shift invariant spaces on lca groups this article generalizes recent results in the extra invariance for shift invariant spaces to the context of lca groups let g be a locally compact abelian lca group and k a closed subgroup of g a closed subspace of l2 g is called k invariant if it is invariant under translations by elements of k assume now that h is a countable uniform lattice in g and m is any closed subgroup of g containing h in this article we study necessary and sufficient conditions for an h invariant space to be m invariant as a consequence of our results we prove that for each closed subgroup m of g containing the lattice h there exists an h invariant space s that is exactly m invariant that is s is not invariant under any other subgroup m containing h we also obtain estimates on the support of the fourier transform of the generators of the h invariant space related to its m invariance

multiresolution expansion approximation order and quasiasymptotic behavior of tempered distributions abstract multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces s r r r n 0 for a function h which is smooth enough and of appropriate decay it is shown that the derivatives of its projections to the corresponding spaces v j j z in a regular multiresolution analysis of l 2 r denoted by h j multiplied by a polynomial weight converge in sup norm i e h j h in s r r as j analogous result for tempered distributions is obtained by duality arguments the analysis of the approximation order of the projection operator within the framework of the theory of shift invariant spaces gives a further refinement of the results the order of approximation is measured with respect to the corresponding space of test functions as an application we give abelian and tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

a novel approach to the 2 d blind deconvolution problem in medical ultrasound the finite frequency bandwidth of ultrasound transducers and the nonnegligible width of transmitted acoustic beams are the most significant factors that limit the resolution of medical ultrasound imaging consequently in order to recover diagnostically important image details obscured due to the resolution limitations an image restoration procedure should be applied the present study addresses the problem of ultrasound image restoration by means of the blind deconvolution techniques given an acquired ultrasound image algorithms of this kind perform either concurrent or successive estimation of the point spread function psf of the imaging system and the original image a blind deconvolution algorithm is proposed in which the psf is recovered as a preliminary stage of the restoration problem as the accuracy of this estimation affects all the following stages of the image restoration it is considered as the most fundamental and important problem the contribution of the present study is twofold first it introduces a novel approach to the problem of estimating the psf which is based on a generalization of several fundamental concepts of the homomorphic deconvolution it is shown that a useful estimate of the spectrum of the psf can be obtained by applying a proper smoothing operator to both log magnitude and phase of the spectra of acquired radio frequency rf images it is demonstrated that the proposed approach performs considerably better than the existing homomorphic cepstrum based deconvolution methods second the study shows that given a reliable estimate of the psf it is possible to deconvolve it out of the rf image and obtain an estimate of the true tissue reflectivity function which is relatively independent of the properties of the imaging system the deconvolution was performed using the maximum a posteriori map estimation framework for a number of statistical priors assumed for the reflectivity function it is shown in a series of in vivo experiments that reconstructions based on the priors which tend to emphasize the sparseness of the tissue structure result in solutions of higher resolution and contrast

the toeplitz theorem and its applications to approximation theory and linear pdes we take an algebraic approach to the problem of approximation by dilated shifts of basis functions given a finite collection d of compactly supported functions in lp ris 1 0 where sh is the hdilate of s we prove that sh h 0 provides lp approximation order r only if s contains all the polynomials of total degree less than r in particular in the case where d consists of a single function p with its moment f p 0 we characterize the approximation order of sh h 0 by showing that the above condition on polynomial containment is also sufficient the above results on approximation order are obtained through a careful analysis of the structure of shift invariant spaces it is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients such a system then can be reduced to an infinite system of linear equations whose solvability is characterized by an old theorem of toeplitz thus the toeplitz theorem sheds light into approximation theory it is also used to give a very simple proof for the well known ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

local approximation by certain spaces of exponential polynomials approximation order of exponential box splines and related interpolation problems local approximation order to smooth complex valued functions by a finite dimensional space 7b h spanned by certain products of exponentials by polynomials is investigated the results obtained together with a suitable quasi interpolation scheme are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline the analysis of a typical 7b h is based here on the identification of its dual with a certain space 7b p of multivariate polynomials this point of view allows us to solve a class of multivariate interpolation problems by the polynomials from 7b p

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

oversampling and reconstruction functions with compact support assume that a sequence of samples of a filtered version of a function in a shift invariant space is given this paper deals with the existence of a sampling formula involving these samples and having reconstruction functions with compact support this is done in the light of the generalized sampling theory by using the oversampling technique a necessary and sufficient condition is given in terms of the smith canonical form of a polynomial matrix finally we prove that the aforesaid oversampled formulas provide nice approximation schemes with respect to the uniform norm

optimal regularity for the no sign obstacle problem in this paper we prove the optimal regularity for a general obstacle type problem under the assumption that is where n is the newtonian potential this is the weakest assumption for which one can hope to get regularity as a by product of the regularity we are able to prove that under a standard thickness assumption on the zero set close to a free boundary point the free boundary is locally a graph close to provided f is dini this completely settles the question of the optimal regularity of this problem which has been the focus of much attention during the last two decades 2012 wiley periodicals inc

approximation order from certain spaces of smooth bivariate splines on a three direction mesh determination de l ordre exact d approximation de certains espaces de fonctions splines regulieres a deux variables

random sampling in shift invariant spaces abstract the set of sampling in a shift invariant space plays an important role in signal processing and has many applications this paper addresses the problem when some randomly chosen samples x x j j j form a set of sampling in a shift invariant space that is when the inequality of the form c p f l p r d p x j x f x j p c p f l p r d p holds uniformly for all functions f in a shift invariant space where c p and c p are positive constants 1 p we prove that with overwhelming probability the above sampling inequality holds for certain compact subsets of the shift invariant space when the sampling size is sufficiently large

optimized kaiser bessel window functions for computed tomography kaiser bessel window functions are frequently used to discretize tomographic problems because they have two desirable properties 1 their short support leads to a low computational cost and 2 their rotational symmetry makes their imaging transform independent of the direction in this paper we aim at optimizing the parameters of these basis functions we present a formalism based on the theory of approximation and point out the importance of the partition of unity condition while we prove that for compact support functions this condition is incompatible with isotropy we show that minimizing the deviation from the partition of unity condition is highly beneficial the numerical results confirm that the proposed tuning of the kaiser bessel window functions yields the best performance

on l factors attached to generic representations of unramified u 2 1 let g be the unramified unitary group in three variables defined over a p adic field with p ne 2 in this paper we establish a theory of newforms for the rankin selberg integral for g introduced by gelbart and piatetski shapiro we describe l and varepsilon factors defined through zeta integrals in terms of newforms we show that zeta integrals of newforms for generic representations attain l factors as a corollary we get an explicit formula for varepsilon factors of generic representations

geometric and energetic criteria for the free boundary regularity in an obstacle type problem we consider an obstacle type problem u f x in d u u 0 on d where d is a given open set in n and is an unknown open subset of d the problem originates in potential theory in connection with harmonic continuation of potentials the qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign using geometric and energetic criteria in delicate combination we show the c1 1 regularity of the solutions and the regularity of the free boundary below the lipschitz threshold for the right hand side

c1 1 regularity in semilinear elliptic problems in this paper we give an astonishingly simple proof of c1 1 regularity in elliptic theory our technique yields both new simple proofs of old results as well as new optical results the setting we ll consider is the following let u be a solution to where b is the unit ball in n f x t is a bounded lipschitz function in x and ft is bounded from below then we prove that u c1 1 b1 2 our method is a simple corollary to a recent monotonicity argument due to caffarelli jerison and kenig 2002 wiley periodicals inc

regular families of kernels for nonlinear approximation abstract this article studies sufficient conditions on families of approximating kernels which provide n term approximation errors from an associated nonlinear approximation space which match the best known orders of n term wavelet expansion these conditions provide a framework which encompasses some notable approximation kernels including splines cardinal functions and many radial basis functions such as the gaussians and general multiquadrics examples of such kernels are given to justify the criteria additionally the techniques involved allow for some new results on n term greedy interpolation of sobolev functions via radial basis functions

collocation discretizations of the transport equation with radial basis functions in this paper a new gridless method for numerically solving hyperbolic partial differential equations is presented this method uses collocation based on hermite interpolation at scattered sites at each time step the basis can be chosen at each time step which makes the approach adaptive and allows flexibility the goal of this paper is to demonstrate the potential of this adaptive flexible model which includes the ability to use scattered sites in a multi dimensional setting with compactly supported radial functions to illustrate the method and to compare it with known results we analyze a certain transport problem in detail when combined with an implicit time discretization and applied to smooth solutions we show that the method yields spectral rates of convergence in the spatial domain the implicit time discretization makes possible time steps of a size appropriate to the smoothness of the solution

a hele shaw problem for tumor growth abstract we consider weak solutions to a problem modeling tumor growth under certain conditions on the initial data solutions can be obtained by passing to the stiff incompressible limit in a porous medium type problem with a lotka volterra source term describing the evolution of the number density of cancerous cells we prove that such limit solutions solve a free boundary problem of hele shaw type we also obtain regularity properties both for the solution and for its free boundary the main new difficulty arises from the competition between the growth due to the source which keeps the initial singularities and the free boundary which invades the domain with a regularizing effect new islands can be generated at singular times