examples of generalized bi circular idempotents on spaces of continuously differentiable functions abstract in this note we investigate the form of a class of idempotents called generalized bi circular idempotents on spaces of continuously differentiable functions defined on the interval 0 1 and with values in the complex numbers

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

generalized tricircular projections on minimal norm ideals in b h abstract a nonzero projection p on a complex banach space x is said to be a generalized tricircular projection if there exist distinct modulus one complex numbers and not equal to 1 and nonzero projections q and r on x such that p q r i and p q r is an isometry we determine the structure of generalized tricircular projections on minimal norm ideals in b h different from the hilbert schmidt class

new tight frames of curvelets and optimal representations of objects with piecewise c2 singularities this paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise c 2 edges conceptually the curvelet transform is a multiscale pyramid with many directions and positions at each length scale and needle shaped elements at fine scales these elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets for instance curvelets obey a parabolic scaling relation which says that at scale 2 j each element has an envelope that is aligned along a ridge of length 2 j 2 and width 2 j we prove that curvelets provide an essentially optimal representation of typical objects f that are c 2 except for discontinuities along piecewise c 2 curves such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object for instance the n term partial reconstruction f c n obtained by selecting the n largest terms in the curvelet series obeys f f c n 2 l2 c n 2 log n 3 n this rate of convergence holds uniformly over a class of functions that are c 2 except for discontinuities along piecewise c 2 curves and is essentially optimal in comparison the squared error of n term wavelet approximations only converges as n 1 as n which is considerably worse than the optimal behavior

a spatial version of the ito stratonovich correction we consider a class of stochastic pdes of burgers type in spatial dimension 1 driven by space time white noise even though it is well known that these equations are well posed it turns out that if one performs a spatial discretization of the nonlinearity in the wrong way then the sequence of approximate equations does converge to a limit but this limit exhibits an additional correction term this correction term is proportional to the local quadratic cross variation in space of the gradient of the conserved quantity with the solution itself this can be understood as a consequence of the fact that for any fixed time the law of the solution is locally equivalent to wiener measure where space plays the role of time in this sense the correction term is similar to the usual ito stratonovich correction term that arises when one considers different temporal discretizations of stochastic odes

multiscale analysis in sobolev spaces on bounded domains we study a multiscale scheme for the approximation of sobolev functions on bounded domains our method employs scattered data sites and compactly supported radial basis functions of varying support radii at scattered data sites the actual multiscale approximation is constructed by a sequence of residual corrections where different support radii are employed to accommodate different scales convergence theorems for the scheme are proven and it is shown that the condition numbers of the linear systems at each level are independent of the level thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points on a bounded domain

spectral projections for hermitian operators abstract the behavior of operators which are both scalar type spectral and hermitian and this includes all hermitian operators on finite dimensional spaces is studied for example it is shown that the resolution of the identity of such an operator being hermitian is equivalent to all the powers of the operator being hermitian or to the operator having a contractive borel functional calculus a sufficient condition for these properties to hold on a finite dimensional space is that the eigenvalues of the operator are rationally independent

embeddings for a weakly differentiable functions on domains abstract we prove that the inhomogeneous estimate of vector fields on balls in r n b d k 1 u n n 1 d x n 1 n c b a u u d x for all u c b r n holds if and only if the linear constant coefficient differential operator a of order k has finite dimensional null space fdn this generalizes the gagliardo nirenberg sobolev inequality on domains and provides the local version of the analogous homogeneous embedding in full space r n d k 1 u n n 1 d x n 1 n c r n a u d x for all u c c r n r n proved by van schaftingen precisely for elliptic and cancelling ec operators building on fundamental l 1 estimates from the works of bourgain and brezis we prove that fdn strictly implies ec and discuss the contrast between homogeneous and inhomogeneous estimates on both algebraic and analytic level

generalized bicircular projections on jb triples abstract let a be a jb triple and let p a a be a linear projection it is proved that p id p is an isometry for some modulus one complex number 1 if and only if either 1 or p is hermitian it is also proved that every rank one bicontractive projection on a is hermitian the particular case when a is a c algebra is discussed through several examples

extending sobolev functions with partially vanishing traces from locally e δ domains and applications to mixed boundary problems we prove that given any k n for each open set rn and any closed subset d of such that is locally an e domain near d there exists a linear and bounded extension operator ek d mapping for each p 1 the space wdk p into wdk p rn here with o denoting either or rn the space wdk p o is defined as the completion in the classical sobolev space wk p o of restrictions to o of functions from cc rn whose supports are disjoint from d in turn this result is used to develop a functional analytic theory for the class wdk p including intrinsic characterizations boundary traces and extensions results interpolation theorems among other things which is then employed in the treatment of mixed boundary value problems formulated in locally e domains finally we also prove extension results on the scales of besov and bessel potential spaces on e domains with partially vanishing traces on ahlfors regular sets and explore some of the implications of such extension results

stieltjes continued fraction and qd algorithm scalar vector and matrix cases the definition in previous studies of vector stieltjes continued fractions in connection with spectral properties of band operators with intermediate zero diagonals left unsolved the question of a direct definition of their coefficients in terms of the original data a vector of stieltjes series the subject was more undefined in the matrix case a new version of the qd algorithm for matrix problem allows to extend to the vector and matrix cases the result of stieltjes expansion of a scalar function in terms of a stieltjes continued fraction beside this connection it solves the inverse miura transform and gives interesting identities between general band matrix and sparse band matrix finally as a consequence we extend to some dynamical systems a method known for toda lattices

analysis of moving least squares approximation revisited in this article the error estimation of the moving least squares approximation is provided for functions in fractional order sobolev spaces the analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details an application to galerkin method for partial differential equations is also supplied

the divergence equation in rough spaces abstract we aim at extending the existence theory for the equation div v f in a bounded or exterior domain with homogeneous dirichlet boundary conditions to a class of solutions which need not have a trace at the boundary typically the weak solutions that we shall consider will belong to some besov space b p q s with s 1 1 p 1 p after generalizing the notion of a solution for this equation we propose an explicit construction by means of the classical bogovski formula this construction enables us to keep track of a marginal information about the trace of solutions in particular it ensures that the trace is zero if f is smooth enough we expect our approach to be of interest for the study of rough solutions to systems of fluid mechanics

hermitian projections on some banach spaces and related topics abstract in the setting of various complex banach spaces we consider the questions of when a square of a hermitian operator is also hermitian and when a hermitian operator has a hermitian square root we also describe the structure of hermitian projections and hermitian square roots of the identity operator

on the equivalence of fractional order sobolev semi norms abstract we present various results on the equivalence and mapping properties under affine transformations of fractional order sobolev norms and semi norms of orders between zero and one main results are mutual estimates of the three semi norms of sobolev slobodeckij interpolation and quotient space types in particular we show that the former two are uniformly equivalent under affine mappings that ensure shape regularity of the domains under consideration

generalized bicircular projections via the operator equation αx2ay2 βxay a 0 abstract the operator equation x 2 ay 2 xay a 0 is used to determine the structure of a projection p such that for some complex unit 1 the mapping p i p is a uav a ua v or a ua t v for some fixed operators u v some applications of these results are also given

projections as convex combinations of surjective isometries on c ω we study properties of operators that are in the convex hull of a finite set of surjective isometries on the banach space of complex valued continuous maps defined on a compact and connected topological space we characterize those projections that are in the convex combination of two surjective isometries and we show that they are generalized bi circular projections

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

hochschild cohomology for von neumann algebras with cartan subalgebras the main result of this paper is that h n m m 0 n 1 for von neumann algebras m with cartan subalgebras and separable preduals

isometries of norm ideals of compact operators abstract let j be a symmetric norm ideal of compact operators on hilbert space h and assume that the finite rank operators are dense in j and that j is not the ideal of hilbert schmidt operators a linear transformation on j is an isometry of j onto itself if and only if there are unitary operators u and v on h such that either x uxv or x uxtv where xt denotes the transpose of x with respect to a fixed orthonormal basis of h

convergence of the discontinuous finite volume method for elliptic problems with minimal regularity this paper investigates convergence of the discontinuous finite volume method dfvm under minimal regularity assumptions on solutions of second order elliptic boundary value problems conventional analysis requires the solutions to be in sobolev spaces h 1 s s 12 here we assume the solutions are in h 1 s s 0 and employ the techniques developed in gudi 2010 18 20 to derive error estimates in a mesh dependent energy norm and the l 2 norm for dfvm the theoretical estimates are illustrated by numerical results which include problems with corner singularity and intersecting interfaces

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

non smooth atomic decompositions traces on lipschitz domains and pointwise multipliers in function spaces abstract we provide non smooth atomic decompositions for besov spaces b p q s r n s 0 0 p q defined via differences the results are used to compute the trace of besov spaces on the boundary of bounded lipschitz domains with smoothness s restricted to 0 s 1 and no further restrictions on the parameters p q we conclude with some more applications in terms of pointwise multipliers

bicircular projections on some matrix and operator spaces a projection on a complex banach space is bicircular if its linear combinations with the complementary projection having coefficients of modulus one are isometries such projections are always bicontractive in this paper we prove structure theorems for bicircular projections acting on the spaces of the full operator algebra symmetric operators and antisymmetric operators

bicircular projections on some banach spaces in this paper we show that the bicircular projections are precisely the hermitian projections and prove some immediate consequences of this result

anisotropic smoothness spaces via level sets it has been understood for sometime that the classical smoothness spaces such as the sobolev and besov classes are not satisfactory for certain problems in image processing and nonlinear pdes their deficiency lies in their isotropy functions in these smoothness spaces must be simultaneously smooth in all directions the anisotropic generalizations of these spaces also have the deficiency that they are biased in coordinate directions while they allow different smoothness in certain directions these directions must be aligned to the coordinate axes in the application areas mentioned above it would be desirable to measure smoothness in new ways that would allow one to have more local control over the smoothness directions we introduce one possible approach to this problem based on defining smoothness via level sets we present this approach in the case of functions defined on r d our smoothness spaces depend on two smoothness indices s1 s 2 the first reflects the smoothness of the level sets of the function while the second index reflects how smoothly the level sets themselves are changing as a motivation we start with d d 2 and investigate besov smooth domains c 2007 wiley periodicals inc

generalized bi circular projections abstract recall that a projection p on a complex banach space x is a generalized bi circular projection if p i p is a surjective isometry for some such that 1 and 1 it is easy to see that every hermitian projection is generalized bi circular a generalized bi circular projection is said to be nontrivial if it is not hermitian botelho and jamison showed that a projection p on c 0 1 is a nontrivial generalized bi circular projection if and only if p i p is a surjective isometry in this article we prove that if p is a projection such that p i p is a surjective isometry for some then either p is hermitian or is an n th unit root of unity we also show that for any n th unit root of unity there are a complex banach space x and a nontrivial generalized bi circular projection p on x such that p i p is an isometry

divergence free kernel methods for approximating the stokes problem we develop a new discretization scheme for the stokes equations using analytically divergence free approximation spaces our scheme is based upon a meshfree method using matrix valued kernels the scheme works in arbitrary space dimension and can produce arbitrarily smooth and hence high order approximations after deriving the scheme we analyze the discretization error in sobolev spaces and give a numerical example

projections as averages of isometries on minimal norm ideals abstract in this paper we consider projections on minimal norm ideals of b h that are represented as the average of two surjective isometries we describe projections of the form p t a 1 tb 1 a 2 tb 2 2 where a 1 a 2 b 1 and b 2 are unitary operators satisfying a commutativity condition we also characterize classes of projections of the form p t at t t b 2 and p t at t t t b 2 with a and b unitary operators satisfying a spectral condition