pinecone/core-2020-05-10-deduplication huggingface dataset:
Local Decomposition of Refinable Spaces and Wavelets

AbstractA convenient setting for studyingmultiscale techniquesin various areas of applications is usually based on a sequence of nested closed subspaces of some function space script F which is often referred to asmultiresolution analysis.The concept ofwaveletsis a prominent example where the practical use of such a multiresolution analysis relies on explicit representations of theorthogonaldifference between any two subsequent spaces. However, many applications prohibit the employment of a multiresolution analysis based on translation invariant spaces on all of open face Rs, say. It is then usually difficult to compute orthogonal complements explicitly. Moreover, certain applications suggest using other types of complements, in particular, those corresponding tobiorthogonal wavelets.The main objective of this paper is therefore to study possibly nonorthogonal but in a certain sensestableand evenlocaldecompositions of nested spaces and to develop tools which are not necessarily confined to the translation invariant setting. A convenient way ofparametrizingsuch decompositions is to reformulate them in terms of matrix relations. This allows one to characterizeallstable or local decompositions by identifying unique matrix transformations that carry one given decomposition into another one. It will be indicated how such a mechanism may help realizing several desirable features of multiscale decompositions and constructing stable multiscale bases with favorable properties. In particular, we apply these results to the identification of decompositions induced by local linear projectors. The importance of this particular application with regard to the construction of multiscale Riesz bases will be pointed out. Furthermore, we indicate possible specializations to orthogonal decompositions of spline spaces relative to nonuniform knot sequences, piecewise linear finite elements and principal shift invariant spaces. The common ground of all these examples as well as other situations of practical and theoretical interest is thatsome initialmultiscale basis is available. The techniques developed here can then be used to generate from such an initial decomposition other ones with desirable properties pertaining to moment conditions or stability properties
Cubic Spline Wavelet Bases of Sobolev Spaces and Multilevel Interpolation

AbstractIn this paper, a semi-orthogonal cubic spline wavelet basis of homogeneous Sobolev spaceH20(I) is constructed, which turns out to be a basis of the continuous spaceC0(I). At the same time, the orthogonal projections on the wavelet subspaces inH20(I) are extended to the interpolating operators on the corresponding wavelet subspaces inC0(I). A fast discrete wavelet transform (FWT) for functions inC0(I) is also given, which is different from the pyramid algorithm and easy to perform using a parallel algorithm. Finally, it is shown that the singularities of a function can be traced from its wavelet coefficients, which provide an adaptive approximation scheme allowing us to reduce the operation time in computation
The Lifting Scheme: A Custom-Design Construction of Biorthogonal Wavelets

AbstractWe present the lifting scheme, a new idea for constructing compactly supported wavelets with compactly supported duals. The lifting scheme uses a simple relationship between all multiresolution analyses with the same scaling function. It isolates the degrees of freedom remaining after fixing the biorthogonality relations. Then one has full control over these degrees of freedom to custom design the wavelet for a particular application. The lifting scheme can also speed up the fast wavelet transform. We illustrate the use of the lifting scheme in the construction of wavelets with interpolating scaling functions
Quasi-Orthogonality and Quasi-Projections

AbstractOur main concern in this paper is the design of simplified filtering procedures for the quasi-optimal approximation of functions in subspaces of L2generated from the translates of a function ϕ(x). Examples of signal representations that fall into this framework are Schoenberg's polynomial splinesof degree n, and the various multiresolution spaces associated with the wavelet transform. After a brief review of the relation between the order of approximation of the representation and the concept of quasi-interpolation (Strang–Fix conditions), we investigate the implication of these conditions on the various basis functions and their duals (vanishing moment and quasi-interpolation properties). We then introduce the notion of quasi-duality and show how to construct quasi-orthogonal and quasi-dual basis functions that are much shorter than their exact counterparts. We also consider the corresponding quasi-orthogonal projection operator at sampling step h and derive asymptotic error formulas and bounds that are essentially the same as those associated with the exact least-squares solution. Finally, we use the idea of a perfect reproduction of polynomials of degree n to construct short kernel quasi-deconvolution filters that provide a well-behaved approximation of an oblique projection operator
On Minimax Wavelet Estimators

AbstractIn the paper minimax rates of convergence for wavelet estimators are studied. The estimators are based on the shrinkage of empirical coefficients βjkof wavelet decomposition of unknown function with thresholds λj. These thresholds depend on the regularity of the function to be estimated. In the problem of density estimation and nonparametric regression we establish upper rates of convergence over a large range of functional classes and global error measures. The constructed estimate is minimax (up to constant) for all Lπerror measures, 0 < π ≤ ∞ simultaneously
On the Support Properties of Scaling Vectors

AbstractC. K. Chui and J. Z. Wang [J. Approx. Theory71(1992), 263–304] derived support properties for a scaling function generating a function space V0⊆ L2(open face R). Motivated by this work, we consider support properties for scaling vectors. T. N. T. Goodman and S. L. Lee [Trans. Amer. Math. Soc.342,No. 1 (Mar. 1994), 307–324] derived necessary and sufficient conditions for the scaling vector {φ1, … , φr}, r ≥1, to form a Riesz basis for V0and develop a general theory for spline wavelets of multiplicity r > 1. We consider conditions under which linear combinations of scaling functions generate V0. These conditions also characterize all other scaling vectors that generate the same V0. In addition, we describe the scaling vectors of minimal support for V0. Next, we give sufficient conditions on the two-scale symbol for scaling vectors under which a given matrix refinement equation can be solved. A spline-wavelet example illustrates these results. For the single scaling function φ, the support of φ is characterized by the degree of the two-scale symbol. The situation is more complicated in the scaling vector case. We prove a result that gives the support of the scaling vector under certain conditions on the coefficient matrices. This result is illustrated by an example of fractal wavelets derived by J. Geronimo, D. Hardin, and P. Massopust [J. Approx. Theory78,No. 3 (1994), 373–401]
Wavelets for Analyzing Scattered Data: An Unbounded Operator Approach

AbstractThe objective of this paper is to provide a general framework for the construction of wavelets that are generated by translates of a given slowly growing function where the translates are defined by a sequence of arbitrarily spaced points. Our approach is operator theoretic in nature and relies only on specific properties of the function (e.g., the order of the singularity of its Fourier transform at the origin) and not on a detailed knowledge of the function itself. The methods used here not only provide a unifying thread for several known results but allow for many new results as well. For example, the first results on stability of certain wavelet bases associated with scattered data on the line are obtained here (the only assumption on the scattered data is boundedness of the global mesh ratio). Many radial basis functions are unbounded on the line and this fact leads to our investigation of certain classes of unbounded operators including their domain identification and their closed extensions. An important result of this study is an exact identification of the L2(open face R) spaces generated by linear combinations of shifts of certain unbounded radial basis functions
Combined Spherical Harmonic and Wavelet Expansion—A Future Concept in Earth's Gravitational Determination

AbstractThe basic theory of spherical singular integrals is recapitulated. Criteria are given for measuring the space-frequency localization of functions on the sphere. The trade-off between “space localization” on the sphere and “frequency localization” in terms of spherical harmonics is described in form of an “uncertainty principle.” A continuous version of spherical multiresolution is introduced, starting from continuous wavelet transform corresponding to spherical wavelets with vanishing moments up to a certain order. The wavelet transform is characterized by least-squares properties. Scale discretization enables us to construct spherical counterparts ofP(acket)-scale discretized andD(aubechies)-scale discretized wavelets. It is shown that singular integral operators forming a semigroup of contraction operators of class (C0) (like Abel–Poisson or Gauß–Weierstraß operators) lead in canonical way to pyramid algorithms. Fully discretized wavelet transforms are obtained via approximate integration rules on the sphere. Finally applications to (geo-)physical reality are discussed in more detail. A combined method is proposed for approximating the “low frequency parts” of a physical quantity by spherical harmonics and the “high frequency parts” by spherical wavelets. The particular significance of this combined concept is motivated for the situation of today's physical geodesy, viz. the determination of the high frequency parts of the earth's gravitational potential under explicit knowledge of the lower order part in terms of a spherical harmonic expansion
On a Family of Filters Arising in Wavelet Construction

AbstractWavelet reconstruction almost always reduces to the study of certain algebraic identities for polynomials. Typically, one polynomial is chosen and another is found which satisfies an identity appropriate for the desired wavelet construction. In this paper we explore one such equation and give applications of it to wavelet construction. The specific equation we study here comes to us from our recent work on iterative interpolation by exponentials. Through this equation we strive to unify several related questions concerning wavelet construction and provide in each case considered estimates for the decay of the Fourier transform of both the refinable functions and corresponding wavelets
Compactly Supported Wavelets in Sobolev Spaces of Integer Order

AbstractWe present a construction of regular compactly supported wavelets in any Sobolev space of integer order. It is based on the existence and suitable estimates of filters defined from polynomial equations. We give an implicit study of these filters and use the results obtained to construct scaling functions leading to multiresolution analysis and wavelets. Their regularity increases linearly with the length of their supports as in theL2case. One technical problem is to prove that the intersection of the scaling spaces is reduced to 0. This is solved using sharp estimates of Littlewood–Paley type
On Dual Wavelet Tight Frames

AbstractA characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton's result on wavelet tight frames inL2(R) is generalized to then-dimensional case. Two ways of constructing certain dual wavelet tight frames inL2(Rn) are suggested. Finally, examples of smooth wavelet tight frames inL2(R) andH2(R) are provided. In particular, an example is given to demonstrate that there is a function ψ whose Fourier transform is positive, compactly supported, and infinitely differentiable which generates a non-MRA wavelet tight frame inH2(R)
Sparse Diagonal Forms for Translation Operators for the Helmholtz Equation in Two Dimensions

AbstractIn the design of fast multipole methods (FMM) for the numerical solution of scattering problems, a crucial step is the diagonalization of translation operators for the Helmholtz equation. These operators have analytically simple, physically transparent, and numerically stable diagonal forms. It has been observed by several researchers that for any given precision ϵ, diagonal forms for the translation operators for the Helmholtz equation are not unique, and that some choices lead to more efficient FMM schemes than others. As is well known, original single-stage FMM algorithms for the Helmholtz equation have asymptotic CPU time requirements of orderO(n3/2), wherenis the number of nodes in the discretization of the boundary of the scatterer; two-stage versions have CPU time estimates of orderO(n4/3); generally,k-stage versions have CPU time estimates of orderO(n(k+2)/(k+1)). However, there exist choices of diagonal forms leading to single-stage FMM algorithms with CPU time requirements of orderO(n4/3), two-stage schemes with CPU time requirementsO(n5/4), etc. In this paper, we construct such diagonal forms in two dimensions. While the construction of this paper is in no sense optimal, it is rigorous and straightforward. Our numerical experiments indicate that it is within a factor of two of being optimal, in terms of the number of nodes required to discretize the translation operator to a specified precision ϵ. The procedure is illustrated with several numerical examples
Using the Discrete Fourier Transform to Analyze the Convergence of Subdivision Schemes

AbstractWhile thecontinuousFourier transform is a well-established standard tool for the analysis of subdivision schemes, we present a new technique based on thediscreteFourier transform instead. We first prove a very general convergence criterion for arbitrary interpolatory schemes, i.e., for nonstationary, globally supported, or even nonlinear schemes. Then we use the discrete Fourier transform as an algebraic tool to transform subdivision schemes into a form suitable for the analysis. This allows us to formulate simple and numerically stable sufficient criteria for the convergence of subdivision schemes of very general type. We analyze some example schemes to illustrate the resulting easy-to-apply criteria which merely require to numerically estimate the maximum of a smooth function on a compact interval
A Multiresolution Strategy for Reduction of Elliptic PDEs and Eigenvalue Problems

AbstractIn many practical problems coefficients of PDEs are changing across many spatial or temporal scales, whereas we might be interested in the behavior of the solution only on some relatively coarse scale. We approach the problem of capturing the influence of fine scales on the behavior of the solution on a coarse scale using the multiresolution strategy. Considering adjacent scales of a multiresolution analysis, we explicitly eliminate variables associated with the finer scale, which leaves us with a coarse-scale equation. We use the term reduction to designate a recursive application of this procedure over a finite number of scales.We present a multiresolution strategy for reduction of self-adjoint, strictly elliptic operators in one and two dimensions. It is known that the non-standard form for a wide class of operators has fast off-diagonal decay and the rate of decay is controlled by the number of vanishing moments of the wavelet. We prove that the reduction procedure preserves the rate of decay over any finite number of scales and therefore results in sparse matrices for computational purposes. Furthermore, the reduction procedure approximately preserves small eigenvalues of self-adjoint, strictly elliptic operators. We also introduce a modified reduction procedure which preserves the small eigenvalues with greater accuracy than the standard reduction procedure and obtain estimates for the perturbation of those eigenvalues. Finally, we discuss potential extensions of the reduction procedure to parabolic and hyperbolic problems
LU Factorization of Non-standard Forms and Direct Multiresolution Solvers

AbstractIn this paper we introduce the multiresolution LU factorization of non-standard forms (NS-forms) and develop fastdirect multiresolutionmethods for solving systems of linear algebraic equations arising in elliptic problems.The NS-form has been shown to provide a sparse representation for a wide class of operators, including those arising in strictly elliptic problems. For example, Green's functions of such operators (which are ordinarily represented by dense matrices, e.g., of sizeNbyN) may be represented by −log ϵ·Ncoefficients, where ϵ is the desired accuracy.The NS-form is not an ordinary[fn9] matrix representation and the usual operations such as multiplication of a vector by the NS-form are different from the standard matrix–vector multiplication. We show that (up to a fixed but arbitrary accuracy) the sparsity of the LU factorization is maintained on any finite number of scales for self-adjoint strictly elliptic operators and their inverses. Moreover, the condition number of matrices for which we compute the usual LU factorization at different scales isO(1). The direct multiresolution solver presents, therefore, an alternative to a multigrid approach and may be interpreted as a multigrid method with a single V-cycle.For self-adjoint strictly elliptic operators the multiresolution LU factorization requires onlyO((−log ϵ)2·N) operations. Combined withO(N) procedures of multiresolution forward and back substitutions, it yields a fast direct multiresolution solver. We also describe direct methods for solving matrix equations and demonstrate how to construct the inverse inO(N) operations (up to a fixed but arbitrary accuracy). We present several numerical examples which illustrate the algorithms developed in the paper. Finally, we outline several directions for generalization of our algorithms. In particular, we note that the multidimensional versions of the multiresolution LU factorization maintain sparsity, unlike the usual LU factorization
Sturm–Liouville Wavelets

AbstractIn this paper, we describe a new construction of wavelet-like functions on a compact interval [a,b] ⊂ R. Our approach of localizing multiscale decomposition of weightedL2-spacesL2,ρ([a,b]) is based on eigenfunctions of regular Sturm–Liouville boundary value problems, and was introduced and analyzed in Depczynski (1995). The asymptotic properties of such eigenfunctions yield localizing and stable bases, which prove to be very useful in time-frequency analysis. For specific types of eigenfunctions, fast algorithms are presented
Spline Interpolation and Wavelet Construction

AbstractThe method of Dubuc and Deslauriers on symmetric interpolatory subdivision is extended to study the relationship between interpolation processes and wavelet construction. Refinable and interpolatory functions are constructed in stages fromB-splines. Their method constructs the filter sequence (its Laurent polynomial) of the interpolatory function as a product of Laurent polynomials. This provides a natural way of splitting the filter for the construction of orthonormal and biorthogonal scaling functions leading to orthonormal and biorthogonal wavelets. Their method also leads to a class of filters which includes the minimal length Daubechies compactly supported orthonormal wavelet coefficients. Examples of “good” filters are given together with results of numerical experiments conducted to test the performance of these filters in data compression
Orthogonality Criteria for Multi-scaling Functions

AbstractNecessary and sufficient conditions for the orthonormality of a multi-scaling function φ with integer dilation factoraand multiplicityrare established. Here φ := (φ1, … , φr)Tand satisfies φ(x) =[formula]Pkφ(ax − k), for some positive integerM, withP0, … ,PMbeingr × rreal matrices andP0,PM≠ 0. These conditions also apply to the setting of the usual uni-scaling functions and uni-wavelets (i.e., multiplicityr= 1). The orthonormality of all the multi-scaling functions and multi-wavelets constructed in the wavelet literature can be verified. Extensive demonstrative examples are given
Asymptotics of Daubechies Filters, Scaling Functions, and Wavelets

AbstractWe study the asymptotic form asp→ ∞ of the Daubechies orthogonal minimum phase filterhp[n], scaling function φp(t), and waveletwp(t). Kateb and Lemarié calculated the leading term in the phase of the frequency responseHp(ω). The infinite productφ&#x0302;p(ω) = ∏Hp(ω/2k) leads us to a problem in stationary phase, for an oscillatory integral with parametert. The leading terms change form with τ =t/pand we find three regions for φp(τ):[formula]The numbers τ0≃ 0.1817 and τ1≃ 0.3515 are known constants. The functionGand its integralG(−1)are independent ofp. Regions 1 and 2 are matched over the intervalp−2/3⪡ τ − τ0⪡ 1.The wavelets have a simpler asymptotic expression because the Airy wavefront is removed by the highpass filter. We also find the asymptotics of the impulse responsehp[n]—a different functiong(ω) controls the three regions.The difficulty throughout is to estimate the phase
The Theory of Multiresolution Analysis Frames and Applications to Filter Banks

AbstractThe notion of a frame multiresolution analysis (FMRA) is formulated. An FMRA is a natural extension to affine frames of the classical notion of a multiresolution analysis (MRA). The associated theory of FMRAs is more complex than that of MRAs. A basic result of the theory is a characterization of frames of integer translates of a function φ in terms of the discontinuities and zero sets of a computable periodization of the Fourier transform of φ. There are subband coding filter banks associated with each FMRA. Mathematically, these filter banks can be used to construct new frames for finite energy signals. As with MRAs, the FMRA filter banks provide perfect reconstruction of all finite energy signals in any one of the successive approximation subspacesVjdefining the FMRA. In contrast with MRAs, the perfect reconstruction filter bank associated with an FMRA can be narrow band. Because of this feature, in signal processing FMRA filter banks achieve quantization noise reduction simultaneously with reconstruction of a given narrow-band signal
Wavelet Transforms That Map Integers to Integers

AbstractInvertible wavelet transforms that map integers to integers have important applications in lossless coding. In this paper we present two approaches to build integer to integer wavelet transforms. The first approach is to adapt the precoder of Laroiaet al.,which is used in information transmission; we combine it with expansion factors for the high and low pass band in subband filtering. The second approach builds upon the idea of factoring wavelet transforms into so-called lifting steps. This allows the construction of an integer version of every wavelet transform. Finally, we use these approaches in a lossless image coder and compare the results to those given in the literature