This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shift-invariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Band-limited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,

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We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudo-spline tight frames and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.

Scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter (in a two-band orthogonal filter bank). For "multifilters" those coefficients are matrices. This gives a new block structure for the filter bank, and leads to multiple scaling functions and wavelets. Geronimo, Hardin, and Massopust constructed two scaling functions that have extra properties not previously achieved. The functions \Phi 1 and \Phi 2 are symmetric (linear phase) and they have short support (two intervals or less), while their translates form an orthogonal family. For any single function \Phi, apart from Haar's piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2 by 2 matrix coefficients while retaining orthogonality. This note derives the properties of \Phi 1 and \Phi 2 from the matrix dilation equation that they satisfy. Then our main step is to construct associated wavelets: two wavelets for two scaling functions....

. The functions f 1 (x); : : : ; fr (x) are refinable if they are combinations of the rescaled and translated functions f i (2x \Gamma k). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth h = 1=2. These refinable spaces have refinable basis functions. The accuracy of the computations depends on p, the order of approximation, which is determined by the degree of polynomials 1; x; : : : ; x p\Gamma1 that lie in V 0 . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions f i (x) are known only through the coefficients c k in the refinement equation---scalars in the traditional case, r \Theta r matrices for multiwavelets. The scalar "sum rules" that determine p are well known. We find the conditions on the matrices c k that yield approximation of order p from V 0 . These are equivalent to the Strang--Fix condition...

Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The “wavelet transform ” maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them — always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including high-definition television). So far the Fourier Transform — or its 8 by 8 windowed version, the Discrete Cosine Transform — is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory. 1. The Haar wavelet To explain wavelets we start with an example. It has every property we hope for, except one. If that one defect is accepted, the construction is simple and the computations are fast. By trying to remove the defect, we are led to dilation equations and recursively defined functions and a small world of fascinating new problems — many still unsolved. A sensible person would stop after the first wavelet, but fortunately mathematics goes on. The basic example is easier to draw than to describe: Figure 1. Scaling function φ(x), wavelet W(x), and the next level of detail.

Refinable function vectors are usually given in the form of an infinite product of their refinement (matrix) masks in the frequency domain and approximated by a cascade algorithm in both time and frequency domains. We provide necessary and sufficient conditions for the convergence of the cascade algorithm. We also give necessary and sufficient conditions for the stability and orthonormality of refinable function vectors in terms of their refinement matrix masks. Regularity of function vectors gives smoothness orders in the time domain, and decay rates at infinity in the frequency domain. Regularity criteria are established in terms of the vanishing moment order of the matrix mask.

Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filter banks leading to wavelet bases, multi-wavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2-channel wavelet systems. After reviewing this recently developed theory, we examine the use of multiwavelets in a filter bank setting for discrete-time signal and image processing. Multiwavelets di#er from scalar wavelet systems in requiring two or more input streams to the multiwavelet filter bank. We describe two methods (repeated row and approximation /deapproximation) for obtaining such a vector input stream from a one-dimensional signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximation-based preprocessing. We describe an additional algorithm for multiwavelet processing of two-dimensional signals, two rows at a time, and develop a new family of multiwavelets (the constr...

. Matrix refinement equations are functional equations of the form f(x) = P N k=0 c k f(2x \Gamma k), where the coefficients c k are matrices and f is a vector-valued function. Refinement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of scalar refinement equations (where the coefficients c k are scalars) are known. This paper considers analogous questions for matrix refinement equations. Conditions for existence and uniqueness of compactly supported distributional solutions are given in terms of the convergence properties of an infinite product of the matrix \Delta = 1 2 P c k with itself. Fundamental differences between solutions of matrix equations and scalar refinement equations are examined. In particular, it is shown that "constrained" solutions of the matrix refinement equation can exist even when the infinite product diverges. The existence of constrained solutions is related to the eigenvalue structure of \Delta; so...

A scaling function is the solution to a dilation equation \Phi(t) = P c k \Phi(2t \Gamma k), in which the coefficients come from a low-pass filter. The coefficients in the wavelet W (t) = P d k \Phi(2t \Gamma k) come from a high-pass filter. When these coefficients are matrices , \Phi and W are vectors: there are two or more scaling functions and an equal number of wavelets. By dilation and translation of the wavelets, we have an orthogonal basis W ijk = W i (2 j t \Gamma k) for all functions of finite energy. Those "multiwavelets " open new possibilities. They can be shorter, with more vanishing moments, than single wavelets. They can be symmetric, which is impossible for scalar wavelets (except for Haar's). We determine the conditions to impose on the matrix coefficients c k in the design of multiwavelets, and we construct a new pair of piecewise linear orthogonal wavelets with two vanishing moments. 1. TWO SCALING FUNCTIONS AND TWO WAVELETS The Haar wavelets are certainly the...