Periodic interpolation and wavelets on sparse grids

Computers & Mathematics with Applications, 1995

Decomposition and reconstruction algorithms for nested spaces of multivariate trigonometric polynomials are described. The scaling functions of these spaces are chosen as fundamental polynomials of Lagrange interpolation. Their properties are crucial for the approach presented here. In particular, FFT{algorithms can be used e ciently. AMS classi cation: 42 A 15, 41 A 10, 41 A 63, 65 D 05

In this paper we present some results regarding wavelet interpolation. After discussing the characterization of interpolating scaling functions, we give two examples of construction of an interpolation operator in the framework of a multiresolu tion analysis, the first obtained by constructing an interpolating scaling function in a given multiresolution analysis and the second based on the autocorrelation function of a compactly supported Daubechies wavelet. For both examples we give an estimate of the order of the interpolation error.

Applied and Computational Harmonic Analysis, 2010

General multivariate periodic wavelets are an efficient tool for the approximation of multidimensional functions, which feature dominant directions of the periodicity.

Wavelet Analysis and Its Applications, 1998

The aim of this paper is the detailed investigation of trigonometric polynomial spaces as a tool for approximation and signal analysis. Sample spaces are generated by equidistant translates of certain de la Vall ee Poussin means. The di erent de la Vall ee Poussin means enable us to choose between better time-or frequency-localization. For nested sample spaces and corresponding wavelet spaces, we discuss di erent bases and their transformations.

Arabian Journal of Mathematics, 2014

We introduce two generalizations, the first of which generalizes the concept of multiresolution analysis. We define the irregular generalized multiresolution analysis (IGMRA). This structure is defined taking translations on sets that are not necessarily regular lattices, for which certain density requirements are required, and without using dilations, also allows each subspace of IGMRA to be generated by outer frames of translations of different functions. The second generalization concerns the concept of association of wavelets to these new structures. We take frames of translations of a countable set of functions, which we called generalized wavelets, and define the concept of association of these generalized wavelets to those previously defined IGMRA. In the next stage, we prove two existence theorems. In the first theorem, we prove existence of IGMRA, and in the second existence of generalized wavelets associated with it. In the latter, we show that we are able to associate frames of translations with optimal localization properties, to IGMRA. In the last section of this paper, concrete examples of these structures are presented for L 2 (R) and for L 2 (R 2). Mathematics Subject Classification 42C40 • 42C30 1 Introduction From the classic concept of multiresolution analysis (MRA), introduced and further developed by Meyer [27,28], and Mallat [24,25], which provides a systematic way to construct orthonormal wavelet bases of L 2 (R), research in this area has been extended in various ways. These concepts are generalized to L 2 (R d) [14],

Advances in Computational Mathematics, 1995

This paper presents a general approach to a multi resolution analysis and wavelet spaces on the interval [-1, I]. Our method is based on the Chebyshev transform, corresponding shifts and the discrete cosine transformation (OCT). For the wavelet analysis of given functions, efficient decomposition and reconstruction algorithms are proposed using fast OCT-algorithms. As examples for scaling functions and wavelets, polynomials and transformed splines are considered.

Advances in Computational Mathematics, 2009

AbstractWe investigate a problem of approximate non-linear sampling recovery of functions on the interval $\mathbb I:=[0,1]$ expressing the adaptive choice of n sampled values of a function to be recovered, and of n terms from a given family of functions Φ. More precisely, for each function f on $\mathbb I$, we choose a sequence $\xi = \{\xi^s\}_{s =1}^n$ of n points in $\mathbb I$, a sequence $a = \{a_s\}_{s =1}^n$ of n functions defined on $\mathbb R^n$ and a sequence $\Phi_n = \{\varphi_{k_s}\}_{s =1}^n$ of n functions from a given family Φ. By this choice we define a (non-linear) sampling recovery method so that f is approximately recovered from the n sampled values f(ξ1), f(ξ2),..., f(ξn), by the n-term linear combination $$S(f\kern1pt) = \ S(\xi, \Phi_n, a, f\kern1pt) := \ \sum_{s =1}^n a_s(f(\xi^1),...,f(\xi^n))\varphi_{k_s}.$$In searching an optimal sampling method, we study the quantity $$\nu_n(f,\Phi)_q := \ \inf_{\Phi_n, \xi, a} \, \|f - S(\xi, \Phi_n, a, f\kern1pt)\|_q,$...

Journal of Fourier Analysis and Applications, 1999

An abstract formulation of generalized multiresolution analyses is presented, and those GMRAs that come from multiwavelets are characterized. As an application of this abstract formulation, a constructive procedure is developed, which produces all wavelet sets in ℝnrelative to an integral expansive matrix.

Mathematics of Computation, 1997

We present a unified approach for the construction of polynomial wavelets. Our main tool is orthogonal polynomials. With the help of their properties we devise schemes for the construction of time localized polynomial bases on bounded and unbounded subsets of the real line. Several examples illustrate the new approach.

A sequence of increasing translation invariant subspaces can be defined by the Haar-system (or generally by wavelets). The orthogonal projection to the subspaces generates a decomposition (multiresolution) of a signal. Regarding the rate of convergence and the number of operations, this kind of decomposition is much more favorable then the conventional Fourier expansion.