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Multiresolution Analysis for Compactly Supported Interpolating Tensor Product Wavelets (long version)

Tommi Höynälänmaa

Abstract

We construct multidimensional interpolating tensor product MRA's of the function spaces $C_0(\mathbb{R}^n,K)$, $K = \mathbb{R}$ or $K = \mathbb{C}$, consisting of real or complex valued functions on $\mathbb{R}^n$ vanishing at infinity and the function spaces $C_\textrm{u}(\mathbb{R}^n,K)$ consisting of bounded and uniformly continuous functions on $\mathbb{R}^n$. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence result from Donoho (1992, Interpolating Wavelet Transforms) to our $n$-dimensional construction.

Multiresolution Analysis for Compactly Supported Interpolating Tensor Product Wavelets (long version)

Abstract

We construct multidimensional interpolating tensor product MRA's of the function spaces , or , consisting of real or complex valued functions on vanishing at infinity and the function spaces consisting of bounded and uniformly continuous functions on . We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence result from Donoho (1992, Interpolating Wavelet Transforms) to our -dimensional construction.

Paper Structure

This paper contains 18 sections, 22 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General
  4. Sequences and Cartesian Products
  5. Vector Spaces
  6. Function Spaces and Sequence Spaces
  7. Tensor Products
  8. General Definitions for a Compactly Supported Interpolating MRA
  9. Mother Scaling Function
  10. General Definitions for the Univariate MRA's
  11. General Definitions for Multivariate MRA's
  12. Compactly Supported Interpolating MRA of ${C_\mathrm{u}({{\mathbb{R}^n}},{K})}$
  13. Interpolating Dual MRA
  14. General
  15. Tensor Product Representation of the Dual MRA
  16. ...and 3 more sections

Key Result

Lemma \oldthetheorem

Under the conditions given at the start of this section (${\varphi^{[{n}]}}$ is an $n$-dimensional tensor product mother scaling function) the conditions (MRA1.1), (MRA1.4), (MRA1.5), and (MRA1.6) are true.

Theorems & Definitions (88)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7
  • definition 8
  • definition 9
  • definition 10
  • ...and 78 more