Michael Frazier

A discrete transform and decompositions of distribution spaces

Abstract We study a representation formula of the form ƒ = ∑ Q 〈ƒ, ϑ Q 〉ψ Q for a distribution ƒ on R n. This formula is obtained by discretizing and localizing a standard Littlewood-Paley decomposition. The map taking ƒ to the sequence {〈ƒ, ϑ Q 〉} Q , with Q running over the dyadic cubes in R n, is called the ϑ-transform. The functions ϑQ and ψQ have a particularly simple form. Moreover, most of the familiar distribution spaces (Lp-spaces, 1 R n. We also consider pointwise multipliers. For the characteristic function of a domain, we obtain boundedness results for a general class of domains which properly includes Lipschitz domains. Several interpolation methods are easily analyzed via the sequence spaces. For real interpolation, we obtain, among other things, an extension to the case p = 0. This in turn gives a new approach to the traditional atomic decomposition of Hardy spaces.

The Boundedness of Calderón Zygmund Operators on the Spaces Fpa,q.

Calderon-Zygmund operators are generalizations of the singular integral operators introduced by Calderon and Zygmund in the fifties [CZ]. These singular integrals are principal value convolutions of the form Tf(x) = lime®0 o|x-y|>e K(x-y) f(y) dy = p.v.K * f(x), where f belongs to some class of test functions.

A new transform for time-frequency analysis

The psi-decomposition of a signal, in which the signal is written as a weighted sum of certain elementary synthesizing functions, is described. The set S of synthesizing functions consists of dilated and translated copies of two parent functions, which are concentrated in both the time and the frequency domains. The weighting constants in the psi-decomposition define a transform called the phi-transform. The phi-transform of a signal captures both the frequency content and the temporal evolution of a nonstationary signal. The phi-transform is linear, continuous, and continuously invertible. The set S of synthesizing functions used in the psi-decomposition is nonorthogonal, hence considerable flexibility is permitted in its construction. It is shown with the help of two examples that the set S is easy to construct. >

The Discrete Orthonormal Wavelet Transform: An Introduction

In this paper z-transform theory is used to develop the discrete orthonormal wavelet transpform for multidimensional signals. The tone is tutorial and expository. Some rudimentary knowledge of z-transforms and vector spaces is assumed. The wavelet transform of a signal consists of a sequence of inner products of a signal computed against the elements of a complete orthonorml set of basis vectors. The signal is recovered as a weighted sum of the basis vectors. This paper addresses the necessary and sufficient conditions that such a basis muct respect. An algorithm for the design of a proper basis is derived from... Read complete abstract on page 2.

Prologue: Compression of the FBI Fingerprint Files

When your local police arrest somebody on a minor charge, they would like to check whether that person has an outstanding warrant, possibly in another state, for a more serious crime. To check, they can send his or her fingerprints to the FBI fingerprint archive in Washington, D.C. Unfortunately, the FBI cannot compare the received fingerprints with their records rapidly enough to make an identification before the suspect must be released. A criminal wanted on a serious charge will most likely have vacated the area by the time the FBI has provided the necessary identification.

Background: Complex Numbers and Linear Algebra

We start by setting some notation. The natural numbers {1, 2, 3, 4,...} will be denoted by ℕ, and the integers {..., −3, −2, −1, 0, 1, 2, 3,...} by ℤ. Complex numbers will be introduced later. We assume familiarity with the real numbers ℝ and their properties, which we briefly summarize here. The basic algebraic properties of ℝ follow from the fact that ℝ is a field.

Wavelets on ℤ

So far we have considered signals (vectors) of finite length, which we have extended periodically to be defined at all integers. In this chapter we deal with infinite signals, which are generally not periodic.

Wavelets and Differential Equations

Many applications of mathematics require the numerical approximation of solutions of differential equations. In this chapter we give a brief introduction to this topic. A thorough discussion is beyond the scope of this text. Instead, by simple examples, we give an idea of the contribution wavelet theory can make to this subject.

Wavelets on ℤN

In this chapter we continue to work with discrete signals z ∈ l 2(ℤ N ). In chapter 2 we noted the two key advantages of the Fourier basis F: (1) translation-invariant linear transformations are diagonalized by F, and (2) the coordinates in the Fourier basis can be computed quickly using the FFT. However, for many purposes in signal analysis and other fields, the Fourier basis has serious limitations. Many of these come from the fact that the Fourier basis elements are not localized in space, in the following sense.

The Discrete Fourier Transform

In chapter 1 we worked with vectors in ℂ N , that is, sequences of N complex numbers. Here we change notation in several ways. First, for reasons that will be more clear later, we index these N numbers over j ∈ {0, 1,..., N − 1} instead of {1, 2,..., N}. Second, instead of writing the components of z as z j , we write them as z(j).