Ten Lectures on Wavelets
This text contains 10 lectures given by the author at the CBMS conference on wavelets organized in June 1990 by the Mathematics Department at the University of Lowell, Massachusetts.
TABLE OF CONTENTS Ten Lectures on Wavelets
Chapter 4: Time-Frequency Density and Orthonormal Bases
This chapter splits naturally into two parts. The first section discusses the role of time-frequency density in wavelet transforms versus windowed Fourier transforms. In particular, for the windowed Fourier transform, orthonormal bases are possible only at the Nyquist density but no such restriction exists for the wavelet case. This leads naturally to the second section, which discusses different possibilities for orthonormal bases in the two cases.
4.1. The role of time-frequency density in wavelet and windowed Fourier frames.
We start with the windowed Fourier case. We mentioned in 3.4.1 that a family of functions 
,
cannot be a frame, whatever the choice of g, if ? 0 t 0 > 2 ?. In fact, for any choice of 
, one can find 
so that f ? 0 but ? f, g m,n ? = 0 for all 
. If, for instance, ? 0 = 2 ?, t 0 = 2, then such a function f is easy to construct: ? f, g m,n ? = 0 for all 
leads to
so that it is sufficient to find f ? 0 for which 
Define now, for 
Clearly, 
However, 
, which turns into its negative upon the substitution ? = 2 n ? ?? ? 1, and therefore equals zero. The same construction can be used for any other pair ? 0, t 0 with product 4 ?; a...
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