TABLE OF CONTENTS This appendix is written for readers who are not familiar with the basic concepts and language of filters. It also provides a larger context for parts of Chapter 3.
theory and definitionsWe begin with a general result about linear operators on 
If F: 
is a continuous linear operator that commutes with translations, then there exists a sequence 
such that
for all 
Furthermore, the function 
is in L ?(0,2 ?), and ?F ? = ?H ? ?. Conversely, if 
is such that H ? L ?(0,2 ?), then (A.1) defines a continuous linear operator that commutes with translations, and ?F ? = ?H ? ?.
Proof. Assume that 
is a continuous linear operator that commutes with translations, and let 
denote the canonical basis for 
defined by e k(n) = 0 if n ? k and e k(k) = 1. Then Fe 0 is an element of 
which we denote by h = ( h k). Since F commutes with translations, we have
for all 
. We go to the spectral domain and define the operator
in the obvious way: For 
in L 2(0,2 ?), define
where ( y k) = F(x k). Since the Fourier transform is an isometry, 
is a bounded linear operator with...
