Convolution Operators and Their Symbols

Abstract

In this chapter we introduce convolution operators on the line, on a half-line, and on finite intervals. The Fourier transform of the convolution kernel is called the symbol of the operator. Shift operators are convolutions by the shifted Dirac delta function and lead to almost periodic symbols. The symbol of the Cauchy singular integral operator is piecewise continuous. The class of semi-almost periodic symbols emerges naturally when considering the algebra generated by shift operators and the Cauchy singular integral operator. Moreover, convolution operators with continuous and piecewise continuous symbols on finite intervals can be transformed into convolution operators on a half-line with>2×2-matrix valued symbols whose entries are almost periodic functions and semi-almost periodic (or even piecewise-almost periodic) functions,respectively.