The Spectral Theory of Bounded Functions
Carl Herz · Transactions of the American Mathematical Society · 1960
The present article is intended as a survey of the title subject.The material is closely intertwined with all branches of harmonic analysis.Historically, the rigorous foundations of the theory arise in Riemann's treatment of trigonometric series, and spectral theory is essentially equivalent to the study of formal multiplication.The original motivation for the modern treatment, due to Wiener, Carleman, and Beurling, came from the study of integral equations with convolution kernels.There are applications to linear partial differential equations with constant coefficients, and there is a very close connection with problems about entire functions bounded on a line.My own concern with the topic commenced with questions about the theory of approximation for multiple Fourier transforms.When I first looked at the subject the basic material did not appear very well organized, and certain elementary facts were not recognized as immediately obvious to experts in the field.Thus, in 1954, I set about to record what was known with certain useful additions.Shortly after the original version of this work was finished in 1956, the paper of Domar appeared.(An author's name in small capitals indicates a reference to the bibliography.)There was a considerable overlap, for, although the two papers were quite independent, both were heavily influenced by unpublished notes of Beurling.I have revised the paper in an attempt to suppress details which may be found elsewhere.Also, I have taken advantage of more recent work by others and myself to improve the material of the last half of the paper.After a preliminary section introducing much of the notation and basic definitions, the contents of this paper are divided into six parts.The headings are:1.The spectrum.2. The point spectrum.3. Potential theory and spectral analysis.4. The spectral synthesis problem. 5. Representations.6. Examples of spectral synthesis.