THE SPECTRAL THEORY OF BOUNDED FUNCTIONS BY CARL S. HERZt1) The present article is intended as a survey of the title subject. The mate- rial 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 ap- proximation 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 immedi- ately 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There was a considerable overlap, for, although the two papers were quite independent, both were heavily influenced by unpublished notes of Beurling.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 re- vised the paper in an attempt to suppress details which may be found else- where. Also, I have taken advantage of more recent work by others and my- self 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. Received by the editors May 18, 1956 and, in revised form, January 28, 1959. (') This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Con- tracts No. AF18(600)-685 and No. AF18(600)-1109. Reproduction in whole or in part is per- mitted for any purpose of the United States Government. 181 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 182 C. S. HERZ [February The first section introduces the fundamental concepts and contains a rough outline of the material to be treated in the later portions. After it, §§2 and 3 may be read independently of the rest of the paper. The exposition here is carried out on a locally compact abelian group. Important ideas, when first introduced, are generally discussed quite fully with the viewpoint of classical analysis in mind, and examples are given, usually at the end of a section, to illustrate what is going on. 0. Preliminaries. Let T be a locally compact Hausdorff space. We denote by C(T) the linear space of bounded continuous functions on T; for <pEC(T), ||<p|| is defined as sup |<p(x)|, where the supremum is taken over all x£r. C(r) has a subspace Co(T) consisting of the functions which vanish at °°, i.e. <pECoiT) it <p is continuous and given e>0 there is a compact subset KCr such that \<pix) \ <e ior x£K, the complement of K. The dual Banach space to Co(r) is B(T), the space of bounded Radon measures. For uEB(T), IImII=f\l1idx)\, the total variation. For our purposes the most useful topology on C(T) is the strict topology (2) which may be defined in either of two equivalent ways. The first is this: con- sider C(T) operating on CoCH by pointwise multiplication; the strict topology is the strong operator topology. For the alternative description we shall de- fine the neighborhoods of 0. Let {e„, K„} be a sequence of pairs of positive numbers e„ and compact sets K„ CT such that e„ / oo and Kn CKn+i- A strict neighborhood of 0 in C(T) consists of all functions <pEC(T) satisfying the inequalities |^>(x)| <e„ for x£Kn. The strict topology is a locally convex topology in which C(T) is a complete topological linear space. The space of strictly continuous linear functionals on C(T) admits a natural identification with B(T). On norm-bounded sets, the strict topology agrees with the one given by uniform convergence on compact sets. Suppose P is another locally compact Hausdorff space and p:T—»P a continuous map. Given <pEC(P) we define p<p as a function on T by p<p(x) = (p(px). (The reason for the notation will become clear in a little while.) It is obvious that one has a map p: C(P)—>C(r) and that ||pv|| ^||^||. What is of paramount importance is that p is continuous with respect to the strict topologies. Here the second definition of the strict topology is most useful; the neighborhood of 0 in C(T) corresponding to the sequence {e„, Kn} con- tains the image of the neighborhood of 0 in C(P) corresponding to the se- quence {e„, pKn}. The dual of the map p: C(P)-+C(T) is a map which we de- note again by p; p: B(T)-*B(P). If uEB(T), the measure ppEB(T) is defined on compact KCP by pp(K) =p(p~1K). At the beginning of §5 we shall return to the examination of these maps. We are mainly concerned with the case where T is a locally compact abelian group. Throughout this paper the word "group" is used to mean (J) The strict topology has been used by Beurling in unpublished work and is discussed in Buck (where it is called "0"). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] THE SPECTRAL THEORY OF BOUNDED FUNCTIONS 183 only "locally compact abelian group." The group operation is written as addi- tion. If P is another group, a map p: T—»P is called a "representation" if it is continuous and is a homomorphism in the algebraic sense. The term "homo- morphism" is reserved for representations in which the image set pr is closed and the image of an open set in Y is relatively open in pr. In particular, a subset A of T is called a "subgroup" if and only if the identity map i: A—*Y is an injective homomorphism of groups. The topology of a group may be described by giving a basis for the neigh- borhoods of 0. We use the term "nucleus" to denote an open neighborhood of 0 with compact closure which satisfies one additional condition. This extra condition is needed only in a few places; it is that we require nuclei to be Jordan measurable. A subset of a group is said to be "Jordan measurable" if its boundary has Haar measure zero. One can always choose a fundamental system of neighborhoods of 0 to be nuclei. To see this, we note that the topol- ogy of a group may be described by a system of continuous invariant pseudo- metrics(2a). An invariant pseudometric on Y is a real valued function 5 with the properties b(x) ^5(0) =0, 5(—x) = S(x), and 8(x+y)^8(x)+8(y). Then 8(x —y) has all the properties of a distance except that 8(x —y) =0 need not imply x = y. It is well known that the topology of a group can be described by a system {5} of continuous invariant pseudometrics where 5(x)=0 for each 8 of the system implies x = 0 and where for each S, the sets [x\ 8(x) ^e] are compact for sufficiently small e. Thus a fundamental system of neighbor- hoods of 0 in T may be given by the sets N(5, £) = {x\ 8(x) <!■} where 5 runs through an admissible system of continuous invariant pseudometrics and for each 5, £ runs through a set of positive numbers tending to 0. Holding 5 fixed, set a(£) = | N(5, f)| where | A| denotes the Haar measure of a set(3) A. Let e be such that {x|5(a:)^e} is compact. The function a is increasing on the interval [0, e]. Hence it has at most a countable set of discontinuities. For each £ in the open interval (0, e) which is a point of right-continuity for a, N(5, £) is a nucleus. More generally, we shall have occasion to consider sets A with the properties that 0(EInt A, the interior of A; the closure A, of A, is compact, and A is Jordan measurable. Such an A will be called a /-set. It is assumed that the reader is familiar with the theory of Fourier analy- sis on groups as exposed, say, in Cartan and Godement. It will be necessary to explain the notation and conventions used in the present paper. If t is a point of the character group £ ol Y, the character itself is denoted by et and we write et(x) =e(tx). The topology of t is equivalent to that inherited by the set of functions {ei|i£f} as a subset of C(r) in the strict topology. Thus whenever p: Y—>P is a representation, the previously defined map p agrees with the natural dual representation on the character groups, />: P—»f.