Multipliers on the space of semiperiodic sequences

Semiperiodic sequences are defined to be the uniform limit of periodic sequences. They form a space of continuous functions on a compact group /\. We study the properties of the Radon measures on / in order to classify the multipliers for the space of semiperiodic sequences, paying special attention to those which can be realized as transference functions of physically constructible filters. Introduction. Almost periodic functions have been the subject of active research since their definition in 1925 by Harald Bohr. Their main interest lies in the fact that they form the smallest Banach space containing all the periodic functions of one real variable, thus becoming the natural background for the study of linear problems involving periodic functions. Subsequently the definition was extended in the following form: a function defined on a locally compact group is almost periodic if it is the uniform limit of a sequence of trigonometric polynomials. In particular, almost periodic sequences correspond to the group Z of integers. Sequences are in some respects a better model than functions for the study of periodic processes. The set p of all the complex-valued periodic sequences, endowed with the usual componentwise sum and product by scalars, is a normed vector space for the supremum norm given by Il(xn)nezlloo = sup lxn nEZ The completion of p does not coincide with the set of almost periodic sequences. Instead, it is a new space sp, which will be called the space of semiperiodic sequences. An element of sp is easily seen as a sequence (xn)nEz with the property that for every E > 0, there exists a "semiperiod" T such that for every m, n C Z we have Ixm+nT-xml < s. Semiperiodic nonperiodic sequences usually arise when we study limits of periodic sequences with unbounded periods. For instance, take el = (...,1,0,1,0,1,0,1,...), e2 = (...,1,0,0,1,0,0,1,...), e3 = (...,1,0,0,0,1,0,0,...), e4 = (...,1,0,0,0,0,1,0,...),.... Then Ek=1 ek/2k is not periodic, since the zero does not appear more than twice in the sequence. Received by the editors February 22, 1984 and, in revised form, October 16, 1984. 1980 Matherr^dics Sutiect Cluss^ficaX7r. Primary 22B10, 43A25, 30A76, 42A18, 28A30.