Besicovitch almost-periodic functions

A class of almost-periodic functions in which the analogue of the Riesz–Fischer theorem is valid: Any trigonometric series

where

is the Fourier series of some -almost-periodic function. The definition of these functions [1], [2] is based on a generalization of the concept of an almost-period, and certain additional ideas must be introduced in it. A set of real numbers is called sufficiently homogeneous if there exists an such that the ratio between the largest number of members of in an interval of length and the smallest number of members in an interval of the same length is less than 2. A sufficiently homogeneous set is also relatively dense. A complex-valued function , , summable to degree on any finite interval of the real axis, is called a Besicovitch almost-periodic function if to each there corresponds a sufficiently homogeneous set of numbers (the so-called -almost-periods of ):

such that for each

and for each

where

Here is a real-valued function, defined, respectively, for a real variable and an integer argument.

References

[1]	A.S. Besicovitch, "On mean values of functions of a complex and of a real variable" Proc. London Math. Soc. (2) , 27 (1927) pp. 373–388
[2]	A.S. Besicovitch, "On Parseval's theorem for Dirichlet series" Proc. London Math. Soc. (2) , 26 (1927) pp. 25–34
[3]	B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)

Comments

Besicovitch developed his theory in [a1], rather than in [1], [2].

As is implicit in the article, for each there is a class of almost-periodic functions, denoted by . The first part of the article deals with , the rest is more general. General references may be found under Almost-periodic function.

References

[a1]	A.S. Besicovitch, "On generalized almost periodic functions" Proc. London Math. Soc. (2) , 25 (1926) pp. 495–512