Besicovitch almost-periodic functions

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

$$ \sum _ { n } a _ {n} e ^ {i \lambda _ {n} x } , $$

where

$$ \sum _ { n } | a _ {n} | ^ {2} < \infty , $$

is the Fourier series of some $ B ^ {2} $- 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 $ E $ of real numbers is called sufficiently homogeneous if there exists an $ L > 0 $ such that the ratio between the largest number of members of $ E $ in an interval of length $ L $ and the smallest number of members in an interval of the same length $ L $ is less than 2. A sufficiently homogeneous set is also relatively dense. A complex-valued function $ f(x) $, $ - \infty < x < \infty $, summable to degree $ p $ on any finite interval of the real axis, is called a Besicovitch almost-periodic function if to each $ \epsilon > 0 $ there corresponds a sufficiently homogeneous set of numbers (the so-called $ (B ^ {p} , \epsilon ) $- almost-periods of $ f(x) $):

$$ {} \dots < \tau _ {-2} < \tau _ {-1} < \tau _ {0} < \tau _ {1} < \dots , $$

such that for each $ i $

$$ \overline{M}\; _ {x} \{ | f (x + \tau _ {i} ) - f (x) | ^ {p} \} < \epsilon ^ {p} , $$

and for each $ c > 0 $

$$ \overline{M}\; _ {x} \overline{M}\; _ {i} { \frac{1}{c} } \int\limits _ { x } ^ { x+c } | f ( \xi + \tau _ {i} ) - f (x) | ^ {p} d \xi < \epsilon ^ {p} , $$

where

$$ \overline{M}\; _ {x} \{ F (x) \} = \ \overline{\lim\limits _ {\tau \rightarrow \infty } } \ { \frac{1}{2 \tau } } \int\limits _ {- \tau } ^ \tau F (x) dx, $$

$$ \overline{M}\; _ {i} \{ F (i) \} = \overline{\lim\limits _ {n \rightarrow \infty } } { \frac{1}{2n+1} } \sum _ { i=-n } ^ { n } F (i). $$

Here $ F(x) $ 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 $ p \geq 1 $ there is a class of almost-periodic functions, denoted by $ B ^ {p} $. The first part of the article deals with $ B ^ {2} $, 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