About Absolute Convergence of Fourier Series of Almost Periodic Functions

Abstract

The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x<sub>1</sub>, x<sub>2,</sub>...x<sub>p</sub>), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x<sub>1</sub>=a<sub>1</sub> t, x<sub>2</sub>=a<sub>2</sub> t,..., x<sub>p</sub>=a<sub>p</sub>t, where a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> - some real, non-zero numbers. Considering the value of the function f(x<sub>1</sub>, x<sub>2,</sub>...x<sub>p</sub>) on this line, he obtains a function of one variable φ(t) = f(a<sub>1</sub> t, a<sub>2</sub> t,...a<sub>p</sub> t) and proves that this function is almost periodic. With some choice of numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> - it may happen that this function is periodic. However, if the numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λ<sub>k</sub>→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.