Bochner, "Continuous mappings of almost automorphic and almost

periodic functions," Proceedings of the National Acadamy of Sciences of the United States of America, vol.

[13] Kokilashvili, V., On estimate of best approximation and modulus of smoothness in Lebesgue spaces of

periodic functions with transformed Fourier series (Russian) Soobshch.Akad.Nauk Gruzin.SSR 35(1965), n:1, 3-8.

Assume ([H.sub.1]) and let also p : R [right arrow] R be a non- constant

periodic function of class [C.sup.2] such that p(t) > 0 for all t [member of] R.

All the elements of A[P.sub.[phi]](R, C) belong to the classical space A[P.sub.1](R, C), the space of Bohr almost

periodic functions, with absolutely (and uniformly) convergent Fourier series (we have called it the Poincare space of almost

periodic functions, due to the fact that he produced the first example of an almost

periodic function, in the sense of Bohr, showing how to find the coefficients of its Fourier series, by the introduction of the mean value on an infinite interval).

By the well-known results [5, 10], we know that the smooth

periodic function [F.sub.p] can be approximated bybivariate trigonometric polynomials very well.

A

periodic function F(t) is such that it couples with the form presented in Equation (2):

The denumerant E([alpha])(t) has a beautiful structure and it has been known since the times of Cayley and Sylvester that E([alpha])(t) is in fact a quasi-polynomial, i.e., it can be written in the form E([alpha])(t) = [[summation].sup.N.sub.i=0][E.sub.i](t)[t.sup.i], where [E.sub.i](t) is a

periodic function of t (a more precise description of the periods of the coefficients [E.sub.i](t) will be given later).

Nanotemplates are structures with nanometer sized

periodic function with high versatility that promise to improve devices and medicine, such as miniaturization of electronics and capturing biological species for detection.

The results provided in the second section show that the accident number that depends on the physical and emotional condition of drivers is a

periodic function. From all possible types of

periodic functions, we will use harmonic functions the differential equation of which is as follows:

where [C.sub.h] is expressed in terms of a very long, involved expression and [[??].sub.h] is a

periodic function.

In the presence of the strict vertical modulation of the composite composition [bar.[[phi]].sub.SL] (z) is the

periodic function along the direction z; consequently, it can be expanded into a Fourier series:

Since [^.[mu].sub.n]([[lambda].sub.j](n)) are uniformly bounded by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the left hand side is an almost

periodic function. Because [a.sub.j](n) [not equal to] 0 for all j, n, this implies [^.[[mu]].sub.nj](n)) = 0.