ABOUT CLASSES OF BASIC FUNCTIONS FOR GENERALIZED TRIGONOMETRIC FUNCTIONS

. This article describes some methods of constructing basic functions with certain differential properties, on which generalized trigonometric functions are given. Functions with certain differential properties are considered, and the construction of such functions through Fourier coefficients with a certain decreasing order; combinations of both methods are also possible. It is shown that the class of basic functions is wide enough to coordinate the choice of these functions for research.


Introduction
In [1], the classes of generalized trigonometric functions were introduced, which, in contrast to the generalized periodic functions (distributions) of Schwartz [2], are given on the classes of basic, finitely differentiated functions. With this approach, the question of constructing such classes of basic functions is important.
Let's consider this question in more detail.
As you know, the trigonometric series is called an expression Trigonometric series can both coincide and diverge. If these series coincide, their sum is periodic with the period  .
In many cases, we have to deal with divergent trigonometric series that do not have a sum in the usual sense [3]. Such series often arise in theoretical research, in the formal differentiation of convergent trigonometric series, etc. For example, in the theoretical study of the Fourier series, it is necessary to study a series of types 1 1 cos 2 k kt  = +  , which is divergent at each point of the interval   0,2 .
One of the approaches to the study of divergent trigonometric series is the approach in which such series are considered generalized periodic functions of slow growth [2]; such functions are set on the classes of basic functions K . As the main functions of the class K periodic, infinitely differentiated functions appear. Thus the functional, which is the result of the action of the generalized periodic function on the basic functions of a class K , is presented in the form of a convergent numerical series. The practicality of this approach is explained by the fact that the numerical series through which the functionals are fed are generalized periodic functions of slow growth in the class K , always coinciding in the usual sense.
However, in our opinion, this convenience significantly limits the use of divergent series, as the requirement of infinite differentiation of basic functions is quite burdensome.
In [1], another approach to the study of divergent trigonometric series was Bernstein and many others [3]. When considering such a series, the methods of generalized summation were used.
Another approach to the study of divergent trigonometric series was proposed by Schwartz [2]. In this approach, divergent trigonometric series are considered objects of a new type called generalized periodic functions; such distributions as functionals on classes of infinitely differentiated basic functions are set. This approach was considered in the works of Mikusinsky, Edwards, Nikolsky, and others. In [1], an approach to the study of divergent trigonometric series was

Main part
1. Basic functions with certain differential properties.
From the finiteness of the spline vector and its normalization, it follows that when 0 h → splines ( , ) r B h t form a delta-like sequence at any finite value r .
For using B -splines as basic functions for constructing generalized trigonometric functions, we need to calculate the Fourier coefficients of these splines. In [9], the following submission was received B -splines: 1 Fig.1.   Fig. 1. Function ( , , ) T r t  when .5

= , and 1 =
It is easy to see that this function is continuous and has a continuous derivative 1 r − -st order.
Here are the Fourier coefficients of this function for some parameter values r , distinguishing between even and odd values of this parameter.
For the odd ones r , 21 rl =+ , ( 0,1, It is easy to see that the order of decline of the obtained coefficients ( , ) k ar  is consistent with the differential properties of the function ( , , )   The descending order of the obtained coefficients ( , ) k r  is consistent with the differential properties of the function ( , , ) P r t  . In addition, when 0 → coefficients ( , ) k r  go to 1 at any value of r ( 1,2, ... r = ). As before, from this fact, it follows that when 0 → functions ( , , ) P r t  form a delta-like sequence.
2. In paragraph 1 were considered the basic finite functions with certain differential properties. However, as we pointed out in [5], another approach is possible, where the main functions will be set through artificially constructed Fourier coefficients; it is clear that the differential properties of such functions are determined by order of decrease of these coefficients. Consider this approach in more detail.
Similarly, it is easy to build other basic functions of this type.
3. When constructing basic functions, you can also use a mixed approach. For example, the coefficients ( )

5.
A mixed approach to constructing Fourier coefficients is considered, which