Uniform Approximation of Periodical Functions by Trigonometric Sums of Special Type

The approximation characteristics of trigonometric sums of special type on the class of ( )-differentiable (in the sense of A. I. Stepanets) periodical functions are studied. Because of agreement between parameters of approximative sums and approximated classes, the solution of Kolmogorov-Nikol’skii problem is obtained in a sufficiently general case. It is shown that in a number of important cases these sums provide higher order of approximation in comparison with Fourier sums, de la Vallée Poussin sums, and others on the class in the uniform metric. The range of parameters in which the sums give the order of the best uniform approximation on the classes is indicated. 1. The Introduction and Problem Definition Let be the space of continuous -periodical functions where the norm is defined by . Let us consider the class [1] of functions such that for given and fixed sequence ( ) of real numbers the series are the Fourier series of some function , where The function is called the -derivative of and denoted by . For , , the class coincides with Weyl-Nagy class , and for coincides with Weyl class . In the case of natural and the class is a class of periodical functions whose th derivatives nearly everywhere do not exceed unity in absolute value. If is a sequence such that then consists of infinitely differentiable functions (see [2, Chapter 1, Section 8]). The example of a sequence satisfying condition (3) is , , . In this case the class will be denoted by . If satisfies the condition consists of analytical functions regularly continuing into the strip of the complex plane. Following [3, page 147], we set by the set of all continuous convex downwards functions , , satisfying the condition and associate each with the characteristic where is an inverse function of . By using we define the next subset of as As is shown in [3, page 153] all functions for which where , belong to . The set of such functions is denoted by . The quantity has a simple geometric interpretation. It is equal to the length of interval where the value of the function decreases by two times. Thereby it is natural to call the function the half-decay period of . The examples of from are the functions , , , , , and others. The set also includes (see [3, page 153]) the subset of all functions for which the characteristic called the modulus of half-decay, tends monotonically to infinity as . If , , , then and that is why Therefore, . Thus, in there exist functions that tend to zero according to the power law as well as the functions that tend to zero faster than any power function. However, the