The Best Approximation of Function Classes Generated by Composite Kernels - Сорич, Віктор; Сорич, Ніна (Кам'янець-Подільський національний університет імені Івана Огієнка, 2025)
The set of periodic functions whose essential supremum of the modulus of their r-th derivatives does not exceed one constitutes a class of convolutions of the Bernoulli kernel of order r with elements of the unit ball in the space of summable, essentially bounded periodic functions with zero mean over a period. In 1936, J. Favard obtained exact values for the best approximations of such classes by trigonometric polynomials of order not exceeding n − 1 in the uniform metric for every natural n. In further studies, when finding upper bounds of the best approximations of classes of convolutions with fractional Bernoulli kernals, generalized Weyl-Nagy kernels, and Poisson kernels by trigonometric polynomials of a given order, both in uniform and integral metrics were considered. In 1938, the Hungarian mathematician B. Nagy proposed a sufficient condition for a kernel of class of convolutions (the so-called Nagy condition): there exists a trigonometric polynomial of order n − 1 that interpolates the kernel at 2n uniformly distributed points on the period, and only at those points, with alternating signs of the difference between the kernel and the polynomial. Satisfying this condition makes it possible to compute the best approximation of the kernel in the integral metric, as well as the best approximation of the corresponding convolution class in both uniform and integral metrics. In 1946, S. Nikolsky generalized Nagy’s condition. Thanks to the results of M. Krein (1938), in most cases it is not difficult to construct a trigonometric polynomial that interpolates a given kernel at 2n uniformly spaced points of the period. The main difficulty lies in proving that no additional interpolation points exist. When studying certain kernels, mathematicians encountered the phenomenon that, besides the «guaranteed» 2n interpolation points, «extra» interpolation points may appear. This motivated researchers to investigate kernels for which the «standard» Nagy condition fails. The present work takes a step in this direction. We establish several sufficient conditions for linear combinations of even kernels and, likewise, for odd kernels, ensuring that they are interpolated by trigonometric polynomials of order n − 1 only at 2n + 2 uniformly distributed points of the period in the even case, and only at a single uniformly distributed set of 2n + 1 points in the odd case, with alternating signs of the difference between the linear combination and the polynomial at the interpolation points. Thus, for these cases, polynomials of order n − 1 provide the best approximation as if they were polynomials of order n. The paper also presents examples of differences of odd kernels (Bernoulli, Bernoulli and Poisson), as well as differences of even Bernoulli kernels, which illustrate the theorems obtained here. As a consequence, the work also determines the values of the best approximations by trigonometric polynomials of order n for certain classes of convolutions with such linear combinations of odd (even) kernels for which the Nagy condition of order n fails, but the corresponding condition of one order higher holds