Abstract
In the metrics C and L we solve the problem of best approximation by trigonometric polynomials in classes of continuous periodic functionsf(x) of the form
where the kernel K(t) is a periodic integral of a linear combination of functions that are absolutely monotonic in the intervals (−∞, 2π) and (0, ∞), and ∥ϕ∥≤1. A particular case of such kernels for any s>0 andαε (−∞, ∞ are kernels of the form
which forα=s generate classes of periodic functions with a bounded s-th derivative in the sense of Weyl, whereas forα=s+1 they generate conjugate classes. For various values of s andα, apart from the case sε (0, 1) andα ε [0, 2]/[s, 2−s], such kernels were studied by various investigators (see [1–12]).
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Translated from Matematicheskie Zametki, Vol. 16, No. 5, pp. 691–701, November, 1974.
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Dzyadyk, V.K. On best approximation in classes of periodic functions defined by integrals of a linear combination of absolutely monotonic kernels. Mathematical Notes of the Academy of Sciences of the USSR 16, 1008–1014 (1974). https://doi.org/10.1007/BF01149788
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DOI: https://doi.org/10.1007/BF01149788