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On best approximation in classes of periodic functions defined by integrals of a linear combination of absolutely monotonic kernels

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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

$$f(x) = \frac{1}{\pi }\int_0^{2\pi } {K(t)} \varphi (x - t)dt,$$

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

$$K(t) = \sum\nolimits_{k = 1}^\infty {\frac{{\cos (kt - {\textstyle{{\alpha \pi } \over 2}})}}{{k^S }}} ,$$

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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Literature cited

  1. J. Favard, “Application de la formula sommatoire d'Euler a la démonstration de quelques propriétés des intégrales des fonctions periodiques ou presque-periodiques,” Matematisk Tidskrift,4, 81–94, Kobenhavn, B. H. (1936).

    Google Scholar 

  2. J. Favard, “Sur les meilleurs procédés d'approximation de certains classes de fonctions par des polynomes trigonométriques,” Bull, de Sci. Math.,61, No. 1, 209–224, 243–256 (1937).

    Google Scholar 

  3. N. I. Akhiezer and M. G. Krein, “On best approximation of differentiable periodic functions by trigonometric sums,” Dokl. Akad. Nauk SSSR,15, 107–112 (1937).

    Google Scholar 

  4. V. K. Dzyadyk, “On best approximation on a class of periodic functions that have a bounded s-th derivative (0<s< 1),” Izv. Akad. Nauk SSSR, Ser. Matem.,17, 135–162 (1953).

    Google Scholar 

  5. V. K. Dzyadyk, “On best approximation on classes of periodic functions defined by kernels which are integrals of absolutely monotonie functions,” Izv. Akad. Nauk SSSR, Ser. Matem.,23, 933–950 (1959).

    Google Scholar 

  6. V. K. Dzyadyk, “On best approximation of absolutely monotonic and some other functions in the metric L with the aid of trigonometric polynomials,” Izv. Akad. Nauk SSSR, Ser. Matem.,25, 173–238 (1961).

    Google Scholar 

  7. B. Nagy, “Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen, L, Periodischer Fall,” Berichte der Math.-Phys. Kl. Akademie d. Wiss. zu Leipzig,90, 103–134 (1938).

    Google Scholar 

  8. S. M. Nikol'skii, “Approximation of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Matem.,10, No. 9, 207–256 (1946).

    Google Scholar 

  9. S. B. Stechkin, “On best approximation of certain classes of periodic functions by trigonometric polynomials,” Izv. Akad. Nauk SSSR, Ser. Matem.,20, 643–648 (1956).

    Google Scholar 

  10. S. B. Stechkin, “On best approximation of conjugate functions by trigonometric polynomials,” Izv. Akad. Nauk SSSR, Ser. Matem.,20, 197–206 (1956).

    Google Scholar 

  11. Sun Yun-Shen, “On best approximation of periodic differentiable functions by trigonometric polynomials,” Vestnik Pekinskogo Un-ta (Peking University Herald),3, 21–25 (1959).

    Google Scholar 

  12. Sun Yun-Shen, “On best approximation of periodic differentiable functions by trigonometric polynomials,” Izv. Akad. Nauk SSSR, Ser. Matem.,25, 143–153 (1961).

    Google Scholar 

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  1. Institute of Mathematics, Academy of Sciences of the USSR, USSR

    V. K. Dzyadyk

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  1. V. K. Dzyadyk
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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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