Generalized Smoothness and Approximation of Periodic Functions in the Spaces Lp, 1 < p < +∞
- 9 Downloads
Norms of images of operators of multiplier type with an arbitrary generator are estimated by using best approximations of periodic functions of one variable by trigonometric polynomials in the scale of the spaces Lp, 1 < p < +∞. A Bernstein-type inequality for the generalized derivative of the trigonometric polynomial generated by an arbitrary generator ψ, sufficient constructive ψ-smoothness conditions, estimates of best approximations of ψ-derivatives, estimates of best approximations of ψ-smooth functions, and an inverse theorem of approximation theory for the generalized modulus of smoothness generated by an arbitrary periodic generator are obtained as corollaries.
Keywordsbest approximation modulus of smoothness generalized derivative
Unable to display preview. Download preview PDF.
The author wishes to express gratitude to the referee for valuable and useful remarks that have led to the improvement of the paper.
- 2.H. Triebel, Higher Analysis (Johann Ambrosius Barth Verlag GmbH, Leipzig, 1992).Google Scholar
- 9.E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Eucliadean Spaces (Princeton Univ. Press, Princeton, NJ, 1971).Google Scholar
- 16.V M. Tikhomirov, Some Questions in Approximation Theory (Izd. Moskov Univ., Moscow, 1976) [in Russian].Google Scholar
- 17.E. Belinski and E. Liflyand, “Approximation properties in L p, 0 < p < 1,” Funct. Approx. Comment. Math. 22, 189–199 (1994).Google Scholar
- 21.M. K. Potapov and B. V. Simonov, “Moduli of smoothness of positive order of functions from the spaces L p, 1 < p < +∞,” in Trudy Mekh.-Mat. Fak. MGU, Modern Problems of Mathematics and Mechanics (Izd. Mekh.-Mat. Fak. MGU, Moscow, 2011), Vol. 7, No. 1 pp. 100–109, [in Russian].Google Scholar