Approximation of integrable functions by linear methods almost everywhere
Article
Received:
- 27 Downloads
Abstract
It is shown that 2π periodic functions whose (r-1)-th derivatives have bounded variation (r > 0) can be approximated by de La Vallée-Poussin sums σ n,m (an ⩽m =m (n) ⩽An,0 <a<A<1) at almost all points with a rate o(n−r). For functions belonging to the class Lip (α, L) (0 <α < 1), any natural N, and a positive ɛ, we have almost everywhere where\(ln_k x = \underbrace {ln \ldots ln x}_k(k = 1, 2, \ldots )\). For any triangular method of summation T with bounded coefficients we construct functions belonging to Lip (α, L) (0 < α < 1) and such that almost everywhere, where the τn(f; x) are the means of the method T.
$$|f(x) - \sigma _{n,m} (f;x)| \leqslant c(f,x)n^{ - \alpha } lnn \ldots ln_N^{1 + \varepsilon } n,$$
$$\mathop {\overline {\lim } }\limits_{n \to \infty } |f(x) - \tau _n (f;x)|n^a (ln n \ldots ln_N n)^{ - a} = \infty $$
Keywords
Periodic Function Integrable Function Linear Method Triangular Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
Literature cited
- 1.N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).Google Scholar
- 2.A. Zygmund, Trigonometric Series, Vol. 1, Monograph. Matem., Warsaw (1935).Google Scholar
- 3.G. Alexits, “A Fourier-sor Cesaro-közepeivel valo approximacio nagysagrendjéröl,” Math. Fiz. Lapok,48, 410–422 (1941).Google Scholar
- 4.G. Alexits, “Sur l'ordre de grandeur de l'approximation d'une fonction periodique par les sommes de Fejér,” Acta Math. Acad. Sci. Hung.,3, 29–40, 41–42 (1952).Google Scholar
- 5.S. B. Stechkin, “Estimate of remainder of Taylor series for certain classes of analytic functions,” Izv. Akad. Nauk SSSR, Ser. Matem.,17, 461–472 (1953).Google Scholar
- 6.A. I. Rubinshtein,” Onω-lacunary series and functions of the classes Hω,n,” Matem. Sb.,65 (107), No. 2, 239–271 (1964).Google Scholar
- 7.A. Zygmund, Trigonometric Series, Vol. 2, Monograph. Matem., Warsaw (1935).Google Scholar
- 8.A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).Google Scholar
- 9.E. M. Stein, “On limits of sequences of operators,” Ann. Math.,74, No. 1, 140–170 (1961).Google Scholar
Copyright information
© Plenum Publishing Corporation 1976