Abstract
Concerning the approximation of periodic functions by means of positive singular integrals, the problem of determining the best asymptotic or Nikolskiĭ constant for the measure of approximation with respect to Lipschitz classes was solved in general provided the corresponding kernel is of Fejér’s type. But there is still a number of important examples to which this theory does not apply. It is shown that for kernels of perturbed Fejér-type the associate Nikolskiĭ constant may be evaluated by a simple comparison theorem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Anghelutza, Une remarque sur lintégrale de Poisson. Bull. Sci. Math. Bibliothèque École Hautes Études = Darboux Bull. (2) 48 (1924), 138–140; FdM 50 (1924), 330.
V. A. Baskakov, The degree of approximation of differentiale functions by certain positive operators (Russ.). Mat. Sb. (N. S.) 76 (118) (1968), 344–361; MR 33#4488.
L. I. Bausov, The order of approximation of functions of class Z a by positive linear polynomial operators (Russ.). Uspehi Mat. Nauk 7, no. 1 (103) (1962), 149–155; MR27#1756.
L. I. Bausov, On the order of approximation of functions of class Z a by linear positive operators (Russ.). Mat. Zametki 4 (1968), 201–210=Transi. Math. Notes 4 (1968), 612-617.
H. Bohman, Approximate Fourier analysis of distribution functions. Ark. Mat. 4 (1960), 99–157; MR 23#A 3963.
P. L. Butzer und E. Görlich, Saturationsklassen und asymptotische Eigenschaften trigonometrischer singulärer Integrale. In: Festschrift zur Gedächtnisfeier für Karl Weierstrass 1815-1965 (Ed. H. Behnke-K. Kopfermann, Wiss. Abh. Arbeitsgemeinschaft für Forschung des Landes Nordrhein—Westfalen 33) Westdeutsch. Verl., Köln—Oplande 1966, 612 pp.; 339 392; MR33#4555.
P. L. Butzer und R. J. Nessel, Fourier Analysis and Approximation. Vol. 1. One-Dimensional Theory. Birkhäuser Verl., Basel—Stuttgart and Academic Press, New York-London 1971, xiv+553 pp.
P. L. Butzer und E. L. Stark, Wesentliche asymptotische Entwicklungen für Approximationsmaße trigonometrischer singulärer Integrale. Math. Nachr. 39 (1969), 223–237; MR 40 #3139.
R. DeVore, Approximation of Continuous Functions by Positive Linear Operators. Preprint (1971).
G. A. Fomin, On the best approximation of functions of classes Z 2 and Z1 by certain linear operators (Russ.). In: Studies of Contemporary Problems of Constructive Theory of Functions (Russ.). (Proc. Second Ail-Union Conf., Baku, 1962, Ed. 1.1. Ibragimov) Izdat. Akad. Nauk Azerbaĭdžan. SSR, Baku 1965, 638 pp., 207–211; MR 33#6243.
M. Ghermanesco, Sur l’intégrale de Poisson. Sur l’intégrale de Poisson (suite). Bull. Sei. École Polytechnique de Timisoara 4,fasc. 3–4 (1932), 159-184, 5, fasc. 1-2 (1933), 41-74; FdM 58 (1932), 1068.
E. Görlich, Über optimale Approximationsprozesse. In: Constructive Function Theory (Proc. Int. Conf., Golden Sands, Varna, 1970, Ed. B. Penkov-D. Vačov) Izdat. Bolg. Akad. Nauk 1972, 363 pp., 187–191.
E. Görlich and E. L. Stark, A unified approach to three problems on approximation by positive linear operators. In: Proceedings of the Conference on Constructive Theory of Functions (Approximation Theory) (Budapest, 1969, Ed. G. Alexits-S. B. Stechkin) Akadémiai Kiadô, Budapest 1972, 538 pp; 201–208.
E. Görlich und E. L. Stark, Über beste Konstanten und asymptotische Entwicklungen positiver Faltungsintegrale und deren Zusammenhang mit dem Saturationsproblem. Jber. Deutsch. Math.-Verein. 72 (1970), 18–61.
B. L. Golinskiȝ, Approximation on the entire number axis of two functions which are conjugate in the sense of Riesz by integral operators of singular type (Russ.). Mat. Sb. (N. S.) 66(108) (1965), 3–34; MR 30 # 2280.
P. P. Korovkin, An asymptotic property of positive methods of summation of Fourier series and best approximation of functions of class Z2 by linear positive polynomial operators (Russ.). Uspehi Mat. Nauk 13, no. 6 (84) (1958), 99–103; MR 21 #253.
P. P. Korovkin, Linear Operators and Approximation (Russ.). Gos. Izdat. Fiz.-Mat. Lit., Moscow 1959, 211 pp. (=Transi. Hindustan Publ. Corp., Delhi 1960, vii + 222 pp.); MR 27#561.
P. P. Korovkin, Asymptotic properties of positive methods of summation of Fourier series (Russ.). Uspehi Mat. Nauk 5, no. 1 (91) (1960), 207-217; MR 22#6975.
Y. Matsuoka, On the degree of approximation of functions by some positive linear operators. Sci. Rep. Kagoshima Univ. 9 (1960), 11–16; MR 23#A 1189.
I. P. Natanson, On the accuracy of representation of continuous periodic functions by singular integrals (Russ.). Dokl. Akad. Nauk SSSR 73 (1950), 273–276; MR 2, 94.
R. J. Nessel, Über Nikolskii-Konstanten von positiven Approximationsverfahren bezüglich Lip schitz-Klassen. Jber. Deutsch. Math.-Verein. 73 (1971), 6–47.
R. J. Nessel, Nikolskiĭ constants of positive operators for Lipschitz classes. In: Constructive Function Theory (Proc. Int. Conf., Golden Sands/Varna, 1970, Ed. B. Penkov-D. Vačov) Izdat. Bolg. Akad. Nauk 1972, 363 pp., 239–244.
S. M. Nikolskiĭ, Sur l’allure asympto tique du reste dans l’approximation au moyen des sommes de Fejér des fonctions vérifiant la condition de Lipschitz (Russ.; French sum.) Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 501–508; MR 2, 279.
I. M. Petrov, Order of approximation of functions belonging to the class Z by some polynomial operators (Russ.). Uspehi Mat. Nauk 3, no. 6 (84), (1958), 127–131; MR 21 #732.
I. M. Petrov, The order of approximation of functions of class Z a by certain polynomial operators (Russ.). Izv. Vyss. Ucebn. Zaved. Matematika 1960, no. 1 (14) (1960), 188–193; MR 24# A 967.
I. M. Petrov, On the order of approximation of class Z 1 by positive linear polynomials (Russ.). Uspehi Mat. Nauk 19, no. 2 (116) (1964), 151–154; MR 29#413.
F. Schurer, Some remarks on the approximation of functions by some positive linear operators. Monatsh. Math. 67 (1963), 353–358; MR 28 #411.
F. Schurer, On linear positive operators in approximation theory (Dutch sum.). Thesis, Technische Hogeschool te Delft, 1965; Uitgeverij Waltman, Delft 1965, iv+79 pp.; MR 34#6389.
F. Schurer and F. W. Steutel, On linear positive operators of the Jackson type. Math. Communication, Technological Univ. Twente 1 (1966) 45 pp.
F. Schurer and F. W. Steutel, On linear positive operators of the Jackson type. Mathematica (Cluj) 9 (32) (1967), 155–184; MR 36#583.
E. L. Stark, Über einige Konstanten der singular en Integrale von Dirichlet, Rogosinski, Fejér und Fejér—Korovkin. Diplomarbeit, Rheinisch-Westfälische Technische Hochschule Aachen, 1966, viii + 105 pp.
E. L. Stark, Über die Approximationsmaße spezieller singulärer Integrale (Engl. sum.). Computing 4 (1969), 153–159; MR 39#7336.
E. L. Stark, On a generalization of Abel—Poisson’s singular integral having kernels of finite oscillation. Studia Sci. Math. Hungar. (to appear).
R. Taberski, Some properties of (K, ϕ)-summability. Bull. Acad. Polon. Sci. Sér. Sei. Math. Astronom. Phys. 9 (1961), 659–666; MR 24#3446.
R. Taberski, More about (K, ϕ)-summability. Bull. Acad. Polon. Sci. Sér. Sei. Math. Astronom. Phys. 9 (1961), 769–774; MR 24#3447.
C. de La Vallée Poussin, Leçons sur L’Approximation des Fonctions d’une Variable Réelle, Gauthier-Villars Ed., Paris 1919, 1952 II, vi + 151 pp.
Zheng Wei-xing, On the extreme property of the operator βσ(f,x) (Chin.). Acta Math. Sinica 5 (1965), 54–62=Transi. Chinese Math. 8 (1965), 353-362; MR 33#473.
Review in Referativnyĭ Žurnal, Matematika 1960, no. 266: Cheng Wee-shing, Asymptotic formula for the approximation of class Z* by polynomials (Chin., Eng. Sum.). Nanjing Daxue Xuebao, Ziran Kexue=Acta Univ. Nankin. Sci. Nat. 1959, no. 3 (1959), 1–6.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1972 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Stark, E.L. (1972). Nikolskiĭ Constants for Positive Singular Integrals of Perturbed Fejér-Type. In: Butzer, P.L., Kahane, JP., Szökefalvi-Nagy, B. (eds) Linear Operators and Approximation / Lineare Operatoren und Approximation. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 20. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7283-6_31
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7283-6_31
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7285-0
Online ISBN: 978-3-0348-7283-6
eBook Packages: Springer Book Archive