Abstract
We discuss the definition and effectiveness of a Padé-type approximation to 2π-periodic finite Baire measures on [-π,π]. In the first two sections we recall the definitions and basic properties of the Padré-type approximants to harmonic functions in the unit disk and to L p -functions on the unit circle. Section 3 deals with the extension of these definitions and properties to a finite 2π-periodic Baire measure. Finally, section 4 is devoted to a study of the convergence of a sequence of such approximants, in the weak star topology of measures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bieberbach, L., Lehrbuch der Funktiontheorie (two volumes), Leipzig, 1927.
Brezinski, C., Accélération de la Convergence en Analyse Numérique, Editions du Laboratoire de Calcul de USTL, Lille, 1973.
Brezinski, C., Padé-type Approximants for Double Power Series, Journal of the Indian Math. Soc., 42(1978), 267-282.
Brezinski, C., Rational Approximation to Formal Power Series, J. Approx. Theory, 25 (1979), 295-317.
Brezinski, C., Padé-type Approximation and General Orthogonal Polynomials, International Series in Numerical Mathematics, Birkhāuser, Basel, 1980.
Brezinski, C., Outlines of Padé Approximation, in Computational Aspects of Complex Analysis, H. Wermer and al. (eds.), D. Reidel Publishing Company, 1981,1-50.
Brezinski, C., Padé Approximants: old and new, Jahrbruch Uberblicke Mathematik, 1983, 37-63.
Brezinski, C., Partial Padé Approximation, J. Approx. Theory, 54(1988), 210-233.
Courant, R., Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience, New York, 1950.
Daras, N. J., Domaine de Convergence d'une Transformation de la Suite des Sommes Partielles d'une Fonction Holomorphe et Applications aux Approximants de tye Padé, Nouvelle Thèse, Université de Lille Flandres-Artois, 1988.
Daras, N. J., The Convergence of Padé-Type Approximants to Holomorphic Functions of Several Complex Variables, Appl. Num. Math., 6(1989/90), 341-360.
Daras, N. J., Two Counter-Examples for the Okada Theorem in C m, Riv. Mat. Univ. Parma, 5(1993), No. 2, 267-278.
Daras, N. J., Continuity of Distributions and Global Convergence of Padé-Type Approximants in Runge Domains, Indian J. Pure Appl. Math., 26(1995), No. 2, 121-130.
Daras, N. J., Summability Transforms of Taylor Series, Math. Jap. 43(1996), No. 2, 497-503.
Daras, N. J., Approximate Quadrature Formulas for Harmonic Functions, submitted.
Daras, N. J., On the Numerical Evaluation of a 2π-Periodic L p-Function by a few of its Fourier Coefficients, submitted.
Daras, N. J., Integral Representations for Padé-Type Operators, submitted.
Daras, N. J., Generalized Padé-Type Approximants and Integral Representations, in preparation.
Daras, N. J., Padé-Type Approximate Integral Equations, in preparation.
Eiermann, M., On the Convergence of Padé-Type Approximants to Analytic Functions, J. Comput, Appl. Math., 10(1984), 219-227.
Eiermann, M. and Niethammer, W., Interpolation Methods for Numerical Analytic Continuation, in International Series in Numerical Mathematics 61(Birkhauser, Basel, 1982), 131-141.
Gawronski, W. and Trautner, R., Verschärfung Eines Satzer von Borel-Okada über Summierbarkeit von Potenzreihen, Period. Math. Hungar, 7(1976), 201-211.
Hoffman, K., Banach Spaces of Analytic Functions, Dover Publications, New York, 1962.
Zygmund, A., Trigonometric Series, 2nd ed., Cambridge University Press, 1959.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Daras, N.J. Interpolation Methods for the Evaluation of 2π-Periodic Finite Baire Measure. Analysis in Theory and Applications 17, 1–27 (2001). https://doi.org/10.1023/A:1015508525910
Issue Date:
DOI: https://doi.org/10.1023/A:1015508525910