Abstract
The rate of best polynomial approximation of an analytic function on a compact Faber set K is characterized in terms of the rate of growth of its Faber coefficients and compared with the rate of approximation by the partial sums of the Faber series. Also the convergence of sequences of interpolating polynomials constructed for various systems of nodes is studied by considering the growth of the interpolated function. Under appropriate assumptions on K the approximation by interpolating polynomials can be incorporated in the characterization theorem. Emphasis is laid on high precision in describing the rate of approximation and on admitting a large class of functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.M. Anderson and J. Clunie, Isomorphisms of the disc algebra and inverse Faber sets, Math. Z. 188 (1985), 545–558.
J.-E. Andersson, On the degree of polynomial and rational approximation of holomorphic functions, Dissertation, University of Göteborg, Göteborg 1975.
A.V. Batyrev, On the question of best approximation of analytic functions by polynomials, (Russian) Dokl. Akad. Nauk SSSR 76 (1951), 173–175.
S. Bernstein, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytique d’une variable réele, Gauthier-Villars, Paris 1926.
F. Beuermann, Wachstumsordnung, Koeffizientenwachstum und Null stellendichte bei Potenzreihen mit endlichem Konvergenzkreis, Math. Z. 33 (1931), 98–108.
J.H. Curtiss, Riemann sums and the fundamental polynomials of Lagrange interpolation, Duke Math. J. 8 (1941), 525–532.
E.M. Dyn’kin, Uniform approximation of functions in Jordan domains, Siberian Math. J. 7 (1977), 548–557.
E.M. Dyn’kin, The uniform polynomial approximation in a complex domain, J. Soviet Math. 9 (1978), 269–271.
G. Faber, über Tschebyscheffsche Polynome, J. Reine Angew. Math. 150 (1919), 79–106.
E. Fabry, Ordre des points singuliers de la série de Taylor, Acta Math. 36 (1913), 69–104.
M. Freund, On the rate of rational approximation to functions with a finite number of poles, to appear.
M. Freund und E. Görlich, Polynomial approximation of an entire function and rate of growth of Taylor coefficients, Proc. Edinburgh Math. Soc. 28 (1985), 341–348.
M. Freund und E. Görlich, The rate of best approximation for entire functions, Analysis 5 (1985), 209–221.
D. Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser, Basel 1980.
J. Hadamard, Essai sur 1’étude des fonctions données par leur développement de Taylor, J. Math. Pures Appl. (4)8 (1892), 101–186.
O.P. Juneja and G.P. Kapoor, Analytic Functions — Growth Aspects, Pitman Adv. Publ. Comp., Boston 1985.
G.P. Kapoor and K. Gopal, On the coefficients of functions analytic in the unit disc having fast rates of growth, Ann. Math. 23 (1978), 337–349.
G.P. Kapoor and A. Nautiyal, On the coefficients of a function analytic in the unit disc having slow rate of growth, Ann. Mat. Pura Appl. (IV) 131 (1982), 281–290.
T. Kövari and C. Pommerenke, On Faber polynomials and Faber expansions, Math. Z. 99 (1967), 193–206.
M.E.Le Roy, Valeurs asymptotiques de certaines séries procedant suivant les puissances entières et positives d’une variable réelle, Bull. Sci. Math. (2) 24 (1900), 245–268.
A.I. Markushevich, Theory of Functions of a Complex Variable, Vol. I-III, Printice-Hall, Inc., Englewood Cliffs, New York 1967.
S.N. Mergelyan, Uniform approximation of functions of a complex variable, Translations Amer. Math. Soc. 3 (1962), 294–391.
J.R. Rice, The degree of convergence for entire functions, Duke Math. J. 38 (1971), 429–440.
S.R.H. Rizvi, On the degree of approximation of analytic functions, Ganita 30 (1979), 12–19.
E.B. Saff, Polynomials of interpolation and approximation to meromorphic functions, Trans. Amer. Math. Soc. 143 (1969), 509–522.
W.E. Sewell, Generalized derivatives and approximation by polynomials, Trans. Amer. Math. Soc. 41 (1937), 84–123.
S.M. Shah, Polynomial approximation of an entire function and generalized orders, J. Approx. Theory 19 (1977), 315–324.
V.l. Smirnov and N.A. Lebedev, Functions of a Complex Variable, M.I.T. Press, Cambridge, Mass. 1968.
P.K. Suetin, Fundamental properties of Faber polynomials, Russian Math. Surveys 19(4) (1964), 121–149.
R.S. Varga, On an extension of a result of S.N. Bernstein, J. Approx. Theory 1 (1968), 176–179.
J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 2nd ed., Amer. Math. Soc. Colloq. Publ., Vol. XX, Providence 1956.
J.L. Walsh, The convergence of sequences of rational functions of best approximation with some free poles, in: Approximation of Functions, pp. 1–16, Ed. H.L. Garabedian, Elsevier Publ. Comp., Amsterdam 1965.
J.L. Walsh and W.E. Sewell, Sufficient conditions for various degrees of approximation by polynomials, Duke Math. J. 6 (1940), 658–705.
T. Winiarski, Approximation and interpolation of entire functions, Ann. Polon. Math. 23 (1970), 259–273.
Author information
Authors and Affiliations
Additional information
The author was supported by the Deutsche Forschungsgemeinschaft under grant No. 251/7-1.
Rights and permissions
About this article
Cite this article
Freund, M. The Degree Of Polynomial Approximation And Interpolation Of Analytic Functions. Results. Math. 13, 81–98 (1988). https://doi.org/10.1007/BF03323397
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03323397