EQUILIBRIUM DISTRIBUTIONS AND DEGREE OF RATIONAL APPROXIMATION OF ANALYTIC FUNCTIONS

A theorem is proved on the degree of rational approximation of sequences of analytic functions given by Cauchy-type integrals of the form The theorem is formulated in terms connected with the equilibrium distribution of the charge on the plates of a capacitor under the assumption that an external field acts on the plate , and this plate satisfies a certain symmetry condition in the field . The theorem is used to solve the problem of the degree of rational approximation of the function on .Bibliography: 44 titles.

[1]  L. Trefethen,et al.  The Carathéodory–Fejér Method for Real Rational Approximation , 1983 .

[2]  J. Andersson.  Approximation of e−x by rational functions with concentrated negative poles , 1981 .

[3]  Thomas Bagby.  On interpolation by rational functions , 1969 .

[4]  R. Varga.  Functional Analysis and Approximation Theory in Numerical Analysis , 1987 .

[5]  G. V. Kuzʹmina.  Moduli of families of curves and quadratic differentials , 1982 .

[6]  A. Gončar,et al.  ON MARKOV'S THEOREM FOR MULTIPOINT PADÉ APPROXIMANTS , 1978 .

[7]  R. Varga,et al.  Chebyshev rational approximations to e−x in [0, +∞) and applications to heat-conduction problems , 1969 .

[8]  Herbert Stahl,et al.  Orthogonal polynomials with complex-valued weight function, I , 1986 .

[9]  K. Scherer,et al.  On the rational approximation ofe−x on [0, ∞) , 1985 .

[10]  A. Gončar,et al.  Zolotarev Problems Connected with Rational Functions , 1969 .

[11]  R. Varga,et al.  Extended numerical computations on the “1/9” conjecture in rational approximation theory , 1984 .

[12]  J. Walsh.  Interpolation and Approximation by Rational Functions in the Complex Domain , 1935 .

[13]  E. Rakhmanov,et al.  ON ASYMPTOTIC PROPERTIES OF POLYNOMIALS ORTHOGONAL ON THE REAL AXIS , 1984 .

[14]  Hrushikesh Narhar Mhaskar,et al.  Where does the sup norm of a weighted polynomial live? , 1985 .

[15]  O G Parfenov,et al.  ESTIMATES OF THE SINGULAR NUMBERS OF THE CARLESON IMBEDDING OPERATOR , 1988 .

[16]  E. Rakhmanov,et al.  EQUILIBRIUM MEASURE AND THE DISTRIBUTION OF ZEROS OF EXTREMAL POLYNOMIALS , 1986 .

[17]  Richard S. Varga.  Topics in polynomial and rational interpolation and approximation , 1982 .

[18]  Arnold Schönhage.  Zur rationalen approximierbarkeit von e−xüber [0, ∞) , 1973 .

[19]  Herbert Stahl,et al.  The structure of extremal domains associated with an analytic function , 1985 .

[20]  V. Peller,et al.  Hankel operators, best approximations, and stationary Gaussian processes , 1982 .

[21]  A. Gončar.  THE RATE OF RATIONAL APPROXIMATION AND THE PROPERTY OF SINGLE-VALUEDNESS OF AN ANALYTIC FUNCTION IN THE NEIGHBORHOOD OF AN ISOLATED SINGULAR POINT , 1974 .

[22]  M. Kreĭn,et al.  ANALYTIC PROPERTIES OF SCHMIDT PAIRS FOR A HANKEL OPERATOR AND THE GENERALIZED SCHUR-TAKAGI PROBLEM , 1971 .

[23]  V. Peller.  HANKEL OPERATORS OF CLASS $ \mathfrak{S}_p$ AND THEIR APPLICATIONS (RATIONAL APPROXIMATION, GAUSSIAN PROCESSES, THE PROBLEM OF MAJORIZING OPERATORS) , 1982 .

[24]  R. Varga,et al.  SOME OPEN PROBLEMS CONCERNING POLYNOMIALS AND RATIONAL FUNCTIONS , 1977 .

[25]  E. Rakhmanov,et al.  On the equilibrium problem for vector potentials , 1985 .

[26]  J Nuttall,et al.  Asymptotics of diagonal Hermite-Padé polynomials , 1984 .