Best approximation by holomorphic functions. Application to the best polynomial approximation of classes of holomorphic functions

We establish necessary and sufficient conditions under which a real-valued function from , 1 ≤ p < ∞, is badly approximable by the Hardy subspace H _p ⁰ := {ƒ ∈ H _p : ƒ(0) = 0}. In a number of cases, we obtain the exact values of the best approximations in the mean of functions holomorphic in the unit disk by functions holomorphic outside this disk. We use the obtained results for finding the exact values of the best polynomial approximations and n-widths of some classes of holomorphic functions. We establish necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1047–1067, August, 2007.     

Rational functions of best L p- approximation and analytic continuability

Sufficient conditions for the analytic (resp. meromorphic) continuability of a function f ∈ L _p([?1, l]), p > 0, in terms of rational functions of best weighted L _p- approximation with an unbounded number of finite poles are established.     

Rational interpolation and best rational approximation

A relation between the degree of convergence (in capacity) of Padé approximants and the degree of best rational approximation is derived for functions in Gon?ar's class R₀. The results contain previous convergence results for Padé approximants proved by Nuttall, Pommerenke, and Gammel and Nuttall.     

Some bounds on best uniform linear approximation

Numerische Mathematik -     

Rational approximation of decay-type functions

Numerically given functions with exponential asymptotic behavior are approximated by rational expressions. The numerators are obtained from the nodes of the functions, and the denominators are powers of polynomials, whose coefficients are obtained by expansion in orthogonal polynomials. Different weights are discussed for uniform absolute or relative errors.An iterative procedure is further described to improve the least-squares approximation to one of Chebyshev-type best fit.The research reported in this paper was sponsored in part by the King Gustaf VI Adolf's 70-Years Fund for Swedish Culture, Knut and Alice Wallenberg's Foundation, The Swedish Natural Science Research Council, The Swedish Technical Research Council, and in part by the Aeronautical Research Laboratory, OAR, through the European Office, Aerospace Research, United States Air Force.Part of the work on this paper was carried out while the author was at the Physics Department, University of Florida, Gainesville, Florida, U.S.A.     

Rational approximation of empirical functions

A method is developed for finding the best rational approximationp(x)/q(x) in the least-squares sense of a functionf(x) known at discrete points of an interval. Especially for functions with a steep tangent the method yields far better results than polynomial approximation. The scheme can easily be extended to functions known at all points of an interval.Part of this work was carried out at SAAB, Linköping. The present form was developed at the Department of Numerical Analysis, Lund.     

Rational approximation of analytic functions

The paper provides an overview of the author’s contribution to the theory of constructive rational approximations of analytic functions. The results presented are related to the convergence theory of Padé approximants and of more general rational interpolation processes, which significantly expand the classical theory’s framework of continuous fractions, to inverse problems in the theory of Padé approximants, to the application of multipoint Padé approximants (solutions of Cauchy-Jacobi interpolation problem) in explorations connected with the rate of Chebyshev rational approximation of analytic functions and to the asymptotic properties of Padé-Hermite approximation for systems of Markov type functions.     

On the best uniform approximation of differentiation operators

We study the existence and uniqueness of an extremal operator for the problem of S. B. Stechkin on the best approximation of differentiation operators by bounded operators.Translated from Matematicheski Zametki, Vol. 5, No. 3, pp. 273–284, March, 1969.     

On best approximation by rational and holomorphic mappings between Banach spaces

After proving a generalized version of Garkavi's theorem, we give as applications proofs of existence results on best approximation by polynomials, and fractional linear and holomorphic operators between Banach spaces. We also obtain theorems on best approximation by some types of rational functions defined in open subsets of Banach spaces. By considering a natural non-normable distance we prove that every mapping bounded on the bounded subsets of a Banach space has best approximation by polynomials of degree less than or equal to a fixed natural number n.     

Rational approximation to harmonic functions

Padé-type approximation is the rational function analogue of Taylor’s polynomial approximation to a power series. A general method for obtaining Padé-type approximants to Fourier series expansions of harmonic functions is defined. This method is based on the Newton-Cotes and Gauss quadrature formulas. Several concrete examples are given and the convergence behavior of a sequence of such approximants is studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some uniform approximation theorems of yk-analytical functions

The paper examines some issues of uniform approximation of y^k -analytical functions infinite simply connected regions in the upper halfplane y >0 whose boundary includes a section of the real axis. Polozhii's integral representation of y^k -analytical functions [3] makes it possible to extend a number of theorems of uniform approximation of analytical functions by polynomials to the class of y^k-analytical functions.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 3–12, 1985.     

On best approximation of infinitely differentiable functions

Dnepropetrovsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 5, pp. 12–28, September–October, 1991.