Integrated continuity conditions and degree of approximation by polynomials or by bounded analytic functions

Integrated Continuity Conditions and Degree of Approximation by Polynomials or by Bounded Analytic Functions^)

Introduction. In the study of uniform approximation to a function of a complex variable by polynomials or by bounded analytic functions, Lipschitz conditions have proved extremely useful in relating degree of approximation to continuity properties of the functions approximated. Parts of this theory are analogous to the older study (S. Bernstein, D. Jackson, de la Vallee Poussin, Montel) of appr...

Approximation of Analytic Functions by Chebyshev Functions

and Applied Analysis 3 where we refer to 1.4 for the am’s and we follow the convention ∏m−1 j m · · · 1. We can easily check that cm’s satisfy the following relation: m 2 m 1 cm 2 − ( m2 − n2 ) cm am 2.2 for any m ∈ {0, 1, 2, . . .}. Theorem 2.1. Assume that n is a positive integer and the radius of convergence of the power series ∑∞ m 0 amx m is ρ > 0. Let ρ0 min{1, ρ}. Then, every solution y ...

Approximation of Analytic Functions by Special Functions

We survey the recent results concerning the applications of power series method to the study of Hyers–Ulam stability of differential equations.

Approximation of Analytic Functions by Kummer Functions

Approximation by Bounded Analytic Functions: Uniform Convergence as Implied by Mean Convergence^) By

In three recent notes [1], [2], [3] I have discussed uniform convergence by polynomials (in the complex variable) to a given function as a consequence of convergence in the mean of those polynomials to the given function, and also convergence in the mean of one order as a consequence of convergence in the mean of a lower order. The present note contains analogs of those results, but now for app...