Approximation in the Mean by Analytic Functions

Best Approximation in the Mean by Analytic and Harmonic Functions

For n ≥ 2, let Bn denote the unit ball in R, and for p ≥ 1 let L denote the Banach space of p-summable functions on Bn. Let L p h(Bn) denote the subspace of harmonic functions on Bn that are p-summable. When n = 2, we often write D instead of B2, and we let A denote the Bergman space of analytic functions in L. Let ω be a function in L. We are interested in finding the best approximation to ω i...

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...