Multiplicative polynomial approximation

A. N. Medvedev^a, N. A. Shirokov<sup>b

a</sup> St. Petersburg Electrotechnical University
^b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: Let $\mathcal{D}$ be a bounded domain on the complex plain $\mathbb{C}$ with sufficiently smooth boundary. We denote by $\Lambda^\alpha(\overline{\mathcal{D}})$, $0<\alpha<1$, the class of analytic functions in $\mathcal{D}$ satisfying the $\alpha$-Hölder condition in $\overline{\mathcal{D}}$. Each function $f\in\Lambda^\alpha(\overline{\mathcal{D}})$ can be factored as $f=FI$ with $F$ an outer function defined in terms of the boundary values of $|f|$, and with $I$ an inner function such that $|I(z)|=1$ a.e. in $\partial\mathcal{D}$.
The following result was obtained.
Theorem. Let $f\in\Lambda^\alpha(\overline{\mathcal{D}})$, $f=F\cdot I$. Then for every $n\in\mathbb{N}$ there exist polynomials $P_n$, $q_n$ of degrees at most $n$ such that the following properties hold.
There exist constants $c_{f,1}$ и $c_{f,2}$ such that for any $z\in\partial\mathcal{D}$
$$ |f(z)-P_n(z)q_n(z)|\leq c_{f,1} \cdot n^{-\alpha},~ |F(z)-P_n(z)|\leq c_{f,2}\cdot n^{-\alpha}. $$
There is constant $c_{\mathcal{D}}$ such that for any $z\in\mathcal{D}$
$$ |q_n(z)|\leq c_{\mathcal{D}}. $$
Finally, for any $z\in\mathcal{D}$
$$ q_n(z) \xrightarrow[n \to \infty]{} I(z). $$


Key words and phrases: polynomials, approximation, Hölder classes, domains with smooth boundary.

UDC: 517.537

Received: 20.06.2025