Inverse theorem for approximation on subsets of a domain with cups
K. А. Sintsova National Research University Higher School of Economics, St. Petersburg School of Economics and Management
Abstract:
Let
$\mathfrak{P}(z)$ be a doubly periodic Weierstrass function with periods
$2\boldsymbol{\omega}_1$,
$2\boldsymbol{\omega}_2$, and let
$ Q$ be the parallelogramm of periods,
$Q = \{z \in \mathbb{C}$ $: z = 2\alpha_1\boldsymbol{\omega}_1 + 2\alpha_2\boldsymbol{\omega}_2, \alpha_1, \alpha_2 \in [0,1)\}$. We consider a simply connected domain
$D, \overline{D} \subset Q$, such that its boundary
$\partial D$ contains cusps, and a function
$f$ that is analytic in
$D$ and continuous on
$\partial D$. We assume that the modulus of continuity
$\omega(t)$ satisfyes the relation
$$ \int\limits_0^x \frac{\omega(t)}{t} dt + x \int\limits_x^\infty \frac{\omega(t)}{t^2} dt \leq c\omega(x).
$$
Let a function
$\Phi$ map conformally the domain
$\mathbb{C} \setminus D$ onto
$\mathbb{C} \setminus \mathbb{D}$ with the normalization $\Phi(\infty) = \infty, \Phi^{\prime}(\infty) > 0$. We put $L_{1+t} = \{z \in \mathbb{C} \setminus D: |\Phi(z)| = 1+t\}, \delta_n(z) = \mathrm{dist} (z, L_{1+\frac{1}{n}}), z \in \partial D$. The main result of the paper is the following statement.
Theorem 1.
Assume that there exists a sequence of polynomials $P_n(u, v)$,
$\deg P_n \leq n$,
such that $$ |f(z) - P_n(\mathfrak{P}(z), \mathfrak{P}^{\prime}(z))| \leq C \delta^{r}_n(z)\omega(\delta_{n}(z)), z \in \partial D. $$
$C$ is independent on $n$ and $z$.
Then $f \in H^{r+\omega}(D)$.
Key words and phrases:
analytic functions, approximation, Weierstrass doubly periodic function.
UDC:
517.537 Received: 11.07.2023