2017
Том 69
№ 9

# Dilnyi V. M.

Articles: 5
Brief Communications (Ukrainian)

### On Invariant Subspaces in Weighted Hardy Spaces

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 853–857

We consider the description of translation invariant subspaces of a weighted Hardy space in the half plane. The obtained result includes the Beurling–Lax theorem for the Hardy space as a special case. We discuss the problem of generalization of the definition of inner function.

Article (Ukrainian)

### On convolution of functions in angular domains

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1155-1164

We obtain analogs of the Parseval theorem, convolution theorem, and some other properties of the convolution of functions from the Hardy – Smirnov spaces in an arbitrary convex unbounded polygon.

Brief Communications (Ukrainian)

### Equivalent definition of some weighted Hardy spaces

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1270–1274

We present the equivalent definition for spaces of functions analytic in the half-plane ${\mathbb C}_+ = \{z: Re z > 0 \}$ for which $$\sup_{|\varphi| < \frac{\pi}2} \left\{\int\limits_0^{+\infty}\left| f(r e^{i\varphi})\right|^p e^{-p\sigma r|\sin \varphi|} dr \right\} < +\infty.$$

Brief Communications (Ukrainian)

### On the equivalence of some conditions for weighted Hardy spaces

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1257–1263

Let $G ∈ H_{σ}^p (ℂ+)$, where $H_{σ}^p (ℂ+)$ is the class of functions analytic in the half plane ℂ+ = {z: Re z > 0} and such that $$\mathop {\sup }\limits_{\left| \varphi \right| < \tfrac{\pi }{2}} \left\{ {\int\limits_0^{ + \infty } {\left| {G(re^{i\varphi } )} \right|^p e^{ - p\sigma r\left| {sin\varphi } \right|} dr} } \right\} < + \infty .$$ In the case where a singular boundary function $G$ is identically constant and $G(z) ≠ 0$ for all $z ∈ ℂ_{+}$, we establish conditions equivalent to the condition $G(z)\exp \left\{ {\frac{{2\sigma }}{\pi }zlnz - cz} \right\} \notin H^p (\mathbb{C}_+ )$, where $H^p (ℂ_{+})$ is the Hardy space, in terms of the behavior of $G$ on the real semiaxis and on the imaginary axis.

Brief Communications (Ukrainian)

### On Completeness of a System of Functions in an Angular Domain

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 255-257

We show that the system {e −λz /(1 + z 2) : λ > 0} is complete in a class of functions analytic in an angle.