Tangent approximation by solutions of the convolution equation

V. V. Volchkov ; Vit. V. Volchkov

Problemy analiza, Tome 11 (2022) no. 3, pp. 125-142.

Résumé

The article for the first time studies the approximation of a function together with its derivatives on the real line by solutions of a multidimensional convolution equation of the form $$g\ast T=0,$$ where $T$ is a given compactly supported radial distribution other than a constant multiplied by the Dirac delta function at zero. The following analogue of the well-known Carleman's theorem on tangent approximation by entire functions is obtained: if $f\in C^k(\mathbb{R})$ for some $k\in\mathbb{Z}_+$ and for any $\nu\in\{0, \ldots, k\}$ there is a finite limit \begin{equation*} \displaystyle \underset{t \to +0 }\lim \left(\frac{d}{dt}\right)^{\nu}f(\mathrm{ln}t), \end{equation*} then for an arbitrary positive continuous function $\varepsilon(t)$ on $\mathbb{R}$ there exists an entire function $\Phi\colon\mathbb{C}^n\to \mathbb{C}$, such that $\Phi\big |_{\mathbb{R}^n}\ast T=0$ and \begin{equation*} \left|f^{(\nu)}(t)-\left(\frac{\partial}{\partial x_1}\right)^{\nu}\Phi(t, 0, \ldots, 0)\right|\varepsilon(e^t) \end{equation*} for all $t\in\mathbb{R}$, $\nu\in\{0, \ldots, k\}$ (Theorem 1). It is shown that when approximating a function on subsets of the real line that do not contain a neighborhood of the point $-\infty$, the condition for the existence of a limit in Theorem 1 can be omitted. In addition, the method of proving Theorem 1 allows one to obtain new results that are of interest for the theory of convolution equations of the indicated type. These are results about the growth of solutions (Corollary 3), on the distribution of values (Theorem 2), and also on the solvability of the interpolation problem for solutions of the convolution equation (Corollary 4).

Détail

Keywords: mean periodicity, Carleman's approximation theorem.
Mots-clés : convolution equation

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