Nikol'skii type inequality; Bessel weight; extremal functions; extremal constants
For $\alpha>-1$ and $1\le q<\infty$, denote by $L_{\alpha}^q$ the space of complex valued measurable functions defined on $[0,\infty)$ for which $\Vert f\Vert_{q,\alpha}:=\left(\int_0^{\infty}\vert f(x)\vert x^{2\alpha+1}dx\right)^{\frac1q}<\infty$. For $\sigma>0$, denote by $\mathcal{E}(\sigma,q,\alpha)$, the set of even entire functions of exponential type at most $\sigma$ which belong to $L_{\alpha}^q$. Consider also the sup norm $\Vert f\Vert_C:=\sup\{\vert f(x)\vert ,\;x\in[0,\infty)\}$. The paper deals with the study of the optimal constants and the corresponding extremal functions which can appear in the following two inequalities:\begin{itemize} \item[(a)] $\Vert f\Vert_C\le M \cdot \Vert f\Vert_{q,\alpha}$ and \item[(b)] $\vert f(0)\vert \le D \cdot \Vert f\Vert_{q,\alpha}$, when $f\in \mathcal{E}(\sigma,q,\alpha)$. \end{itemize} The main result says that, for $\alpha\ge-\frac12$ and $1\le q<\infty$, we have $M(\alpha,q,\sigma)=D(\alpha,q,\sigma)$, where $M(\alpha,q,\sigma)$ and $D(\alpha,q,\sigma)$ are the optimal constants in the inequalities (a) and (b), respectively. In addition, the sets of corresponding extremal functions in (a) and (b) coincide and there is given a characterization of the extremal functions. For $1<q<\infty$, the extremal function is unique (excepting a multiplicative factor). Moreover, the uniform norm of extremal functions for inequality (a) is attained only at the point $x=0$.
scientific article; zbMATH DE number 6865435
scientific article