Nikol’skii type inequality; Bessel weight; extremal functions; extremal constants. (English) Zbl 1424.47026
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:
- (a)
- \(\Vert f\Vert_C\le M \cdot \Vert f\Vert_{q,\alpha}\) and
- (b)
- \(\vert f(0)\vert \le D \cdot \Vert f\Vert_{q,\alpha}\), when \(f\in \mathcal{E}(\sigma,q,\alpha)\).
Reviewer: Radu Păltănea (Braşov)
MSC:
47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |
41A44 | Best constants in approximation theory |
41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
Keywords:
entire function of exponential type; Nikol’skii-type inequality; Bessel generalized translationReferences:
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