Inequalities of Carlson-Taikov-Shadrin type in $$$L_{2,r;\alpha,\beta}((-1,1))$$$ and $$$L_{2,e^{-t^2}}(\mathbb{R})$$$ spaces

Abstract

We consider the problem of finding sharp inequalities for the norms of derivatives of the functions. This classical problem arises in Approximation Theory in the beginning of XX century in works of E. Landau, J. Hadamard, G.H. Hardy and J.E. Littlewood. A thorough overview of many known results and related problems can be found in surveys [1, 2] and the book [3].
Recall that $$$L_{2,r;\alpha,\beta}((-1,1))$$$, $$$r\in \mathbb{N}$$$ and $$$\alpha,\beta> -1$$$, is the space of measurable functions $$$x:(-1,1)\to\mathbb{R}$$$ such that $$$\|x\|_{2,r;\alpha,\beta} := \int_{-1}^{1} |x(t)|^2(1-t)^{\alpha+r}(1+t)^{\beta+r}\,{\rm d}t < \infty$$$, and $$$L_{2,e^{-t^2}}(\mathbb{R})$$$ is the space of measurable functions $$$x:\mathbb{R}\to\mathbb{R}$$$ such that $$$\|x\|_{2,e^{-t^2}} := \int_{-\infty}^{+\infty} |x(t)|^2e^{-t^2}\,{\rm d}t < \infty$$$. S.Z. Rafalson [7], S.Z. Rafalson and I.V. Berdnikova [5] obtained analogues of Hardy-Littlewood-Polya inequalities for the norms of derivatives of functions in spaces $$$L_{2,r;\alpha,\beta}((-1,1))$$$ and $$$L_{2,e^{-t^2}}(\mathbb{R})$$$. Namely, they established sharp inequalities that estimate $$$\left\|x^{(k)}\right\|_{2,k;\alpha\beta}$$$, $$$k\in\mathbb{N}$$$ and $$$0 < k < r$$$, in terms of $$$\|x\|_{2,0;\alpha,\beta}$$$ and $$$\left\|x^{(r)}\right\|_{2,r;\alpha,\beta}$$$, and sharp inequalities that estimate $$$\left\|x^{(k)}\right\|_{2,e^{-t^2}}$$$ in terms of $$$\left\|x\right\|_{2,e^{-t^2}}$$$ and $$$\left\|x^{(r)}\right\|_{2,e^{-t^2}}$$$.
In this paper we obtain the analogues of Taikov-Shadrin inequalities for the norms of derivatives in spaces $$$L_{2,r;\alpha,\beta}((-1,1))$$$ and $$$L_{2,e^{-t^2}}(\mathbb{R})$$$. Namely, we obtain sharp inequalities that estimate $$$\left|x^{(k)}(t_0)\right|$$$, $$$t_0\in(-1,1)$$$, $$$k\in\mathbb{Z}_+$$$ and $$$k < r$$$, in terms of $$$\|x\|_{2,0;\alpha,\beta}$$$ and $$$\left\|x^{(r)}\right\|_{2,r;\alpha,\beta}$$$, and sharp inequalities that estimate $$$\left|x^{(k)}(t_0)\right|$$$, $$$t_0\in \mathbb{R}$$$, $$$k\in\mathbb{Z}_+$$$ and $$$k < r$$$, in terms of $$$\|x\|_{2,e^{-t^2}}$$$ and $$$\left\|x^{(r)}\right\|_{2,e^{-t^2}}$$$.