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Estimates of derivatives in Sobolev spaces in terms of hypergeometric functions. (English. Russian original) Zbl 1479.46039

Math. Notes 109, No. 4, 527-533 (2021); translation from Mat. Zametki 109, No. 4, 500-507 (2021).
Summary: The paper deals with sharp estimates of derivatives of intermediate order \(k\le n-1\) in the Sobolev space \(\mathring{W}^n_2[0;1]\), \(n\in\mathbb N\). The functions \(A_{n,k}(x)\) under study are the smallest possible quantities in inequalities of the form \[|y^{(k)}(x)|\le A_{n,k}(x)\|y^{(n)}\|_{L_2[0;1]}.\] The properties of the primitives of shifted Legendre polynomials on the interval \([0;1]\) are used to obtain an explicit description of these functions in terms of hypergeometric functions. In the paper, a new relation connecting the derivatives and primitives of Legendre polynomials is also proved.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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