Geodesic
Estimates of Derivatives in Sobolev Spaces in Terms of Hypergeometric Functions
Matematičeskie zametki, Tome 109 (2021) no. 4, pp. 500-507

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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.
Keywords: Sobolev space, embedding constants, analytic inequalities, hypergeometric functions.
Mots-clés : Legendre polynomials 
@article{MZM_2021_109_4_a1,
     author = {T. A. Garmanova},
     title = {Estimates of {Derivatives} in {Sobolev} {Spaces} in {Terms} of {Hypergeometric} {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {500--507},
     publisher = {mathdoc},
     volume = {109},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a1/}
}
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T. A. Garmanova. Estimates of Derivatives in Sobolev Spaces in Terms of Hypergeometric Functions. Matematičeskie zametki, Tome 109 (2021) no. 4, pp. 500-507. http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a1/