Approximation in~$L_2$ by Partial Integrals of the Fourier Transform over the Eigenfunctions of the Sturm--Liouville Operator

D. V. Gorbachev; V. I. Ivanov

Geodesic

Parcourir par

Approximation in~$L_2$ by Partial Integrals of the Fourier Transform over the Eigenfunctions of the Sturm--Liouville Operator

D. V. Gorbachev ; V. I. Ivanov

Matematičeskie zametki, Tome 100 (2016) no. 4, pp. 519-530

Voir la notice de l'article provenant de la source Math-Net.Ru

Résumé

For approximations in the space $L_2(\mathbb{R}_+)$ by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator, we prove Jackson's inequality with exact constant and optimal argument in the modulus of continuity. The optimality of the argument in the modulus of continuity is established using the Gauss quadrature formula on the half-line over the zeros of the eigenfunction of the Sturm–Liouville operator.

Keywords: Sturm–Liouville operator on the half-line, the space $L_2$, Jackson's inequality
Mots-clés : Fourier transform, Gauss quadrature formula.

@article{MZM_2016_100_4_a4,
     author = {D. V. Gorbachev and V. I. Ivanov},
     title = {Approximation in~$L_2$ by {Partial} {Integrals} of the {Fourier} {Transform} over the {Eigenfunctions} of the {Sturm--Liouville} {Operator}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {519--530},
     publisher = {mathdoc},
     volume = {100},
     number = {4},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_100_4_a4/}
}

TY  - JOUR
AU  - D. V. Gorbachev
AU  - V. I. Ivanov
TI  - Approximation in~$L_2$ by Partial Integrals of the Fourier Transform over the Eigenfunctions of the Sturm--Liouville Operator
JO  - Matematičeskie zametki
PY  - 2016
SP  - 519
EP  - 530
VL  - 100
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2016_100_4_a4/
LA  - ru
ID  - MZM_2016_100_4_a4
ER  -

%0 Journal Article
%A D. V. Gorbachev
%A V. I. Ivanov
%T Approximation in~$L_2$ by Partial Integrals of the Fourier Transform over the Eigenfunctions of the Sturm--Liouville Operator
%J Matematičeskie zametki
%D 2016
%P 519-530
%V 100
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2016_100_4_a4/
%G ru
%F MZM_2016_100_4_a4

D. V. Gorbachev; V. I. Ivanov. Approximation in~$L_2$ by Partial Integrals of the Fourier Transform over the Eigenfunctions of the Sturm--Liouville Operator. Matematičeskie zametki, Tome 100 (2016) no. 4, pp. 519-530. http://geodesic.mathdoc.fr/item/MZM_2016_100_4_a4/