Geodesic
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

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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/}
}
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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/