Abstract:
For approximations in the space L2(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 L2, Fourier transform, Jackson's inequality, Gauss quadrature formula.
This work was supported
by the Ministry of Education and Science of the Russian Federation
(grant no. 5414GZ and no. 1.1333.2014K),
by the Russian Foundation for Basic Research
under grant 16-01-00308,
and the “Dynasty” Foundation of D. Zimin.
Citation:
D. V. Gorbachev, V. I. Ivanov, “Approximation in L2 by Partial Integrals of the Fourier Transform over the Eigenfunctions of the Sturm–Liouville Operator”, Mat. Zametki, 100:4 (2016), 519–530; Math. Notes, 100:4 (2016), 540–549
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Linking options:
https://www.mathnet.ru/eng/mzm11110
https://doi.org/10.4213/mzm11110
https://www.mathnet.ru/eng/mzm/v100/i4/p519
This publication is cited in the following 5 articles:
D. V. Gorbachev, “Konstanty Nikolskogo - Bernshteina dlya neotritsatelnykh tselykh funktsii eksponentsialnogo tipa na osi”, Tr. IMM UrO RAN, 24, no. 4, 2018, 92–103
D. V. Gorbachev, V. I. Ivanov, E. P. Ofitserov, O. I. Smirnov, “Vtoraya ekstremalnaya zadacha Logana dlya preobrazovaniya Fure po sobstvennym funktsiyam operatora Shturma–Liuvillya”, Chebyshevskii sb., 19:1 (2018), 57–78
D. V. Gorbachev, V. I. Ivanov, “Nekotorye ekstremalnye zadachi dlya preobrazovaniya Fure po sobstvennym funktsiyam operatora Shturma–Liuvillya”, Chebyshevskii sb., 18:2 (2017), 34–53
D. V. Gorbachev, V. I. Ivanov, E. P. Ofitserov, O. I. Smirnov, “Nekotorye ekstremalnye zadachi garmonicheskogo analiza i teorii priblizhenii”, Chebyshevskii sb., 18:4 (2017), 140–167
D. V. Gorbachev, V. I. Ivanov, R. A. Veprintsev, “Approximation in L2 by partial integrals of the multidimensional Fourier transform in the eigenfunctions of the Sturm–Liouville operator”, Proc. Steklov Inst. Math. (Suppl.), 300, suppl. 1 (2018), 97–113
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