Abstract
For approximations in the space L 2(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.
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References
A. G. Babenko, “Exact Jackson-Stechkin inequality in the space L 2(Rm),” Tr. Inst. Mat. Mekh. (Ekaterinburg) 5, 183–198 (1998).
A. V. Moskovskii, “Jackson theorems in the spaces L p(Rn) and L p, λ(R+),” Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform. 3 (1), 44–70 (1997).
D. V. Gorbachev, “Extremum problems for entire functions of exponential spherical type,” Mat. Zametki 68 (2), 179–187 (2000) [Math. Notes 68 (1–2), 159–166 (2000)].
C. Frappier and P. Oliver, “A quadrature formula involving zeros of Bessel functions,” Math. Comp. 60 (201), 303–316 (1993).
G. R. Grozev and Q. I. Rahman, “A quadrature formula with zeros of Bessel functions as nodes,” Math. Comp. 64, 715–725 (1995).
V. V. Arestov and N. I. Chernykh, “On the L 2-approximation periodic functions by trigonometric polynomials,” in Approximation and Functions Spaces (North-Holland, Amsterdam, 1981), pp. 25–43.
E. E. Berdysheva, “Two related extremal problems for entire functions of several variables,” Mat. Zametki 66 (3), 336–350 (1999) [Math. Notes 66 (3–4), 271–282 (1999)].
A. V. Ivanov, “Some extremal problems for entire functions in weighted spaces,” Izv. Tul. Gos. Univ. Estestv. Nauki, No. 1, 26–44 (2010).
A. V. Ivanov, “The Logan problem for entire functions of several variables and the Jackson constants in weighted spaces,” Izv. Tul. Gos. Univ. Estestv. Nauki, No. 2, 29–58 (2011).
A. V. Ivanov and V. I. Ivanov, “Optimal arguments in Jackson’s inequality in the power-weighted space L 2(Rd),” Mat. Zametki 94 (3), 338–348 (2013) [Math. Notes 94 (3–4), 320–329 (2013)].
V. I. Ivanov and A. V. Ivanov, “Optimal arguments in the Jackson–Stechkin inequality in L 2(Rd) with Dunkl weight,” Mat. Zametki 96 (5), 674–686 (2014) [Math. Notes 96 (5–6), 666–677 (2014)].
D. V. Gorbachev, V. I. Ivanov, and R. A. Veprintsev, “Optimal argument in the sharp Jackson inequality in the space L 2 with hyperbolic weight,” Math. Notes 96 (5–6), 904–913 (2014).
R. A. Veprintsev, “Approximation of the multidimensional Jacobi transform in L 2 by partial integrals,” Mat. Zametki 97 (6), 815–831 (2015) [Math. Notes 97 (5–6), 831–845 (2015)].
D. V. Gorbachev and V. I. Ivanov, “Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm-Liouville problem, which are exact for entire functions of exponential type,” Mat. Sb. 206 (8), 63–98 (2015) [Sb. Math. 206 (8), 1087–1122 (2015)].
B. M. Levitan, Theory of Generalized Shift Operators (Nauka, Moscow, 1973) [in Russian].
B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory (Nauka, Moscow, 1970) [in Russian].
B. M. Levitan and I. S. Sargsyan, Sturm–Liouville and Dirac Operators (Nauka, Moscow, 1988) [in Russian].
M. Flensted-Jensen and T. H. Koornwinder, “Jacobi functions. The addition formula and the positivity of dual convolution structure,” Ark. Mat. 17, 139–151 (1979).
N. P. Korneichuk, Extremal Problems of Approximation Theory (Nauka, Moscow, 1976) [in Russian].
V. V. Arestov and V. Yu. Popov, “Jackson inequalities on sphere in L 2,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 8, 13–20 (1995) [Russian Math. (Iz. VUZ) 39 (8), 11–18 (1995)].
B. F. Logan, “Extremal problems for positive-definite bandlimited functions I. Eventually positive functions with zero integral,” SIAM J. Math. Anal. 14 (2), 249–252 (1983).
V. A. Yudin, “Multidimensional Jackson theorem in L 2,” Mat. Zametki 29 (2), 309–315 (1981) [Math. Notes 29 (1–2), 158–162 (1981)].
B. Ya. Levin, Distribution of Zeros of Entire Functions (Gostekhizdat, Moscow, 1956) [in Russian].
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Original Russian Text © D. V. Gorbachev, V. I. Ivanov, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 4, pp. 519–530.
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Gorbachev, D.V., Ivanov, V.I. Approximation in L 2 by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator. Math Notes 100, 540–549 (2016). https://doi.org/10.1134/S000143461609025X
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DOI: https://doi.org/10.1134/S000143461609025X