Abstract
We give a solution to the Bohman extremal problem for nonnegative even entire functions of exponential type that are Jacobi transforms of compactly supported functions. We prove that the extremal function is unique. The Gauss quadrature formula on the half-line over zeros of the Jacobi function is used.
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References
D. V. Gorbachev, “Bohman extremal problem for the Fourier–Hankel transform,” Izv. Tul’sk. Gos. Univ., Ser. Estestv. Nauki, No. 4, 5–10 (2014).
D. V. Gorbachev, “Multidimensional extremal Logan’s and Bohman’s problems,” in Methods of Fourier Analysis and Approximation Theory, Ed. by M. Ruzhansky and S. Tikhonov (Birkhäuser, Basel, 2016), Ser. Applied and Numerical Harmonic Analysis, pp. 43–58.
D. Gorbachev and V. Ivanov, “Extremal Bohman’s problem for Dunkl transform,” in Abstracts of the 5th Workshop on Fourier Analysis and Related Fields, Budapest, Hungary, 2015 (MTA Rényi Institute, Budapest, 2015), p. 6.
H. Bohman, “Approximate Fourier analysis of distribution functions,” Ark. Mat. 4, 99–157 (1960).
W. Ehm, T. Gneiting, and D. Richards, “Convolution roots of radial positive definite functions with compact support,” Trans. Amer. Math. Soc. 356, 4655–4685 (2004).
M. de Jeu, “Paley–Wiener theorems for the Dunkl transform,” Trans. Amer. Math. Soc. 358 (10), 4225–4250 (2006).
T. H. Koornwinder, “Jacobi functions and analysis on noncompact semisimple Lie groups,” in Special Functions: Group Theoretical Aspects and Applications, Ed. by R. A. Askey, T. H. Koornwinder, and W. Schempp (Reidel, Dordrecht, 1984), pp. 1–85.
B. M. Levitan and I. S. Sargsyan, Sturm–Liouville and Dirac Operators (Nauka, Moscow, 1988; Kluwer, Dordrecht, 1991).
T. H. Koornwinder, “A new proof of a Paley–Wiener type theorem for the Jacobi transform,” Ark. Mat. 13 (1), 145–159 (1979).
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,” Sb. Math. 206 (8), 1087–1122 (2015).
M. Flensted-Jensen and T. H. Koornwinder, “The convolution structure for Jacobi function expansions,” Ark. Mat. 11 (1), 245–262 (1973).
D. V. Gorbachev, V. I. Ivanov, and R. A. Veprintsev, “Optimal argument in the sharp Jackson inequality in the space L2 with hyperbolic weight,” Math. Notes 96 (5–6), 904–913 (2014).
B. Ya. Levin, Distribution of Zeros of Entire Functions (Gostekhizdat, Moscow, 1956; AMS, New York, 1964).
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Original Russian Text © D.V.Gorbachev, V.I. Ivanov, 2016, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, Vol. 22, No. 4, pp. 126–135.
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Gorbachev, D.V., Ivanov, V.I. Bohman Extremal Problem for the Jacobi Transform. Proc. Steklov Inst. Math. 300 (Suppl 1), 88–96 (2018). https://doi.org/10.1134/S0081543818020098
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DOI: https://doi.org/10.1134/S0081543818020098