Interpolation formulas with derivatives in de Branges spaces
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Abstract:
The purpose of this paper is to prove an interpolation formula involving derivatives for entire functions of exponential type. We extend the interpolation formula derived by J. Vaaler in 1985 to general $L^p$ de Branges spaces. We extensively use techniques from de Branges’ theory of Hilbert spaces of entire functions, but a crucial passage involves the Hilbert-type inequalities as derived by Carneiro, Littmann, and Vaaler. We give applications to homogeneous spaces of entire functions that involve Bessel functions and we prove a uniqueness result for extremal one-sided band-limited approximations of radial functions in Euclidean spaces.References
- Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR 1184139, DOI 10.1007/b97238
- A. D. Baranov, Differentiation in the Branges spaces and embedding theorems, J. Math. Sci. (New York) 101 (2000), no. 2, 2881–2913. Nonlinear equations and mathematical analysis. MR 1784683, DOI 10.1007/BF02672176
- A. D. Baranov, On estimates for the $L^p$-norms of derivatives in spaces of entire functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 303 (2003), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 31, 5–33, 321 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 129 (2005), no. 4, 3927–3943. MR 2037529, DOI 10.1007/s10958-005-0330-9
- Frank Bowman, Introduction to Bessel functions, Dover Publications, Inc., New York, 1958. MR 0097539
- Louis de Branges, Homogeneous and periodic spaces of entire functions, Duke Math. J. 29 (1962), 203–224. MR 148917
- Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
- Emanuel Carneiro and Vorrapan Chandee, Bounding $\zeta (s)$ in the critical strip, J. Number Theory 131 (2011), no. 3, 363–384. MR 2739041, DOI 10.1016/j.jnt.2010.08.002
- E. Carneiro, V. Chandee, F. Littmann and M. B. Milinovich, Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function, J. Reine Angew. Math. (to appear).
- Emanuel Carneiro, Vorrapan Chandee, and Micah B. Milinovich, Bounding $S(t)$ and $S_1(t)$ on the Riemann hypothesis, Math. Ann. 356 (2013), no. 3, 939–968. MR 3063902, DOI 10.1007/s00208-012-0876-z
- Emanuel Carneiro, Vorrapan Chandee, and Micah B. Milinovich, A note on the zeros of zeta and $L$-functions, Math. Z. 281 (2015), no. 1-2, 315–332. MR 3384872, DOI 10.1007/s00209-015-1485-9
- Emanuel Carneiro and Felipe Gonçalves, Extremal problems in de Branges spaces: the case of truncated and odd functions, Math. Z. 280 (2015), no. 1-2, 17–45. MR 3343896, DOI 10.1007/s00209-015-1411-1
- Emanuel Carneiro and Friedrich Littmann, Bandlimited approximations to the truncated Gaussian and applications, Constr. Approx. 38 (2013), no. 1, 19–57. MR 3078273, DOI 10.1007/s00365-012-9177-8
- Emanuel Carneiro and Friedrich Littmann, Extremal functions in de Branges and Euclidean spaces, Adv. Math. 260 (2014), 281–349. MR 3209354, DOI 10.1016/j.aim.2014.04.007
- Emanuel Carneiro and Friedrich Littmann, Extremal functions in de Branges and Euclidean Spaces II, Preprint.
- Emanuel Carneiro, Friedrich Littmann, and Jeffrey D. Vaaler, Gaussian subordination for the Beurling-Selberg extremal problem, Trans. Amer. Math. Soc. 365 (2013), no. 7, 3493–3534. MR 3042593, DOI 10.1090/S0002-9947-2013-05716-9
- Emanuel Carneiro and Jeffrey D. Vaaler, Some extremal functions in Fourier analysis. II, Trans. Amer. Math. Soc. 362 (2010), no. 11, 5803–5843. MR 2661497, DOI 10.1090/S0002-9947-2010-04886-X
- Emanuel Carneiro and Jeffrey D. Vaaler, Some extremal functions in Fourier analysis. III, Constr. Approx. 31 (2010), no. 2, 259–288. MR 2581230, DOI 10.1007/s00365-009-9050-6
- Vorrapan Chandee and K. Soundararajan, Bounding $|\zeta (\frac 12+it)|$ on the Riemann hypothesis, Bull. Lond. Math. Soc. 43 (2011), no. 2, 243–250. MR 2781205, DOI 10.1112/blms/bdq095
- William S. Cohn, Radial limits and star invariant subspaces of bounded mean oscillation, Amer. J. Math. 108 (1986), no. 3, 719–749. MR 844637, DOI 10.2307/2374661
- P. X. Gallagher, Pair correlation of zeros of the zeta function, J. Reine Angew. Math. 362 (1985), 72–86. MR 809967, DOI 10.1515/crll.1985.362.72
- D. A. Goldston and S. M. Gonek, A note on $S(t)$ and the zeros of the Riemann zeta-function, Bull. Lond. Math. Soc. 39 (2007), no. 3, 482–486. MR 2331578, DOI 10.1112/blms/bdm032
- Felipe Gonçalves, Michael Kelly, and Jose Madrid, One-sided band-limited approximations of some radial functions, Bull. Braz. Math. Soc. (N.S.) 46 (2015), no. 4, 563–599. MR 3436559, DOI 10.1007/s00574-015-0104-z
- S. W. Graham and Jeffrey D. Vaaler, A class of extremal functions for the Fourier transform, Trans. Amer. Math. Soc. 265 (1981), no. 1, 283–302. MR 607121, DOI 10.1090/S0002-9947-1981-0607121-1
- Jeffrey J. Holt and Jeffrey D. Vaaler, The Beurling-Selberg extremal functions for a ball in Euclidean space, Duke Math. J. 83 (1996), no. 1, 202–248. MR 1388849, DOI 10.1215/S0012-7094-96-08309-X
- Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
- M. Krein, A contribution to the theory of entire functions of exponential type, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 11 (1947), 309–326 (Russian, with English summary). MR 0022252
- Friedrich Littmann, Entire approximations to the truncated powers, Constr. Approx. 22 (2005), no. 2, 273–295. MR 2148534, DOI 10.1007/s00365-004-0586-1
- Friedrich Littmann, Entire majorants via Euler-Maclaurin summation, Trans. Amer. Math. Soc. 358 (2006), no. 7, 2821–2836. MR 2216247, DOI 10.1090/S0002-9947-06-04121-3
- Friedrich Littmann, Quadrature and extremal bandlimited functions, SIAM J. Math. Anal. 45 (2013), no. 2, 732–747. MR 3038107, DOI 10.1137/120888004
- Friedrich Littmann and Mark Spanier, Extremal functions with vanishing condition, Constr. Approx. 42 (2015), no. 2, 209–229. MR 3392488, DOI 10.1007/s00365-015-9304-4
- Yurii I. Lyubarskii and Kristian Seip, Weighted Paley-Wiener spaces, J. Amer. Math. Soc. 15 (2002), no. 4, 979–1006. MR 1915824, DOI 10.1090/S0894-0347-02-00397-1
- Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543, DOI 10.1090/cbms/084
- H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, J. London Math. Soc. (2) 8 (1974), 73–82. MR 337775, DOI 10.1112/jlms/s2-8.1.73
- Joaquim Ortega-Cerdà and Kristian Seip, Fourier frames, Ann. of Math. (2) 155 (2002), no. 3, 789–806. MR 1923965, DOI 10.2307/3062132
- M. Plancherel and G. Pólya, Fonctions entières et intégrales de fourier multiples, Comment. Math. Helv. 10 (1937), no. 1, 110–163 (French). MR 1509570, DOI 10.1007/BF01214286
- Atle Selberg, Collected papers. Vol. II, Springer-Verlag, Berlin, 1991. With a foreword by K. Chandrasekharan. MR 1295844
- Jeffrey D. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 183–216. MR 776471, DOI 10.1090/S0273-0979-1985-15349-2
- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
Additional Information
- Felipe Gonçalves
- Affiliation: IMPA - Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Rio de Janeiro, RJ, Brazil 22460-320
- Email: ffgoncalves@impa.br
- Received by editor(s): October 20, 2014
- Received by editor(s) in revised form: January 23, 2015
- Published electronically: March 1, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 805-832
- MSC (2010): Primary 46E22, 30D10, 41A05, 41A30, 33C10
- DOI: https://doi.org/10.1090/tran6672
- MathSciNet review: 3572255