On the Fourier transform of Bessel functions over complex numbers—II: The general case

Qi, Zhi

doi:10.1090/tran/7710

On the Fourier transform of Bessel functions over complex numbers—II: The general case
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Trans. Amer. Math. Soc. 372 (2019), 2829-2854 Request permission

Abstract:

In this paper, we prove an exponential integral formula for the Fourier transform of Bessel functions over complex numbers, along with a radial exponential integral formula. The former will enable us to develop the complex spectral theory of the relative trace formula for the Shimura–Waldspurger correspondence and extend the Waldspurger formula from totally real fields to arbitrary number fields.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
Ehud Moshe Baruch and Orr Beit-Aharon, A kernel formula for the action of the Weyl element in the Kirillov model of $SL(2,\Bbb {C})$, J. Number Theory 146 (2015), 23–40. MR 3267110, DOI 10.1016/j.jnt.2014.02.001
Ehud Moshe Baruch and Zhengyu Mao, Bessel identities in the Waldspurger correspondence over a $p$-adic field, Amer. J. Math. 125 (2003), no. 2, 225–288. MR 1963685, DOI 10.1353/ajm.2003.0008
Ehud Moshe Baruch and Zhengyu Mao, Bessel identities in the Waldspurger correspondence over the real numbers, Israel J. Math. 145 (2005), 1–81. MR 2154720, DOI 10.1007/BF02786684
Ehud Moshe Baruch and Zhengyu Mao, Central value of automorphic $L$-functions, Geom. Funct. Anal. 17 (2007), no. 2, 333–384. MR 2322488, DOI 10.1007/s00039-007-0601-3
Roelof Wichert Bruggeman and Yoichi Motohashi, A note on the mean value of the zeta and $L$-functions. XIII, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 6, 87–91. MR 1913937
Roelof W. Bruggeman and Yoichi Motohashi, Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field, Funct. Approx. Comment. Math. 31 (2003), 23–92. MR 2059538, DOI 10.7169/facm/1538186640
J. Chai and Z. Qi, Bessel identities in the Waldspurger correspondence over the complex numbers. arXiv:1802.01229, to appear in Israel J. Math., 2018.
J. Chai and Z. Qi, On the Waldspurger formula and the metaplectic Ramanujan conjecture over number fields, arXiv:1808.05398, to appear in J. Funct. Anal.
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0065685
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
Yoichi Motohashi, Mean values of zeta-functions via representation theory, Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Sympos. Pure Math., vol. 75, Amer. Math. Soc., Providence, RI, 2006, pp. 257–279. MR 2279942, DOI 10.1090/pspum/075/2279942
Z. Qi, Theory of fundamental Bessel functions of high rank, arXiv:1612.03553, to appear in Mem. Amer. Math. Soc., 2016.
Zhi Qi, On the Kuznetsov trace formula for $\textrm {PGL}_2(\Bbb {C})$, J. Funct. Anal. 272 (2017), no. 8, 3259–3280. MR 3614169, DOI 10.1016/j.jfa.2017.01.011
Zhi Qi, On the Fourier transform of Bessel functions over complex numbers—I: The spherical case, Monatsh. Math. 186 (2018), no. 3, 471–479. MR 3817014, DOI 10.1007/s00605-017-1108-0
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746