Decay of Fourier transforms and summability of eigenfunction expansions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Volume 29 (2000) no. 3, p. 611-638
@article{ASNSP_2000_4_29_3_611_0,
     author = {Brandolini, Luca and Colzani, Leonardo},
     title = {Decay of Fourier transforms and summability of eigenfunction expansions},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 29},
     number = {3},
     year = {2000},
     pages = {611-638},
     mrnumber = {1817712},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2000_4_29_3_611_0}
}
Brandolini, Luca; Colzani, Leonardo. Decay of Fourier transforms and summability of eigenfunction expansions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Volume 29 (2000) no. 3, pp. 611-638. http://www.numdam.org/item/ASNSP_2000_4_29_3_611_0/

[1] S.A. Alimov - V.A. Il'In - E.M. Nikishin, Convergence problems of multiple trigonometric series and spectral decompositions, I, II, Russian Math. Surveys 31 (1976), 29-86, 32 (1977), 115-139. | Zbl 0376.42002

[2] P. Bérard, On the wave equation on a manifold without conjugate points, Math. Z. 155 (1977), 249-276. | MR 455055 | Zbl 0341.35052

[3] P. Bérard, Riesz means on Riemannian manifolds, Amer. Math. Soc. Proc. Symp. Pure Math. XXXVI (1980), 1-12. | MR 573426 | Zbl 0443.58023

[4] S. Bochner, Summation of multiple Fourier series by spherical means, Trans. Amer. Math. Soc. 40 (1936), 175-207. | JFM 62.0293.03 | MR 1501870 | Zbl 0015.15702

[5] L. Brandolini - L. Colzani, Localization and convergence of eigenfunction expansions, J. Fourier Anal. Appl. 5 (1999), 431-447. | MR 1755098 | Zbl 0938.42015

[6] L. Brandolini - L. Colzani - G. Travaglini, Average decay of Fourier transforms and integer points in polyhedra, Ark. Mat. 35 (1997), 253-275. | MR 1478780 | Zbl 0937.11043

[7] L. Colzani - M. Vignati, The Gibbs phenomenon for multiple Fourier integrals, J. Approx. Th. 80 (1995), 119-131. | MR 1308597 | Zbl 0815.42008

[8] L. De Michele - D. Roux, Approximate units and Gibbs phenomenon, Boll. Un. Mat. Ital. A (7) (1997), 739-746. | MR 1489045 | Zbl 0898.42002

[9] L. De Michele - D. Roux, The Gibbs phenomenon for L1 loc kernels, J. Approx. Th. 100 (1999), 144-156. | MR 1710557 | Zbl 0953.42006

[10] L. De Michele - D. Roux, The Gibbs phenomenon for multiple Fourier integrals and series: restriction theorems, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 351-360. | MR 1645726 | Zbl 0916.42008

[11] G.H. Hardy, On the expression of a number as a sum of two squares, Quart. J. Math. 46 (1915), 263-283. | JFM 45.1253.01

[12] E. Hlawka, Uber Integrale auf convexen Korpen, I & II, Monats. Math. 54 (1950), 1-36, 81-99. | MR 37003 | Zbl 0036.30902

[13] C. Herz, Fourier transform related to convex sets, Ann. of Math. 75 (1962), 81-92. | MR 142978 | Zbl 0111.34803

[14] L. Hörmander, On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, In: "Some recent advances in the basic sciences", Yeshiva University 1966, pp. 155-202. | MR 257589

[15] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218. | MR 609014 | Zbl 0164.13201

[16] L. Hörmander, "The analysis of linear partial differential operators", I, II, III, IV, Springer Verlag, 1985-1985. | MR 781536 | Zbl 0601.35001

[17] J.P. Kahane, Le phénomène de Pinsky et la géométrie des surfaces, C. R. Acad. Sci. Paris 321 (1995), 1027-1029. | MR 1360566 | Zbl 0841.42004

[18] D.G. Kendall, On the number of lattice points inside a random oval, Quart. J. Math. Oxford 19 (1948), 1-26. | MR 24929 | Zbl 0031.11201

[19] C. Meaney, On almost-everywhere convergent eigenfunction expansions of the Laplace-Beltrami operator, Math. Proc. Cambridge Philos. Soc. 92 (1982), 129-131. | MR 662968 | Zbl 0495.58030

[20] M.A. Pinsky, Pointwise Fourier inversion and related eigenfunction expansions, Comm. Pure Appl. Math. 47 (1994), 653-681. | MR 1278348 | Zbl 0802.42010

[21] M.A. Pinsky, Fourier inversion in the piecewise smooth category, In: "Fourier Analysis, analytic and geometric aspects", W. O. Bray - P. S. Milojevic' - C. V. Stanojevic' (eds.), Marcel Dekker (1994). | MR 1277831 | Zbl 0815.42007

[22] M.A. Pinsky - M. Taylor, Pointwise Fourier inversion: a wave equation approach, J. Fourier Anal. Appl. 3 (1997), 647-703. | MR 1481629 | Zbl 0901.42008

[23] A.N. Podkorytov, The asymptotic of Fourier transform of a convex curve, Vestnik Leningr. Univ. Mat. 24 (1991), 57-65. | MR 1166380 | Zbl 0741.42012

[24] C.D. Sogge, Concerning the Lp norm of spectral clusters for second order elliptic differential operators on compact manifolds, J. Funct. Anal. 77 (1988), 123-134. | MR 930395 | Zbl 0641.46011

[25] C.D. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of Math. 126 (1987), 439-447. | MR 908154 | Zbl 0653.35068

[26] E.M. Stein - G. Weiss, "Introduction to Fourier analysis on Euclidean spaces", Princeton University Press, 1971. | MR 304972 | Zbl 0232.42007

[27] M.E. Taylor, Pointwise Fourier inversion on tori and other compact manifolds, J. Fourier Anal. Appl. 5 (1999), 449-463. | MR 1755099 | Zbl 0938.42007

[28] M.E. Taylor, Pointwise Fourier inversion - an addendum, Proc. Amer. Math. Soc., to appear. | MR 1825908 | Zbl 0967.42006

[29] M.E. Taylor, The Dirichlet-Jordan test and multidimensional extensions, Proc. Amer. Math. Soc., to appear. | MR 1709767 | Zbl 0969.42006

[30] M.E. Taylor, Eigenfunction expansions and the Pinsky phenomenon on compact manifolds, preprint. | MR 1845101 | Zbl 1065.58023

[31] A. Torlaschi, Sviluppi in armoniche sferiche di funzioni regolari a tratti, Tesi di Laurea, Università degli Studi di Milano (1998).

[32] A.N. Varchenko, Number of lattice points in families of homethetic domains in Rn, Funktional An. 17 (1983), 1-6. | MR 705041 | Zbl 0522.10031

[33] G.N. Watson, "A treatise on the theory of Bessel functions", Cambridge University Press, 1944. | MR 10746 | Zbl 0063.08184

[34] H. Weyl, Die Gibbsche Erscheinung in der Theorie der Kugelfunktionen, Rendiconti Circ. Mat. Palermo 29 (1910), 308-323. | JFM 41.0528.01

[35] H. Weyl, Über die Gibbsche Erscheinung und verwandte Konvergenzphänomene, Rendiconti Circ. Mat. Palermo 30 (1910), 377-407. | JFM 41.0528.02