On algebras with convolution structures for Laguerre polynomials

Kanjin, Yūichi

doi:10.1090/S0002-9947-1986-0833709-3

On algebras with convolution structures for Laguerre polynomials

Author: Yūichi Kanjin
Journal: Trans. Amer. Math. Soc. 295 (1986), 783-794
MSC: Primary 43A32; Secondary 42C10, 43A45, 43A46
DOI: https://doi.org/10.1090/S0002-9947-1986-0833709-3
MathSciNet review: 833709
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we treat the convolution algebra connected with Laguerre polynomials which was constructed by Askey and Gasper [1]. For this algebra, we study the maximal ideal space, Wiener's general Tauberian theorem, spectral synthesis and Helson sets. We also study Sidon sets and idempotent measures for the algebras with dual convolution structures.

[1] Richard Askey and George Gasper, Convolution structures for Laguerre polynomials, J. Analyse Math. 31 (1977), 48–68. MR 0486692, https://doi.org/10.1007/BF02813297
[2] Charles F. Dunkl, Operators and harmonic analysis on the sphere, Trans. Amer. Math. Soc. 125 (1966), 250–263. MR 0203371, https://doi.org/10.1090/S0002-9947-1966-0203371-9
[3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. II, McGraw-Hill, New York, 1953.
[4] George Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math. (2) 95 (1972), 261–280. MR 0310536, https://doi.org/10.2307/1970800
[5] E. Görlich and C. Markett, A convolution structure for Laguerre series, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 2, 161–171. MR 662652
[6] S. Igari, Certain Banach algebras and Jacobi polynomials, Lecture Note Ser. No. 110, RIMS, Kyoto, 1971, pp. 36-46. (Japanese)
[7] Satoru Igari and Yoshikazu Uno, Banach algebra related to the Jacobi polynomials, Tôhoku Math. J. (2) 21 (1969), 668–673; correction, ibid. (2) 22 (1970), 142. MR 0433123, https://doi.org/10.2748/tmj/1178242910
[8] Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin-New York, 1970 (French). MR 0275043
[9] Benjamin Muckenhoupt, Mean convergence of Hermite and Laguerre series. I, II, Trans. Amer. Math. Soc. 147 (1970), 419-431; ibid. 147 (1970), 433–460. MR 0256051, https://doi.org/10.1090/S0002-9947-1970-99933-9
[10] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834
[11] Alan Schwartz, The structure of the algebra of Hankel transforms and the algebra of Hankel-Stieltjes transforms, Canad. J. Math. 23 (1971), 236–246. MR 0273312, https://doi.org/10.4153/CJM-1971-023-x
[12] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1975.
[13] G. N. Watson, Another note in Laguerre polynomials, J. London Math. Soc. 14 (1939), 19-22.