Convergence of Fourier series expansion related to free groups
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Abstract:
In $[0,\pi ]$ we consider the complete orthogonal system ${P_n}$ associated to the weight function $\psi = r(2r - 1){\pi ^{ - 1}}{\rm {si}}{{\rm {n}}^2}\theta {({r^2} - (2r - 1){\rm {co}}{{\rm {s}}^2}\theta )^{ - 1}}$ and we study mean and pointwise convergence of series expansions with respect to the system ${P_n}$ in ${L^p}([0,\pi ],d\psi )$. This weight function, and the corresponding system ${P_n}$ arise from the study of Gelfand transforms of radial functions on a finitely generated free group ${F_r}$ and our results can be interpreted in terms of multipliers theory on ${F_r}$.References
- Jean-Pierre Arnaud, Fonctions sphériques et fonctions définies positives sur l’arbre homogène, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 2, A99–A101 (French, with English summary). MR 563948 W. Betori and M. Pagliacci, Harmonic analysis for groups on trees, preprint
- P. Cartier, Harmonic analysis on trees, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 419–424. MR 0338272
- Joel M. Cohen, Operator norms on free groups, Boll. Un. Mat. Ital. B (6) 1 (1982), no. 3, 1055–1065 (English, with Italian summary). MR 683492
- Joel M. Cohen and Leonede De-Michele, The radial Fourier-Stieltjes algebra of free groups, Operator algebras and $K$-theory (San Francisco, Calif., 1981) Contemp. Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1982, pp. 33–40. MR 658507
- Bernd Dreseler and Paolo M. Soardi, A Cohen type inequality for ultraspherical series, Arch. Math. (Basel) 38 (1982), no. 3, 243–247. MR 656190, DOI 10.1007/BF01304783 A. Erdelyi, Higher tracendental functions, Vols. I-III, Bateman Manuscript Project, McGrawHill, New York, 1955.
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628
- Alessandro Figà-Talamanca and Massimo A. Picardello, Spherical functions and harmonic analysis on free groups, J. Functional Analysis 47 (1982), no. 3, 281–304. MR 665019, DOI 10.1016/0022-1236(82)90108-2
- Alessandro Figà-Talamanca and Massimo A. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker, Inc., New York, 1983. MR 710827
- Uffe Haagerup, An example of a nonnuclear $C^{\ast }$-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293. MR 520930, DOI 10.1007/BF01410082
- Richard A. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967) Southern Illinois Univ. Press, Carbondale, Ill., 1968, pp. 235–255. MR 0238019
- R. A. Kunze, $L_{p}$ Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89 (1958), 519–540. MR 100235, DOI 10.1090/S0002-9947-1958-0100235-1 A. Iozzi and M. Picardello, Spherical functions on graphs, preprint.
- Fritz Mayer-Lindenberg, Zur Dualitätstheorie symmetrischer Paare, J. Reine Angew. Math. 321 (1981), 36–52 (German). MR 597978, DOI 10.1515/crll.1981.321.36 C. Meaney, Almost everywhere divergent Jacobi polynomial series, preprint.
- Harry Pollard, The mean convergence of orthogonal series. II, Trans. Amer. Math. Soc. 63 (1948), 355–367. MR 23941, DOI 10.1090/S0002-9947-1948-0023941-X
- Harry Pollard, The mean convergence of orthogonal series. III, Duke Math. J. 16 (1949), 189–191. MR 28459
- Harry Pollard, The convergence almost everywhere of Legendre series, Proc. Amer. Math. Soc. 35 (1972), 442–444. MR 302973, DOI 10.1090/S0002-9939-1972-0302973-7
- T. Pytlik, Radial functions on free groups and a decomposition of the regular representation into irreducible components, J. Reine Angew. Math. 326 (1981), 124–135. MR 622348, DOI 10.1515/crll.1981.326.124
- Robert J. Stanton and Peter A. Tomas, Polyhedral summability of Fourier series on compact Lie groups, Amer. J. Math. 100 (1978), no. 3, 477–493. MR 622197, DOI 10.2307/2373834
- W. Forrest Stinespring, Integration theorems for gages and duality for unimodular groups, Trans. Amer. Math. Soc. 90 (1959), 15–56. MR 102761, DOI 10.1090/S0002-9947-1959-0102761-9
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 31-36
- MSC: Primary 43A50; Secondary 20E05, 33A45, 42C10
- DOI: https://doi.org/10.1090/S0002-9939-1984-0749884-9
- MathSciNet review: 749884