Orthogonal bases for spaces of complex spherical harmonics

• Autorzy: Menegatto V. A. , Oliveira C. P.
• Języki publikacji: EN

Abstrakty

EN

This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω 2q of Cq. The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L²(Ω 2q). As a bypass, we study the inner product [f,g] = f(D)(g(z))(0) over the space P(Cq) of polynomials in the variables [wzór], in which f(D) is the differential operator with symbol f(z). On the spaces of spherical harmonics, it is shown that the inner product [. , .] reduces to a multiple of the L²(Ω 2q) inner product. Bi-orthogonality in (F(Cq), [. , .] ) is fully investigated.

Słowa kluczowe

EN

spherical harmonics; sphere; orthogonal basis; generating function; addition formula; Funk-Hecke formula

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2005
• Tom: Vol. 11, nr 1
• Strony: 113--132
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Menegatto V. A.

• Departamento de Matemática, ICMC-USP — Säo Carlos, Caixa Postal 668, 13560-970 Säo Carlos SP, Brasil

autor

Oliveira C. P.

• Universidade Federal de Itajubá, ICE-DMC, Caixa Postal 50, 37500-903 Itajubá MG, Brasil

Bibliografia

• [1] Axler, S., Bourdon, P., Ramey, W., Harmonic Function Theory (2nd edition), Graduate Texts in Mathematics 137, Springer-Verlag, New York, 2001.
• [2] Koornwinder, T. H., The addition formula for Jacobi polynomials, II. The Laplace, type integral representation and the product formula, Math. Centrum Afd. Toegepaste Wisk., Report TW133 (1972).
• [3] Koornwinder, T. H., The addition formula for Jacobi polynomials, III. Completion of the proof Math. Centrum Afd. Toegepaste Wisk., Report TW135 (1972).
• [4] Lang, S.,Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass, 1965.
• [5] Menegatto, V. A., Oliveira, C. P., Annihilating properties of convolution operators on complex spheres, Anal. Math. 31 (2005), 13-30.
• [6] Müller, C, Analysis of Spherical Symmetries in Euclidean Spaces, Appl. Math. Sci. 129, Springer-Verlag, New York, 1998.
• [7] Quinto, E. T., Injectivity of rotation invariant Radon transforms on complex hyper-planes in C”, „Integral geometry” (Brunswick, Maine, 1984), 245-260, Contemp. Math. 63, Amer. Math. Soc, Providence, RI, 1987.
• [8] Rudin, W., Function Theory in the Unit Ball of Cn, Grundlehren Math. Wiss. 241, Springer-Verlag, New York-Berlin, 1980.
• [9] Stein, E. M., Weiss, G., Introduction to Fourier Analysis on Euclidea,n Spaces, Princeton Math. Ser. 32, Princeton Univ. Press, Princeton, NJ, 1971.