Volume 11, Issue Supplement 1, October 2004, Pages 167 - 173
A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms
Agata Bezubik, Agata Dbrowska, Aleksander Strasburger
Available Online 1 October 2004.
- https://doi.org/10.2991/jnmp.2004.11.s1.22How to use a DOI?
- This article summarizes a new, direct approach to the determination of the expansion into spherical harmonics of the exponential ei(x|y) with x, y Rd . It is elementary in the sense that it is based on direct computations with the canonical decomposition of homogeneous polynomials into harmonic components and avoids using any integral identities. The proof makes also use of the standard representation theoretic properties of spherical harmonics and the explicit form of the reproducing kernels for these spaces by means of classical Gegenbauer polynomials. In the last section of the paper a new method of computing the Fourier transforms of SO(d)-finite functions on the unit sphere is presented which enables us to reobtain both the classical Bochner identity and generalizations of it due to one of the present authors and F. J. Gonzalez Vieli.
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Cite this article
TY - JOUR AU - Agata Bezubik AU - Agata Dbrowska AU - Aleksander Strasburger PY - 2004 DA - 2004/10 TI - A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms JO - Journal of Nonlinear Mathematical Physics SP - 167 EP - 173 VL - 11 IS - Supplement 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2004.11.s1.22 DO - https://doi.org/10.2991/jnmp.2004.11.s1.22 ID - Bezubik2004 ER -