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Integration of a zonal function related to the local harmonic analysis on spheres

Authors:	Sun Limin

Institution:	(1) Department of Mathematics, Hangzhou University, 310028 Hangzhou, PRC

Abstract:	In studying local harmonic analysis on the sphere Sⁿ, R.S. Strichartz introduced certain zonal functions ϕ₂(d(x, y)) which satisfy the equation $\Delta _z \varphi _h \left( {d\left( {x,y} \right)} \right) = \mu \cdot \varphi _h \left( {d\left( {x,y} \right)} \right) + a\left( \lambda \right) \cdot \delta _{ - y} \left( x \right), \mu = \sigma ^2 - \lambda ^2 ,0 = \frac{{n - 1}}{2}$ , where Δ_z is the Laplace operator and δ_−y the Dirac measure. The explicit expression of the constant a (λ) is given by R.S. Strichartz in the case that n is odd. Appyling the Apéry identity, we show in this paper that $a\left( \lambda \right) = \frac{2}{\pi }( - 1)^{\pi + \frac{1}{2}} \cdot \omega _{\pi - 1} \gamma _\pi ^2 \left\{ {\prod\limits_{k = 0}^{\pi - 3/2} {\left( {\lambda ^2 - (k + \frac{1}{2})^2 } \right)^{} } } \right\}^{ - 1} \cdot \cos \left( {\pi \lambda } \right)$ for n even, where w_n-1 is the surface area of S^n-1, $\gamma _n = \left( {n - 2} \right) \cdot \left( {n - 4} \right) \cdots 2$ . The author's research was supported by a grant from NSFC.

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