Sampling expansion for the continuous bessel transform

Abstract

For arbitrary complex numbers b and a≠O,-1,-2,…, the Bessel polynomials {y_n(z;a,b)}^∞ _n=0 are orthogonal with respect to a complex measure dρ(z) on any closed contour enclosing the origin. If f(z)εL¹(T) where T is the unit circle , we can associate with it the formal series

where

and

we call (*)the discrete Bessel transform of f (z) . In this article , we introduce hat we call the continuous Bessel transform, which is a generalization of (*) to the case where the discrete parameter n is replaced by a continuous real parameter λ Moreover, we show that , for a = 2 = b, i f f(z) i s a nicely behaved function, then one can obtain a sampling theorem for its continuous Bessel transform F(λ) in the form

for some given functions S_n(λ).

AMS(MOS):

• 94A11
• 44A15