2020/05/12 by Kenan Uriostegui, Uriostegui, Kenan, Kurt Bernardo Wolf +1 · 1 citation
Mathematics · #Mathematical functions and polynomials #Algebraic and Geometric Analysis #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2005.06076
We present a straightforward discretization of the Bessel functions Jn(x) to discrete counterparts B(N)n(xm), of N integer orders n on N integer points xm ≡ m, that we call discrete Bessel functions. These are built from a Bessel integral generating function, restricting the Fourier transform over the circle to N points. We show that the discrete Bessel functions satisfy several linear and quadratic relations, particularly Graf's product-displacement formulas, that are exact analogues of well-known relations between the continuous functions. It is noteworthy that these discrete Bessel functions approximate very closely the values of the continuous functions in ranges n + |m| < N. For fixed N, this provides an N-point transform between functions of order and of position,fn and \widetildefm, which is efficient for the Fourier analysis of finite decaying signals.