Fast Hankel transform by fast sine and cosine transforms: the Mellin connection

Fast Hankel transform by fast sine and cosine transforms: the Mellin connection

The Hankel transform of a function by means of a direct Mellin approach requires sampling on an exponential grid, which has the disadvantage of coarsely undersampling the tail of the function. A novel modified Hankel transform procedure, not requiring exponential sampling, is presented. The algorithm proceeds via a three-step Mellin approach to yield a decomposition of the Hankel transform into...

Fast Hankel Transforms

JOHANSEN, H. K., and SORENSEN, K., 1979, Fast Hankel Transforms, Geophysical Prospecting 27, 876-901. Inspired by the linear filter method introduced by D. P. Ghosh in rg7o we have developed a general theory for numerical evaluation of integrals of the Hankel type: m g(r) = Sf(A)hJ,(Ar)dh; v > I. II Replacing the usual sine interpolating function by sinsh (x) = a . sin (xx)/sinh (UTW), where th...

The Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms

It is known that the discrete Fourier transform (DFT) used in digital signal processing can be characterized in the framework of representation theory of algebras, namely as the decomposition matrix for the regular module C[Zn] = C[x]/(x − 1). This characterization provides deep insight on the DFT and can be used to derive and understand the structure of its fast algorithms. In this paper we pr...

Discrete Cosine and Sine Transforms

Fractional Cosine and Sine Transforms

The fractional cosine and sine transforms – closely related to the fractional Fourier transform, which is now actively used in optics and signal processing – are introduced and their main properties and possible applications are discussed.