A method is described for validating fast Fourier transforms (FFTs) based on the use of simple input fiinctions whose discrete Fourier transforms can be evaluated in closed form. Explicit analytical results are developed for one-dimensional and two-dimensional discrete Fourier transforms. The analytical results are easily generalized to higher dimensions. The results offer a means for validatin...

We present the role that spectral methods play in the development of the most impressive quantum algorithms, such as the polynomial time number factoring algorithm by Shor. While the fast transform algorithms reduce the number of operations needed in obtaining the transforms from O(2) to O(n2), quantum transforms are in comparison super-fast. The Quantum Fourier Transform can be performed in O(...

We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit exp...

Based on the recently introduced fractional Taylor’s formula, a fractional heat conduction constitutive equation is formulated by expanding the single-phase lag model using the fractional Taylor’s formula. Combining with the energy balance equation, the derived fractional heat conduction equation has been shown to be capable of modeling diffusion-to-Thermal wave behavior of heat propagation by ...

Abstract. This paper introduces a new fast algorithm for uniform-resolution filtering of functions defined on the sphere. We use a fast summation algorithm based on Nonequispaced Fast Fourier Transforms, building on previous work that used Fast Multipole Methods. The resulting algorithm performs a triangular truncation of the spectral coefficients while avoiding the need for fast spherical Four...

SnFFT is an easy to use software library written in the Julia language to facilitate Fourier analysis on the symmetric group (set of permutations) of degree n, denoted Sn and make it more easily deployable within statistical machine learning algorithms. Our implementation internally creates the irreducible matrix representations of Sn, and efficiently computes fast Fourier transforms (FFTs) and...