The Fourier Transform is an important mathematical tool in a wide variety of fields of science and engineering [1, p. XI]. In this paper, by the use of some integral identities and inequalities developed in [2](see also [3]), we point out some approximations of the Fourier transform in terms of the complex exponential mean, E (z, w) (see Section 2) and study the error of approximation for diffe...

The novel methods for binary discrete Fourier transform (DFT) computation over the finite field have been proposed. are based on a trace calculation and use cyclotomic DFT. direct DFT computational complexity has reduced due to using function functions of which stored in small tables. inverse representation elements with respect normal basis. proposed can be used encoding/decoding subfield subc...

There are numerous directions from which one can approach the subject of the fast Fourier Transform (FFT). It can be explained via numerous connections to convolution, signal processing, and various other properties and applications of the algorithm. We (along with Geddes/Czapor/Labahn) take a rather simple view from the algebraic manipulation standpoint. As will be apparent shortly, we relate ...

Abstract. A complete set of d + 1 mutually unbiased bases exists in a Hilbert spaces of dimension d, whenever d is a power of a prime. We discuss a simple construction of d+1 disjoint classes (each one having d−1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. Such a construction works properly for prime dimension. We investigate an alternative construc...

Let Z/n denote the integers mod n and let % denote the finite Fourier transform on L\Z/n). We let ©2$), » F operate on ©2L2(Z/n) and show that ©2L2(Z/n) can be given a graded algebra structure (with no zero divisors) such that f(/g) ■" ^U)^(g). We do this by establishing a natural isomorphism with the algebra of theta functions with period i. In addition, we find all algebra structures on 02L2(...

The finite Fourier transform of a family of orthogonal polynomials is the usual transform of these polynomials extended by 0 outside their natural domain of orthogonality. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families. 2010 Mathematics subject classification: primary 33C45; secondary 44A38, 33C47, 33C10, 42C10.

We present an in-depth study of the Quantum Fourier Transform for finite groups and the underlying mathematics. The study includes a look at the most salient results linking the Quantum Fourier Transform to the Hidden Subgroup Problem. This provides a useful context for determining the extent to which the Fourier transform can serve to recognize periodicity of a function on a finite group. Résu...

There are numerous directions from which one can approach the subject of the fast Fourier Transform (FFT). It can be explained via numerous connections to convolution, signal processing, and various other properties and applications of the algorithm. We (along with Geddes/Czapor/Labahn) take a rather simple view from the algebraic manipulation standpoint. As will be apparent shortly, we relate ...