This paper is part 2 in a series of papers about the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). The focus of this paper is on a fast implementation of the DFT, called the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform). The implementation is based on a well-known algorithm, called the Radix 2 FFT, and requires that its' input d...

Legendre coefficients of an integrable function f(x) are proved to coincide with the Fourier a nonnegative index suitable Abel-type transform itself. The numerical computation N can thus be carried out efficiently in O(NlogN) operations by means single fast f(x). Symmetries associated exploited further reduce computational complexity. dual problem calculating sum expansions at prescribed set po...

We consider a forced differential difference equation and by the use of Laplace Transform Theory generate non-hypergeometric type series which we prove may be expressed in closed form.

This paper is part 3 in a series of papers about the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). The focus of this paper is on computing the Power Spectral Density (PSD) of the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform). The implementation is based on a well-known algorithm, called the decimation in time Radix 2 FFT, and re...

Given a multivariate compactly supported distribution φ, we derive here a necessary and sufficient condition for the global linear independence of its integer translates. This condition is based on the location of the zeros of φ̂ = The Fourier-Laplace transform of φ. The utility of the condition is demonstrated by several examples and applications, showing in particular, that previous results on...

In this paper, we generalize the windowed Fourier transform to the windowed linear canonical transform by substituting the Fourier transform kernel with the linear canonical transform kernel in the windowed Fourier transform definition. It offers local contents, enjoys high resolution, and eliminates cross terms. Some useful properties of the windowed linear canonical transform are derived. Tho...

In this paper, the lognormal distribution is studied, and a new series representation proposed. This uses powers of bilinear function. From it, simplified form obtained used to compute Laplace transform distribution.

This work is devoted to the nondestructive evaluation of materials using stimulated infrared thermography. The proposed approach provides simple analytical solutions to evaluate the lateral extent, and the thickness of a plane flaw in a three-dimensional (3D) heat transfer configuration. It is based on the application of a Laplace transform on the time variable t, then a double Fourier transfor...