A general analytical scheme for computing the n-fold convolution of exponentialsum distribution functions has been developed in this paper. The n-fold convolution is first expressed by multiple sums of recursive integrals. These recursive integrals are then reconstructed with a series of delta functions to avoid separations of integrations. Part of the recursive integrals has been solved analyt...

A new approach to derive transparent boundary conditions (TBCs) for wave, Schrödinger, heat and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform of the exterior solution. To obtain a numerical algorithm, a Möbius transform is appli...

This application note describes the basics of the Fast Fourier Transform and how to use a Digital Signal Processing Library. The library and design example use Microchip’s dsPIC30F Series Digital Signal Controller, specifically the dsPIC30F3013. The Transform Analysis discusses the differences between the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT), and how the FFT is ...

Algorithms for the rapid computation of the forward and inverse discrete Fourier transform for points which are nonequispaced or whose number is unrestricted are presented. The computational procedure is based on approximation using a local Taylor series expansion and the fast Fourier transform (FFT). The forward transform for nonequispaced points is computed as the solution of a linear system ...

This paper looks at one of the jewels of mathematics, the Fourier series / transform and briefly surveys its applications and looks at possible ways of enhancing the performance of the fast Fourier transform

In the present study, the boundary value problems in generalized thermodiffusive elastic medium has been investigated as a result of inclined load. The inclined load is assumed to be a linear combination of normal load and tangential load. Laplace transform with respect to time variable and Fourier transform with respect to space variable are applied to solve the problem. As an application of t...

Recently, regime shifts or structure breaks had acquired very high attention in analyzing financial time series data. Abrupt of changes in the government policy, financial crises and many of challenges lead to change in the behavior of the financial time series data. In addition, wavelet transform also becomes very famous in the financial sector and it has better advantages than the other filte...

Consider a Markov process with countably many states. In order to find a one-state occupation time distribution, we use a combination of Fourier and Laplace transforms in the way that allows for the inversion of the Fourier transform. We derive a closed-form expression for the occupation time distribution in the case of a simple continuous time random walk on Z and represent the one state occup...

We consider the problem of convergence of Fourier series when we make a change of variable. Under a certain reasonable hypothesis, we give a necessary and sufficient condition for a homeomorphism of the circle to transform absolutely convergent Fourier series into uniformly convergent Fourier series.

Previous research has shown that computation of convolution in the frequency domain provides a significant speedup versus traditional convolution network implementations. However, this performance increase comes at the expense of repeatedly computing the transform and its inverse in order to apply other network operations such as activation, pooling, and dropout. We show, mathematically, how co...