The Fast Fourier Transform, the FFT, is one of the most widely used algorithms in science and engineering. It is used both in signal processing and for the solution of partial differential equations. For instance, FFTs are used in so–called Fast Poisson solvers. It is also used in combination with cyclic reduction for the FACR (Fourier Analysis Cyclic Reduction) [4] method for Poisson’s equatio...

This paper derives sharp l∞ and l1 estimates of the error arising from an explicit approximation of the constant coefficient 1D convection/diffusion equation with Dirac initial data. The analysis embeds the discrete equations within a semi-discrete system of equations which can be solved by Fourier analysis. The error estimates are then obtained though asymptotic approximation of the integrals ...

Toxicity bioassays are important tools to determine biological effects of chemical agents on species. The questions remained on, what effects have been imposed on each of the different molecular site of cells by chemical exposure and how to find a pattern for chemical toxicity. To address the questions, HepG2 cell lines were exposed to the different concentrations of cisplatin for 24 hours to r...

Conventional Fourier analysis has many schemes for different types of signals. They are Fourier transform (FT), Fourier series (FS), discrete-time Fourier transform (DTFT), and discrete Fourier transform (DFT). The goal of this correspondence is to develop two absent schemes of fractional Fourier analysis methods. The proposed methods are fractional Fourier series (FRFS) and discrete-time fract...

The 2-D Fourier transform has been generalized into the 2-D separable fractional Fourier transform (replaces 1-D Fourier transform by 1-D fractional Fourier transform for each variable) and the 2-D separable canonical transform (further replaces the fractional Fourier transform by canonical transform) in [3]. It also has been generalized into the 2-D unseparable fractional Fourier transform wit...

The mathematical study of the diatonic and chromatic universes in the tradition of David Lewin (9) and John Clough (6) is a point of departure for several recent investigations. Surprisingly, Lewin’s original idea to apply finite Fourier transform to musical structures has not been further investigated for four decades. It turns out that several music-theoretically interesting properties of cer...

Suppose we have a function s(t) that measures the sound level at time t of an analog audio signal. We assume that s(t) is piecewise-continuous and of finite duration: s(t) = 0 when t is outside some interval a ≤ t ≤ b. Make a change of variable x = (t− a)/(b− a) and set f(x) = s(t). Then 0 ≤ x ≤ 1 when a ≤ t ≤ b, and f(x) is a piecewise continuous function of x. We convert f(x) into a digital s...