In this paper we will discuss the version of the Fourier analysis and pseudodifferential operators on the torus. Using the toroidal Fourier transform we will show several simplifications of the standard theory. We will also discuss the corresponding toroidal version of Fourier integral operators. To distinguish them from those defined using the Euclidean Fourier transform, we will call them Fou...

Abstract. In this article, we derive a series expansion of the multivariate normal probability integrals based on Fourier series. The basic idea is to transform the limits of each integral from hi to ∞ to be from −∞ to ∞ by multiplying the integrand by a periodic square wave that approximates the domain of the integral. This square wave is expressed by its Fourier series expansion. Then a Chole...

We generalize the windowed Fourier transform to the variable-windowed Fourier transform. This generalization brings the Gabor transform and the wavelet transform under the same framework. Using frame theory we characterize frames and orthonormal bases for the variable windowed Fourier series (VWFS). These characterizations are formulated explicitly in terms of window functions. Therefore they c...

This paper is part 4 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 spectral leakage. Spectral leakage applies to all forms of DFT, including the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform). We demonstrate an approach to mitigating spectral leakage based on windowing...

Abstract The conventional Fourier transform has a well-known uncertainty relation that is defined in terms of the first and second moments of both a function and its Fourier transform. It is also well known that Gaussian functions, when translated to an arbitrary centre and supplemented by a linear phase factor, provide a complete set of minimum uncertainty states (MUSs) that exactly satisfies ...

In the paper, a technique of the numerical inversion of multidimensional Laplace transforms (nD NILT), based on a complex Fourier series approximation is elaborated in light of a possible ralative error achievable. The detailed error analysis shows a relationship between the numerical integration of a multifold Bromwich integral and a complex Fourier series approximation, and leads to a novel f...

1 For multiple Fourier integrals in the place of Fourier series this formula has been previously estabLished and made use of in the paper: S. Bochner, "Ein Konvergenzsatz fur mehrvariablige Fouriersche Integrale," Math. Zeit., 34, 440-447 (1931); see also A. C. Berry, "The Fourier Transform Theorem," Jour. Math. Phys. Mass. Inst. Technology, 8, 106-118 (1929). 2 See Bochner, S., "Properties of ...

The majority of musical feature extraction applications are based on the Fourier transform in various disguises. This is despite the fact that this transform is subject to a series of restrictions, which admittedly ease the computation and interpretation of transform coefficients, but also imposes arguably unnecessary limitations on the ability of the transform to extract and identify features....

Fourier analysis started even before Fourier, but in the context of Fourier’s work on the Analytic Theory of Heat, it began with a claim regarding expansions of functions as trigonometrical series, what we now call Fourier series. Such expansions form the framework for writing down solutions to the heat equation in terms of initial temperature distributions. [refer to Enrique A. Gonzalez-Velasc...

In this report, we propose a novel automatic and computationally efficient method of Fourier imaging analysis using Fourier transform. Besides Fourier transform’s many applications, one can use Fourier transform to select significant frequencies of an observed noisy signal, which can be applied as a model selection tools of (weighted) Fourier series analysis of medical images. Both simulated da...