Application of the discrete Fourier transform to solving the Cauchy integral equation

The uniquely solvable system of the Cauchy integral equation of the first kind and index 1 and an additional integral condition is treated. Such a system arises, for example, when solving the skew derivative problem for the Laplace equation outside an open arc in a plane. This problem models the electric current from a thin electrode in a semiconductor film placed in a magnetic field. A fast and accurate numerical method based on the discrete Fourier transform is proposed. Some computational tests are given. It is shown that the convergence is close to exponential.     

The differential Fourier transform method

We suggest the two new discrete differential sine and cosine Fourier transforms of a complex vector which are based on solving by a finite difference scheme the inhomogeneous harmonic differential equations of the first order with complex coefficients and of the second order with real coefficients, respectively. In the basic version, the differential Fourier transforms require by several times less arithmetic operations as compared to the basic classicalmethod of discrete Fourier transform. In the differential sine Fourier transform, the matrix of the transformation is complex,with the real and imaginary entries being alternated, whereas in the cosine transform, the matrix is purely real. As in the classical case, both matrices can be converted into the matrices of cyclic convolution; thus all fast convolution algorithms including the Winograd and Rader algorithms can be applied to them. The differential Fourier transform method is compatible with the Good–Thomas algorithm of the fast Fourier transform and can potentially outperform all available methods of acceleration of the fast Fourier transform when combined with the fast convolution algorithms.     

Fourier transform versus Hilbert transform

Certain relations between the Fourier transform of a function and the Hilbert transform of its derivative are revealed. They concern the integrability/non-integrability of both transforms. Certain applications are discussed.     

The Fourier transform of multiradial functions

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form \(\varPhi (x)=\phi (|x_1|, \dots , |x_m|), x_i\in \mathbf R^{n_i}\) , in terms of the Fourier transform of the function \(\phi \) on \(\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}\) , where \(r_i\) is either \(1\) or \(2\) .     

The Fourier inversion and the Riemann functional equation

We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.     

Ring structures and the discrete Fourier transform

Since the pioneering work of J. W. Cooley and J. W. Tukey Math. Comp.19 1965 297–301, a great deal of effort has been devoted to developing efficient algorithms for computing finite Fourier transform. Among the new methods suggested over the past several years, methods depending on ring-theoretic structures have received special attention. This approach, originally suggested by C. Rader (Proc. IEEE5, 6 1968, 1107–1108), was really developed by S. Winograd (“Arithmetic Complexity of Computations,” CBMS-NSF Regional Conference Series in Applied Mathematics 1980 into a powerful new algorithm. Winograd's real contribution was to realize that there are efficient algorithms to evaluate cyclic convolution. The purpose of this article is twofold: first, to make more explicit the interplay between ring-theoretic structures and the algorithms for the finite Fourier transform; second, to use this new insight to construct new algorithms for evaluating the finite Fourier transform on the groups $\text{Z}\text{(p}{}^{\mathrm{s}}\text{Z)}\text{⊕}\text{Z}\text{(p}{}^{\mathrm{s}}\text{Z)}\text{⊕ … ⊕}\text{Z}\text{(p}{}^{\mathrm{s}}\text{Z)}$.     

A hybrid Fourier transform

A Fourier transform akin to Sneddon's R-transform is introduced. It is shown that the Hilbert transform links the two in much the same way as it connects the classical Fourier sine and cosine transforms.     

Computation of the fractional Fourier transform

In this paper we make a critical comparison of some programs for the digital computation of the fractional Fourier transform that are freely available and we describe our own implementation that filters the best out of the existing ones. Two types of transforms are considered: first, the fast approximate fractional Fourier transform algorithm for which two algorithms are available. The method is described in [H.M. Ozaktas, M.A. Kutay, G. Bozda i, IEEE Trans. Signal Process. 44 (1996) 2141–2150]. There are two implementations: one is written by A.M. Kutay, the other is part of package written by J. O'Neill. Second, the discrete fractional Fourier transform algorithm described in the master thesis by Ç. Candan [Bilkent University, 1998] and an algorithm described by S.C. Pei, M.H. Yeh, and C.C. Tseng [IEEE Trans. Signal Process. 47 (1999) 1335–1348].     

On the analyticity of the Fourier original and Fourier transform inside alternate angles

We prove that a Fourier image is an analytic function inside two alternate angles if the Fourier preimage possesses the same property. Odessa Academy of Building and Architecture, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 703–707, May, 1999.