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This paper is concerned with the Cauchy problem connected with the Helmholtz equation. We investigate the denseness of Fourier–Bessel functions and propose a numerical method for approximating the solution to the problem. The convergence and stability are analyzed with truncated singular value decomposition method. Numerical experiments are also presented to show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Abstract: Two-dimensional hypersingular equations over a disc are considered. A spectral method is developed, using Fourier series in the azimuthal direction and orthogonal polynomials in the radial direction. The method is proved to be convergent. Then, Tranter’s method is discussed. This method was devised in the 1950s to solve certain pairs of dual integral equations. It is shown that this method is also convergent because it leads to the same algebraic system as the spectral method. [Copyright &y& Elsevier]

Abstract: Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. When constructed in a particular manner, Fourier extensions share many of the same features of a standard Fourier series. In particular, one can compute Fourier extensions which converge spectrally fast whenever the function is smooth, and geometrically fast if the function is analytic, much the same as the Fourier series of a smooth/analytic and periodic function. With this in mind, the purpose of this paper is to describe, analyse and explain the observation that Fourier extensions, much like classical Fourier series, also have excellent resolution properties for representing oscillatory functions. The resolution power, or required number of degrees of freedom per wavelength, depends on a user-controlled parameter and, as we show, it varies between 2 and . The former value is optimal and is achieved by classical Fourier series for periodic functions, for example. The latter value is the resolution power of algebraic polynomial approximations. Thus, Fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation. [Copyright &y& Elsevier]

Abstract: We discuss the numerical computation of the cosine lemniscate function and its inverse, the lemniscate integral. These were previously studied by Bernoulli, Euler, Gauss, Abel, Jacobi and Ramanujan. We review general elliptic formulas for this special case and provide some novelties. We show that a Fourier series by Ramanujan converges twice as fast as the standard elliptic cosine Fourier series specialized to this case. The so-called imbricate series, however, converges geometrically fast over the entire complex plane. We derive two new expansions. The rational-plus-Fourier series converges much faster than Ramanujan’s: for real : each term is asymptotically 12,400 times smaller than its immediate predecessor: where is the elliptic nome, is the complete elliptic integral of the first kind for a modulus and . The rational imbricate series is uniformly valid over the complex plane, but converges twice as fast as the sech-imbricate series: where is the period in both the real and imaginary directions. We devise a new approximation for the lemniscate integral for real argument as the arccosine of a Chebyshev series and show that 17 terms yield about 15 digits of accuracy. For complex argument, we show that the lemniscate integral can be found to near machine precision (assumed as sixteen decimal digits) by computing the roots of a polynomial of degree thirteen. Alternatively, Newton’s iteration converges in three iterations with an initialization, accurate to four decimal places, that is the chosen root of a cubic equation. [ABSTRACT FROM AUTHOR]

Abstract: For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in is the celebrated Carleson theorem, proved in 1966 (and extended to by Hunt in 1967). In this paper, we take the system (with being the Bessel function of the first kind and of the order ), which is orthonormal in , and whose Fourier series are the so-called Fourier–Neumann series. We study the almost everywhere convergence of Fourier–Neumann series for functions in and we show that, surprisingly, the proof is relatively simple (inasmuch as the mean convergence has already been established). [Copyright &y& Elsevier]

Abstract: We discuss the effect of a particular sequence acceleration method, the process, on the partial sums of Fourier series. We show that for a very general class of functions, this method fails on a dense set of points; not only does it not speed up convergence, it turns the sequence of partial sums into a sequence with multiple limit points. [Copyright &y& Elsevier]