Generalized Hardy identity and relations to Gibbs-Wilbraham and Pinsky phenomena

We consider the Fourier series of the indicator functions of several dimensional balls. For the spherical partial sum of the Fourier series, we extract the Gibbs-Wilbraham (or Gibbs), Pinsky and the third phenomena as an extension of Hardy's identity. The third phenomenon has been shown by Kuratsubo recently. We prove the Gibbs-Wilbraham phenomenon for all dimensions and give another proof of the Pinsky phenomenon. Pinsky gave the order of the divergence for the Fourier inversion at the origin. We give the order of the divergence of the Fourier series at the origin and show that both orders coincide. We also investigate the uniform convergence for the Fourier series and the Fourier inversion.     

A summability for Meyer wavelets

The Gibbs’ phenomenon in the classical Fourier series is well-known. It is closely related with the kernel of the partial sum of the series. In fact, the Dirichlet kernel of the Fourier series is not positive. The poisson kernel of Cesaro summability is positive. As the consequence of the positiveness, the partial sum of Cesaro summability does not exhibit the Gibbs’ phenomenon. Most kernels associated with wavelet expansions are not positive. So wavelet series is not free from the Gibbs’ phenomenon. Because of the excessive oscillation of wavelets, we can not follow the techniques of the Fourier series to get rid of the unwanted quirk. Here we make a positive kernel for Meyer wavelets and as the result the associated summability method does not exhibit Gibbs’ phenomenon for the corresponding series.     

Gibbs phenomenon for Fourier series on finite sets

The Gibbs phenomenon is described for the Fourier series of a function at its jump, the function being defined along the finite circle ?/p?.The phenomenon consists in the fact, that the jump of the truncated Fourier series is greater, than that of the original function (being approximately 1+1/p times greater).     

A Sharp Error Bound in Terms of an Averaged Modulus of Smoothness for Fourier Lagrange Coefficients

This paper discusses the approximation of Fourier coefficients by Fourier Lagrange coefficients. It gives an error bound in terms of an averaged modulus of smoothness. The sharpness of this estimate is shown as an application of a quantitative resonance principle by utilizing the aliasing phenomenon that occurs in the context of discrete Fourier transformation. The scenario is used to compare the averaged modulus with classical uniform and integral moduli of smoothness.     

Gibbs phenomenon and its removal for a class of orthogonal expansions

We detail the Gibbs phenomenon and its resolution for the family of orthogonal expansions consisting of eigenfunctions of univariate polyharmonic operators equipped with homogeneous Neumann boundary conditions. As we establish, this phenomenon closely resembles the classical Fourier Gibbs phenomenon at interior discontinuities. Conversely, a weak Gibbs phenomenon, possessing a number of important distinctions, occurs near the domain endpoints. Nonetheless, in both cases we are able to completely describe this phenomenon, including determining exact values for the size of the overshoot.     

On the stability of the unsmoothed Fourier method for hyperbolic equations

Summary. It has been a long open question whether the pseudospectral Fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs. In this work we answer this question with a detailed stability analysis of prototype cases of the Fourier method. We show that due to weighted -stability, the -degree Fourier solution is algebraically stable in the sense that its amplification does not exceed . Yet, the Fourier method is weakly -unstable in the sense that it does experience such amplification. The exact mechanism of this weak instability is due the aliasing phenomenon, which is responsible for an amplification of the Fourier modes at the boundaries of the computed spectrum. Two practical conclusions emerge from our discussion. First, the Fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon. Otherwise, the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones. Second -- independent of whether smoothing was used or not, the small scale information contained in the highest modes of the Fourier solution will be destroyed by their amplification. Happily, with enough resolution nothing worse can happen. Received December 14, 1992/Revised version received March 1, 1993     

Gibbs phenomenon removal by adding Heaviside functions

We define a kind of spectral series to filter off completely the Gibbs phenomenon without overshooting and distortional approximation near a point of discontinuity. The construction of this series is based on the method of adding the Fourier coefficients of a Heaviside function to the given Fourier partial sums. More precisely, we prove the uniform convergence of the proposed series on the class of piecewise smooth functions. Also, we attach two numerical examples which illustrate the uniform convergence of the suggested series in comparison with the Fourier partial sums.     

Some relationships for the double modified generalized analytic function space Fourier–Feynman transform and its applications

In this paper we first introduce the concept of a double modified analytic function space Fourier–Feynman transform using the double modified analytic function space integral. We then proceed to establish the existence of the modified analytic function space Fourier–Feynman transform for all functionals in the Banach algebra. Finally we use this double modified analytic function space transform to explain various physical phenomenon.     

Paley�CWiener Functions with a Generalized Spectral Gap

It is well known that real functions whose Fourier transform vanishes around the origin must have many sign changes. We show that a similar phenomenon occurs for real Paley–Wiener functions whose Fourier transform is “smoother” at the origin than elsewhere. We also show that if the Fourier transform of a function is “less smooth” in a neighborhood of the origin than elsewhere, then the function cannot have too many real zeros.     

Continuous and Discrete Nonlinear Approximations Based on Fourier Series

Continuous and discrete nonlinear extensions of Fourier seriesare derived by the method of generating functions, and examplesof their use are presented. Gibbs' phenomenon of the continuousnonlinear approximation is investigated.     

Gibbs-Wilbraham splines

If a function with a jump discontinuity is approximated in the norm ofL ²[–1,1] by a periodic spline of orderk with equidistant knots, a behavior analogous to the Gibbs-Wilbraham phenomenon for Fourier series occurs. A set of cardinal splines which play the role of the sine integral function of the classical phenomenon is introduced. It is then shown that ask becomes large, the phenomenon for splines approaches the classical phenomenon.Communicated by Ronald A. DeVore.     

Random Homeomorphisms and Fourier Expansions

We prove that a random change of variable in general improves convergence properties of the Fourier expansions, and we give a precise quantitative estimate of the phenomenon. Submitted: December 1997, final version: May 1998     

Periods and Young’s diagram of Fermat–Euler’s geometrical progressions of residues

The Gibbs phenomenon is described for the Fourier series of a function at its jump, the function being defined along the finite circle ℤ/pℤ.     

Lanczos-Like σ-Factors for Reducing the Gibbs Phenomenon in General Orthogonal Expansions and Other Representations

The first attempt for reducing the Gibbs phenomenon in an orthogonalexpansion, besides the usual one of Fourier series, is due to Cooke in1927–1928 for the Fourier Bessel series.However, his work was limited tothe well-known Fejer averaging of the series. For the past 10 years or so,we have tried a parallel to the more effective Lanczos-type localaveraging method of the Fourier series. As expected, such efforts werehindered by the lack of realizable tools for the general orthogonalexpansion that parallels the familiar simple ones of the Fourier seriesand transforms. During the past 3 years, we have succeeded in developinga simple direct method of filtering the Gibbs phenomenon in Fourier--Besselseries, Hankel transforms representation, and a number of orthogonalpolynomials series expansions. This parallels the equivalent result ofLanczos, which he obtained for his local averaging with the help ofthe (Fourier) convolution theorem.     

Fourier-Padé approximations and filtering for spectral simulations of an incompressible Boussinesq convection problem

In this paper, we present rational approximations based on Fourier series representation. For periodic piecewise analytic functions, the well-known Gibbs phenomenon hampers the convergence of the standard Fourier method. Here, for a given set of the Fourier coefficients from a periodic piecewise analytic function, we define Fourier-Padé-Galerkin and Fourier-Padé collocation methods by expressing the coefficients for the rational approximations using the Fourier data. We show that those methods converge exponentially in the smooth region and successfully reduce the Gibbs oscillations as the degrees of the denominators and the numerators of the Padé approximants increase.

Numerical results are demonstrated in several examples. The collocation method is applied as a postprocessing step to the standard pseudospectral simulations for the one-dimensional inviscid Burgers' equation and the two-dimensional incompressible inviscid Boussinesq convection flow.

     

Pseudospectral time-domain method for pressure-release rough surface scattering using a surface transformation and image method

When numerically analyzing acoustic scattering at a pressure-release rough surface, the conventional pseudospectral time domain (PSTD) method using Fourier transform requires rigorous stability conditions in order to solve the spatial derivative in the wave equation on the irregular boundaries between the two media due to the Gibbs phenomenon and short wavelength in air. To eliminate such disadvantages, a new algorithm is proposed based on the Fourier PSTD method utilizing a surface boundary transformation and an image method. Irregular surface boundaries are flattened by transformation and then an image method is applied to the half-space domain. The efficiency and accuracy of the proposed PSTD method are better than the conventional Fourier PSTD method. Numerical results are presented for a sloped and a sinusoidal pressure-release surface.     

Manipulating Gibbs'' Phenomenon for Fourier Interpolation

The Fourier interpolation polynomials of a periodic function with an isolated jump discontinuity at a node exhibit for growing order a Gibbs phenomenon. By a suitable definition of the function value at the jump the over- and undershoots on one side may be minimized.     

On inverse methods for the resolution of the Gibbs phenomenon

When Fourier expansions, or more generally spectral methods, are used for the representation of nonsmooth functions, then one has to face the so-called Gibbs phenomenon. Considerable progresses have been made these last years to overcome the Gibbs phenomenon, using direct or inverse approaches, both in the discrete or continuous framework. A discrete inverse method for the global or local reconstruction of a non-smooth function starting from its oscillating (trigonometric) polynomial interpolant is introduced and both its capabilities and limits are emphasized.     

Salem sets in local fields, the Fourier restriction phenomenon and the Hausdorff-Young inequality

We establish the existence of Salem sets in the ring of integers of any local field and study the Fourier restriction phenomenon on such sets. Optimal extension of the Hausdorff-Young inequality, initially attained for the torus by G. Mockenhaupt and W. Ricker, is also established in the local field setting.     

Generalized Gibbs phenomenon for Fourier partial sums and de la Vallée-Poussin sums

It is well-known that the Fourier partial sums of a function exhibit the Gibbs phenomenon at a jump discontinuity. We study the same question for de la Vallée-Poussin sums. Here we find a new Gibbs function and a new Gibbs constant. When the function is continuous, a behavior similar to the Gibbs phenomenon also occurs at a kink. We call it the “generalized Gibbs phenomenon”. Let $F_{n}(x):=\frac{k_{n}(g,x)-g(x)}{k_{n}(g,x_{0})-g(x_{0})}$ , where x ₀ is a kink and where k _n (g,x) represents Fourier partial sums and de la Vallée-Poussin sums. We show that F _n (x) exhibits the “generalized Gibbs phenomenon”. New universal Gibbs functions for both sums are derived.