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.     

Dirichlet characters,Gauss sums and arithmetic Fourier transforms

In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic(continuous) functions is developed based on Dirichlet characters, Gauss sums and the generalized M¨obius transform. It permits the direct extraction of the Fourier cosine and sine coefficients. Three special cases of our algorithm are presented. A VLSI architecture is presented and the error estimates are given.     

Lebesgue inequalities for poisson integrals

We obtain estimates for the deviations of the Fourier partial sums on the sets of the Poisson integrals of functions from the spaceL _p ,p≥1, that are expressed in terms of the values of the best approximations of such functions by trigonometric polynomials in the metric ofL _p . We show that the estimates obtained are unimprovable on some important functional subsets. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 798–808, June, 2000     

On sharp estimates of the convergence of double Fourier–Bessel series

The problem of approximation of a differentiable function of two variables by partial sums of a double Fourier–Bessel series is considered. Sharp estimates of the rate of convergence of the double Fourier–Bessel series on the class of differentiable functions of two variables characterized by a generalized modulus of continuity are obtained. The proofs of four theorems on this issue, which can be directly applied to solving particular problems of mathematical physics, approximation theory, etc., are presented.     

Convergence Properties of Generalized Fourier Series on a Parallel Hexagon Domain

A new Rogosinski-type kernel function is constructed using kernel function of partial sums Sn（f; t） of generalized Fourier series on a parallel hexagon domain Ω associating with threedirection partition. We prove that an operator Wn（f; t） with the new kernel function converges uniformly to any continuous function f（t） ∈ Cn（Ω） （the space of all continuous functions with period Ω） on Ω. Moreover, the convergence order of the operator is presented for the smooth approached function.     

Spaces of Functions of Generalized Bounded Variation. II: Questions of Uniform Convergence of Fourier Series

Tests are given for uniform convergence of Fourier series for spaces of functions of generalized bounded variation; along with the well-known tests (of Salem–Oskolkov–Young, Chanturiya, and Waterman) we suggest new tests. We show that the Waterman test for uniform convergence of Fourier series is strongest and unimprovable. We present a theorem on exact estimates for the Fourier coefficients for spaces of functions of bounded variation which contains classical results, improves several well-known results, and gives some new results.     

Generalized Pseudo-Butterworth Refinable Functions

In this paper we propose the generalized pseudo-Butterworth refinable functions which involve pseudo-splines of type I and II, Butterworth refinable functions, pseudo-Butterworth refinable functions, and almost all symmetric and causal fractional B-splines. Furthermore, the convergence of cascade algorithms associated with the new masks is proved, and Riesz wavelet bases in L ₂(?) corresponding to the parameters are constructed. The regularity of the generalized pseudo-Butterworth refinable functions is also analyzed by Fourier analysis.     

Some remarks concerning the Fourier transform in the space L 2(ℝ n )

For the Fourier transform in the space L ₂(?²) of square integrable multivariable functions, two practically useful estimates are proved in certain classes of functions characterized by a generalized continuity modulus.     

Approximations in spaces of locally integrable functions

We study approximations of functions from the sets $\hat L_\beta ^\psi \mathfrak{N}$ , which are determined by convolutions of the following form: $$f\left( x \right) = A_0 + \int\limits_{ - \infty }^\infty {\varphi \left( {x + t} \right)\hat \psi _\beta \left( t \right)dt, \varphi \in \mathfrak{N}, \hat \psi _\beta \in L\left( { - \infty ,\infty } \right),} $$ where η is a fixed subset of functions with locally integrablepth powers (p≥1). As approximating aggregates, we use the so-called Fourier operators, which are entire functions of exponential type ≤ σ. These functions turn into trigonometric polynomials if the function ?(·) is periodic (in particular, they may be the Fourier sums of the function approximated). The approximations are studied in the spacesL _p determined by local integral norms ∥·∥_-p . Analogs of the Lebesgue and Favard inequalities, wellknown in the periodic case, are obtained and used for finding estimates of the corresponding best approximations which are exact in order. On the basis of these inequalities, we also establish estimates of approximations by Fourier operators, which are exact in order and, in some important cases, exact with respect to the constants of the principal terms of these estimates.     

Inversion of the Short-Time Fourier Transform Using Riemannian Sums

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂ and modulation spaces. In the present note the Riemannian sums of the inverse short-time Fourier transform are investigated. Under some conditions on the window functions we prove that the Riemannian sums converge to f in the modulation spaces and inWiener amalgam norms, hence also in the L_p sense.     

Estimates of The Fourier Widths of the Classes Of Periodic Functions With Given Majorant of the Mixed Modulus of Smoothness

We obtain some order-sharp estimates for the Fourier widths of Nikol'skii–Besov and Lizorkin–Triebel function classes with given majorant of the mixed modulus of smoothness in the Lebesgue space for a few relations between the parameters of the class and the space. The upper bounds follow from estimates of the approximation of functions of these classes by special partial sums of their Fourier series with respect to the multiple system of periodized Meyer wavelets.     

On Strong Summability of Fourier Series of Summable Functions

In the metric of L, we obtain estimates for the generalized means of deviations of partial Fourier sums from an arbitrary summable function in terms of the corresponding means of its best approximations by trigonometric polynomials.     

Some estimates for the error in Fourier–Legendre expansions of functions of one variable

Some issues concerning expansions of functions in Fourier–Legendre series is considered in L₂[?1, 1]. In particular, the rate of their convergence in the classes of functions characterized by the generalized modulus of continuity are estimated, and estimates of the remainder terms are obtained.     

How well does the finite Fourier transform approximate the Fourier transform?

We show that the answer to the question in the title is “very well indeed.” In particular, we prove that, throughout the maximum possible range, the finite Fourier coefficients provide a good approximation to the Fourier coefficients of a piecewise continuous function. For a continuous periodic function, the size of the error is estimated in terms of the modulus of continuity of the function. The estimates improve commensurately as the functions become smoother. We also show that the partial sums of the finite Fourier transform provide essentially as good an approximation to the function and its derivatives as the partial sums of the ordinary Fourier series. Along the way we establish analogues of the Riemann‐Lebesgue lemma and the localization principle. © 2004 Wiley Periodicals, Inc.     

Some Fourier-type operators for functions on unbounded intervals

In order to approximate functions defined on the real line or on the real semiaxis by polynomials, we introduce some new Fourier-type operators, connected to the Fourier sums of generalized Freud or Laguerre orthonormal systems. We prove necessary and sufficient conditions for the boundedness of these operators in suitable weighted L ^p -spaces, with 1 < p < ∞. Moreover, we give error estimates in weighted L ^p and uniform norms.     

Certain weighted averages of generalized Ramanujan sums

We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Tóth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose a generalization of Ramanujan sum given by E. Cohen and derive the weighted averages corresponding to the versions of the weighted averages established by Tóth.     

On strong Carleman means of multiple trigonometric Fourier series

We prove the strong Carleman summability of the Fourier series of continuous functions on the m-dimensional torus, with partial sums constructed over polyhedra of a certain class.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 275–279, February, 1992.     

Reduction of the Gibbs phenomenon for smooth functions with jumps by the -algorithm

Recently, Brezinski has proposed to use Wynn's ε-algorithm in order to reduce the Gibbs phenomenon for partial Fourier sums of smooth functions with jumps, by displaying very convincing numerical experiments. In the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions, and obtain the rate of column convergence for such functions, possibly perturbed by another sufficiently differentiable function. We also analyze the connection to Padé–Fourier and Padé–Chebyshev approximants, including those recently studied by Kaber and Maday.     

高维广义Kuramoto-Sivashinsky型方程Fourier拟谱方法的大时间误差估计

在很多物理问题中出现如下方程:Kuramoto在研究反应扩散系统耗散结构时导出了上述方程,Sivashinsky在模拟火焰传播时也得到了它.此外,它还出现在粘性层流和Navier-Stokes方程的分枝解中.在[5-8]中,作者研究了一维情形下周期初值问题的整体吸引子和分枝解;[9]提出了广义KS型方程;[10-14]中研究了它的光滑解的存在性和t→+∞时的渐近性     

Sharp estimates for the convergence rate of Fourier series in terms of orthogonal polynomials in L 2((a, b), p(x))

Sharp estimates are given for the convergence rate of Fourier series in terms of classical orthogonal polynomials in some classes of functions characterized by a generalized modulus of continuity in the space L ₂((a, b), p(x)). Expansions in terms of Laguerre, Hermite, and Jacobi polynomials are considered.