Inequalities for Alternating Trigonometric Sums

In 1970, J.B. Kelly proved that $$\begin{array}{ll}0 \leq \sum\limits_{k=1}^n (-1)^{k+1} (n-k+1)|\sin(kx)| \quad{(n \in \mathbf{N}; \, x \in \mathbf{R})}.\end{array}$$ We generalize and complement this inequality. Moreover, we present sharp upper and lower bounds for the related sums $$\begin{array}{ll} & \sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1) | \cos(kx) | \quad {\rm and}\\ & \quad{\sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1)\bigl( | \sin(kx) | + | \cos(kx)| \bigr)}.\end{array}$$      

Three Problems on Trigonometric Sums

Let Λ ? R~n be a uniformly discrete set and let C_Λ be the vector space consisting of all mean periodic functions whose spectrum is simple and contained in Λ. If Λ is a gentle set then for every f ∈ C_Λ we have f(x) = O(ω_Λ(x)) as |x| →∞ and ω_Λ(x) can be estimated(Theorem 4.1). This line of research was proposed by Jean-Pierre Kahane in 1957.     

On Zeros of Real Trigonometric Sums

The problem of calculating the number of zeros of a real trigonometric sum of an arbitrary form on a given interval is considered. Upper and lower bounds for this number are obtained by using the argument principle and are illustrated by examples.     

Lattice Paths and Positive Trigonometric Sums

A trigonometric polynomial generalization to the positivity of an alternating sum of binomial coefficients is given. The proof uses lattice paths, and identifies the trigonometric sum as a polynomial with positive integer coefficients. Some special cases of the q -analogue conjectured by Bressoud are established, and new conjectures are given. January 22, 1997. Date revised: July 9, 1997.     

Bernstein Fractal Trigonometric Approximation

Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein \(\alpha \)-fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein \(\alpha \)-fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function \(f\) defined on \([-l,l]\). The Bernstein fractal Fourier series converges to \(f\) even if the magnitude of the scaling factors does not approach zero. Existence of the \(\mathcal{C}^{r}\)-Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of \(\mathcal{C}^{r}\)-Bernstein fractal functions.

     

Pairs of Lattice Paths and Positive Trigonometric Sums

Ismail et al. (Constr. Approx. 15:69–81, 1999) proved the positivity of some trigonometric polynomials with single binomial coefficients. In this paper, we prove some similar results by replacing the binomial coefficients with products of two binomial coefficients.     

Approximation of Periodic Analytic Functions by Interpolation Trigonometric Polynomials

We obtain asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials on classes of convolutions of periodic functions admitting a regular extension to a fixed strip of the complex plane.     

The Bernoulli Spline and Approximation by Trigonometric Blending Polynomials

Using an extension of the notion of Bernoulli spline to a multivariate setting, a Jackson-Favard estimate is derived for approximation of 2π-periodic test functions by trigonometric blending polynomials. Techniques involved in the proof include properties of periodic distributions and of their Fourier transforms. The usual Jackson-Favard estimates from the literature can be derived from our result if limits of test functions are considered.     

Chebyshev Approximation to Data by Positive Sums of Exponentials

The problem is considered of calculating Chebyshev approximationsto given data by sums of exponentials with positive coefficients,where the number of terms in the sum has to be obtained as partof the process. An exchange procedure based on linear programmingis developed for the estimation of the exponents, and this ismade efficient by the use of postoptimality theory and the applicationof the dual simplex algorithm. Rapid convergence to a best approximationcan then be obtained by the application of Newton's method tothe characterization conditions interpreted as a nonlinear systemof equations. The Newton step can be determined through thesolution of a quadratic programming problem, and advantage istaken of the structure so that the calculation can be simplifiedwithout inhibiting a second-order convergence rate. Numericalresults are presented for the application of an algorithm basedon these ideas to a number of data sets which have appearedin the literature.     

一类三角插值多项式在Besov空间中的逼近

利用K泛函的定义首次研究了在Besov空间中,一类三角插值多项式的逼近和饱和问题,确定了逼近的饱和类与饱和阶.     

On Nonlinear Kloosterman Sums

Doklady Mathematics - Additive shift is a widely used tool for estimating exponential sums and character sums. According to it, the summation variable n is replaced by an expression of the type n +...     

Greedy Algorithm and m -Term Trigonometric Approximation

We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form , where is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients of function f . We compare the efficiency of this method with the best m -term trigonometric approximation both for individual functions and for some function classes. It turns out that the operator G _m provides the optimal (in the sense of order) error of m -term trigonometric approximation in the L _p -norm for many classes. September 23, 1996. Date revised: February 3, 1997.     

Approximation of Convolution Classes by Fourier Sums. New Results

We present a survey of new results related to the investigation of the rate of convergence of Fourier sums on the classes of functions defined by convolutions whose kernels have monotone Fourier coefficients.     

On Simultaneous Approximation by Iterated Boolean Sums of Bernstein Operators

The paper presents upper estimates of the error of weighted and unweighted simultaneous approximation by the Bernstein operators and their iterated Boolean sums. The estimates are stated in terms of the Ditzian–Totik modulus of smoothness or appropriate K-functionals.     

Approximation by Partial Sums of Fourier Series and the Best Approximation of Certain Function Classes

Continuing investigations of S. A. Telyakovskii, S. B. Stechkin, A. I. Stepanets and R. M. Trigub, the author studies two questions about approximation of the class of periodic functions with a bounded -derivative by polynomials in the spaces C and L.First, the asymptotics of approximation by partial sums of Fourier series has been found without limitations for the decreasing order of . Second, the exact decreasing order of the best approximation of the class has been found. New examples are also given.     

Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums

Journal of Fourier Analysis and Applications - In this paper, we study the convergence of adaptive Fourier sums for real-valued $$2\pi $$ -periodic functions. For this purpose, we approximate the...     

一类三角级数的最佳逼近度

本文在复值连续函数空间中给出一类具有O-正则变化拟单调系数之三角级数的最佳逼近度的一个完备结果.特别是第3节的定理3.2,即使在实连续函数空间中亦是一个新的结论,而且也是Xie与Zhou[6]的定理2在一个方向上的本质推广.最后,在第4节给出一个应用.