A polynomial approximation for arbitrary functions

We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, for approximating arbitrary functions. We show that the polynomial coefficients in the Legendre expansion, and thus, the whole series, converge to zero much more rapidly compared to those in the Taylor expansion of the same order. Furthermore, using numerical analysis with a sixth-order polynomial expansion, we demonstrate that the Legendre polynomial approximation yields an error at least an order of magnitude smaller than that of the analogous Taylor series approximation. This strongly suggests that Legendre expansions, instead of Taylor expansions, should be used when global accuracy is important.     

On a Result of Telyakovskii and Multiple Hilbert Transforms with Polynomial Phases

In this paper, we prove a multiple analog of the theorem proved by Arkhipov and the author in 1987, which provides an estimate for the discrete Hilbert transform with polynomial phase. For the linear case, the corresponding estimates of the sum of multiple trigonometric series was proved by Telyakovskii.     

Extreme value theory for stochastic integrals of Legendre polynomials

We study in this paper the extremal behavior of stochastic integrals of Legendre polynomial transforms with respect to Brownian motion. As the main results, we obtain the exact tail behavior of the supremum of these integrals taken over intervals [0,h] with h>0 fixed, and the limiting distribution of the supremum on intervals [0,T] as T→∞. We show further how this limit distribution is connected to the asymptotic of the maximally selected quasi-likelihood procedure that is used to detect changes at an unknown time in polynomial regression models. In an application to global near-surface temperatures, we demonstrate that the limit results presented in this paper perform well for real data sets.     

求解混合三角多项式方程组的直接GBQ方法

许多科学与工程领域,我们经常需要求混合三角多项式方程组的全部解.一般来说,混合三角多项式方程组可以通过变量替换及增加二次多项式转化为多项式方程组,进而利用数值方法进行求解,但这种转化会增大问题的规模从而增加计算量.在本文中,我们不将问题转化,考虑利用直接同伦方法求解,并给出基于GBQ方法构造的初始方程组及同伦定理的证明.数值实验结果表明我们构造的直接同伦方法较已有的直接同伦方法更加有效.     

A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators

We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p − 1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of −1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums, and orthogonality of trigonometric functions.     

Control curves and knot insertion for trigonometric splines

We introduce control curves for trigonometric splines and show that they have properties similar to those for classical polynomial splines. In particular, we discuss knot insertion algorithms, and show that as more and more knots are inserted into a trigonometric spline, the associated control curves converge to the spline. In addition, we establish a convex-hull property and a variation-diminishing result.     

Proof of two conjectures of Móricz on double trigonometric series

We use a unified approach to obtain several integrability theorems of Boas [R.P. Boas, Integrability of trigonometric series, I, Duke Math. J. 18 (1951) 787-793]. In particular, we settle two conjectures of Móricz [F. Móricz, On the integrability of double cosine and sine series, II, J. Math. Anal. Appl. 154 (1991) 466-483] concerning double trigonometric series.     

Readjustment of the Paper [J.Kaur and S.S.Bhatia,Integrability and L~1-Convergence of Double Cosine Trigonometric Series,Anal.Theory Appl.,27(1)(2011),pp.32–39.]

In this paper, we show that new modified double cosine trigonometric sums introduced in [1] are inappropriate, the class of double sequences Jd introduced there is unusable for such sums and consequently the results obtained in it are completely incorrect. We here introduce appropriate modified double cosine trigonometric sums making the class Jd usable considering a particular double cosine trigonometric series.     

Measuring the efficiency of trigonometric series estimates of a density

Some types of density estimators, particularly those based on trigonometric series, converge reasonably quickly to their limit except in the neighbourhood of one or two singularities. In this situation the mean integrated square error, the traditional measure of the efficiency of a density estimator, is an unsatisfactory measure. The notion of partial mean integrated square error is introduced and used to compare the performance of trigonometric series estimators. The results lead to consideration of some new estimators which have excellent properties from the points of view of both efficiency and ease of computation.     

Shifted Legendre direct method for variational problems

The shifted Legendre polynomial series is employed to solve variational problems. The solution is carried out by using an operational matrix for integrating the shifted Legendre polynomial vector. Variational problems are reduced to solving algebraic equations. Two illustrative examples are given, and the computational results obtained by Legendre series direct method are compared with the exact solutions.     

On orthogonal polynomial approximation with the dimensional expanding technique for precise time integration in transient analysis

We use four orthogonal polynomial series, Legendre, Chebyshev, Hermite and Laguerre series, to approximate the non-homogeneous term for the precise time integration and incorporate them with the dimensional expanding technique. They are applied to various structures subjected to transient dynamic loading together with Fourier and Taylor approximation proposed in previous works. Numerical examples show that all six methods are efficient and have reasonable precision. In particular, Legendre approximation has much higher precision and better convergence; Chebyshev approximation is also good, but only slightly inferior to Legendre approximation. The other four approximation methods usually produce results with errors hundreds of thousands of times larger. Hermite and Laguerre approximation may be useful for some special non-homogeneous terms, but do not work sufficiently well in our numerical examples. Other contributions of this paper include, a Dynamic Programming scheme for computing series coefficients, a general formula to find the assistant matrix for any polynomial series.     

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [−1, 1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.     

On the Lebesgue summability of double trigonometric series

We recall that the Lebesgue summability of a single trigonometric series is defined in terms of the symmetric differentiability of the sum of the formally integrated trigonometric series in question. In this paper, we present another proof of the theorem given in Zygmund's monograph. Then we define the notion of Lebesgue summability of a double trigonometric series and extend the theorem of Fatou and Zygmund from single to double trigonometric series.     

The characteristic polynomial of a multiarrangement

Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates ‘global’ data to ‘local’ data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.     

THE FRACTAL GEOMETRY AND TRIGONOMETRIC SERIES

Mathemstics is used to study the nature. Straight lines, circles, ellipses,continuous and differentiable curves and surfaces etc. are the first approximations of forms of concrete objects. But in reality, these forms are very irregular. Consequentily B. Mandebrot introduces since 1975 fractals and the fractal geometry to study the second approximaions of such forms. Si     

A Condition for Convergence in Mean of Trigonometric Series

We construct an example of a trigonometric series showing that a certain condition closely related to convergence in mean of trigonometric series cannot be sharpened.     

On the detection of singularities of a periodic function

We discuss the problem of detecting the location of discontinuities of derivatives of a periodic function, given either finitely many Fourier coefficients of the function, or the samples of the function at uniform or scattered data points. Using the general theory, we develop a class of trigonometric polynomial frames suitable for this purpose. Our methods also help us to analyze the capabilities of periodic spline wavelets, trigonometric polynomial wavelets, and some of the classical summability methods in the theory of Fourier series. This revised version was published online in June 2006 with corrections to the Cover Date.     

Interlacing of Hurwitz series

In this paper, we introduce the notion of interlacing of Hurwitz series. We begin by reviewing some important properties of the ring of Hurwitz series over a commutative ring A of arbitrary characteristic, and we introduce and investigate properties of the maps exp and log. We show that solutions of linear homogeneous differential equations with constant coefficients from the ring A can be described simply as interlacings of solutions of a first order system of differential equations. We give several examples to illustrate this result, and we conclude by defining and investigating properties of trigonometric functions using interlacings of Hurwitz series.     

Trigonometric series with general monotone coefficients

We study trigonometric series with general monotone coefficients. Convergence results in the different metrics are obtained. Also, we prove a Hardy-type result on the behavior of the series near the origin.     

Method of summation of some slowly convergent series

A new method of summation of slowly convergent series is proposed. It may be successfully applied to the summation of generalized and basic hypergeometric series, as well as some classical orthogonal polynomial series expansions. In some special cases, our algorithm is equivalent to Wynn’s epsilon algorithm, Weniger transformation [E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10 (1989) 189-371] or the technique recently introduced by ?í?ek et al. [J. ?í?ek, J. Zamastil, L. Skála, New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field, Journal of Mathematical Physics 44 (3) (2003) 962-968]. In the case of trigonometric series, our method is very similar to the Homeier’s H transformation, while in the case of orthogonal series — to the K transformation. Two iterated methods related to the proposed method are considered. Some theoretical results and several illustrative numerical examples are given.