Decompositions of metrizable compact spaces admittingT-systems of complex-valued functions

Upper semicontinuous decompositions into continua of a metrizable compact space admitting a Chebyshev system of continuous complex-valued functions are considered. It is proved that the cyclic elements of the Moore decomposition space can be embedded in the two-dimensional sphere. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 259–267, August, 1997. Translated by O. V. Sipacheva     

扰动Chebyshev结点上的Hermite－Fejer插值

本文发现当Ｃｈｅｂｙｓｈｅｖ结点产生某些扰动时，只要扰动量不超过，则基于扰动后的Ｃｈｅｂｙｓｈｅｖ结点的Ｈｅｒｍｉｔｅ－Ｆｅｊｅｒ插值过程仍然保持对［－１，１］上任意连续函数的一致收敛性，此外，文中还给出了这种收敛性的收敛速度估计。     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

A Numerical Approach of the sentinel method for distributed parameter systems

In this paper we consider the problem of detecting pollution in some non linear parabolic systems using the sentinel method. For this purpose we develop and analyze a new approach to the discretization which pays careful attention to the stability of the solution. To illustrate convergence properties we give some numerical results that present good properties and show new ways for building discrete sentinels.      

Spectral radial basis functions for full sphere computations

The singularity of cylindrical or spherical coordinate systems at the origin imposes certain regularity conditions on the spectral expansion of any infinitely differentiable function. There are two efficient choices of a set of radial basis functions suitable for discretising the solution of a partial differential equation posed in either such geometry. One choice is methods based on standard Chebyshev polynomials; although these may be efficiently computed using fast transforms, differentiability to all orders of the obtained solution at the origin is not guaranteed. The second is the so-called one-sided Jacobi polynomials that explicitly satisfy the required behavioural conditions. In this paper, we compare these two approaches in their accuracy, differentiability and computational speed. We find that the most accurate and concise representation is in terms of one-sided Jacobi polynomials. However, due to the lack of a competitive fast transform, Chebyshev methods may be a better choice for some computationally intensive timestepping problems and indeed will yield sufficiently (although not infinitely) differentiable solutions provided they are adequately converged.     

On a quadrature formula of Micchelli and Rivlin

Micchelli and Rivlin (1972) obtained a quadrature formula of highest algebraic degree of precision for the Fourier-Chebyshev coefficients A_n(f), which is based on the divided differences of f′ at the zeros of the Chebyshev polynomial T_n(x). We give here a simple approach to questions of this type, which applies to the coefficients in arbitrary orthogonal expansion of f. As an auxiliary result we obtain a new interpolation formula and a new representation of the Turán quadrature formula.     

Efficient modelling of rotating disks and cylinders using a cartesian grid

Calculations are presented of flow characteristics in the vicinity of disks and cylinders rotating at speeds typical of those found in modern mechatronics machinery. The rotational speeds are slow or intermittent, and the generated boundary layers are laminar and transitional. Comparison is made with existing experimental data and exact, though idealised, analytical solutions. A three-dimensional finite volume procedure with time dependence was employed as the solution method, and two grid geometries were used, namely, axisymmetric and cartesian. Use of a cartesian grid is very important, as it is compatible with the design of the interiors of mechatronics machinery, and present practice is to model these interiors with computationally economical cartesian grids. Expanding grids were generated normal to surfaces for each of the grid geometries so as to capture the thin boundary layers. To alleviate numerical difficulties, when using the cartesian geometry, an expanding and contracting grid was generated normal to the axis of the disks and cylinders with the grid spacing based on a shifted Chebyshev polynomial.     

The Minimality Properties of Chebyshev Polynomials and Their Lacunary Series

By considering four kinds of Chebyshev polynomials, an extended set of (real) results are given for Chebyshev polynomial minimality in suitably weighted Hölder norms on [–1,1], as well as (L ) minimax properties, and best L ₁ sufficiency requirements based on Chebyshev interpolation. Finally we establish best L _p , L and L ₁ approximation by partial sums of lacunary Chebyshev series of the form _i=0 a ⁱ _b ⁱ(x) where _n (x) is a Chebyshev polynomial and b is an odd integer 3. A complete set of proofs is provided.     

Interpolating polynomial wavelets on [−1,1]

The present paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. As is so often the case in classical approximation, the authors follow the pattern provided by the trigonometric polynomial case. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolation properties of de la Vallée Poussin kernels with respect to the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast discrete cosine and sine transforms. Dedicated to Prof. Guiseppe Mastroianni on the occasion of his 65th birthday.AMS subject classification 65D05, 65T60     

Symmetrized Chebyshev polynomials

We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. As a corollary we find that and are positive definite functions. We further show that a Central Limit Theorem holds for the coefficients of our polynomials.