On modified Chebyshev polynomials

Two elegant representations are derived for the modified Chebyshev polynomials discussed by Witula and Slota [R. Witula, D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl. 324 (2006) 321-343].     

子孔径拼接法检测大口径光学镜面的精度分析

讨论了在用子孔径拼接法检测大口径光学镜面时的精度问题，并分别从理论上和实验中对拼接检验的精度做了定量分析。分别用平面消差、在非圆域内的Zemike多项式系数拟合和在圆域内的Zemike多项式系数拟合等三种方法对检测得到的波面面形数据进行数据处理。通过对这三种方法计算结果的分析，发现利用平面消差法拟合波面数据能更好地保存原始波面信息，且拟合误差较小。最后通过对实验数据的分析和处理验证了平面消差法拟合得到的波面更接近原始波面，拼接效果较好。     

Pietsch's factorization theorem for dominated polynomials

We prove that, like in the linear case, there is a canonical prototype of a p-dominated homogeneous polynomial through which every p-dominated polynomial between Banach spaces factors.     

APPROXIMATION OF GENERALIZED BI－AXIALLY SYMMETRIC POTENTIALS WITH FAST RATES OF GROWTH

ＡＰＰＲＯＸＩＭＡＴＩＯＮＯＦＧＥＮＥＲＡＬＩＺＥＤＢＩ－ＡＸＩＡＬＬＹＳＹＭＭＥＴＲＩＣＰＯＴＥＮＴＩＡＬＳＷＩＴＨＦＡＳＴＲＡＴＥＳＯＦＧＲＯＷＴＨ￥Ｈ．Ｓ．Ｋａｓａｎａ（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＢｉｒｌａＩｕｓｔ．ｏｆＴｅｃｈ．＆Ｓｃｉ．，Ｐ...     

Fractional integral formulae involving a general class of polynomials and the multivariableH-function

We obtain two fractional integral formulae involving a general class of polynomials and the multivariableH-function. On account of the most general nature of the polynomials and the multivaribleH-function involved herein, our findings provide interesting unifications and extensions of a number of (known and new) results. We have mentioned here only two such results.     

Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

The convergence in L²( ) of the even approximants of the Wall continued fractions is extended to the Cesàro–Nevai class CN, which is defined as the class of probability measures σ with lim_n→∞ ∑ⁿ⁻¹_k=0 |a_k|=0, {a_n}_n0 being the Geronimus parameters of σ. We show that CN contains universal measures, that is, probability measures for which the sequence {|_n|² dσ}_n0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the “opposite” Szeg class which consists of measures with ∑^∞_n=0 (1−|a_n|²)^1/2<∞ and describe it in terms of Hessenberg matrices.     

Local zeta functions of general quadratic polynomials

This paper is concerned with the kind of local zeta functions now often called Igusa’s local zeta functions: A simple closed form of such a zeta function for an arbitrary quadratic form, its variants, and an application are given. Dedicated to the memory of Professor K G Ramanathan     

Matrix polynomials and the index problem for elliptic systems

The main new results of this paper concern the formulation of algebraic conditions for the Fredholm property of elliptic systems of P.D.E.'s with boundary values, which are equivalent to the Lopatinskii condition. The Lopatinskii condition is reformulated in a new algebraic form (based on matrix polynomials) which is then used to study the existence of homotopies of elliptic boundary value problems. The paper also contains an exposition of the relevant parts of the theory of matrix polynomials and the theory of elliptic systems of P.D.E.'s.

     

Computation of special functions by Padé approximants with orthogonal polynomial denominators

This paper deals with the computation of special functions of mathematics and physics in the complex domain using continued fraction (one-point or two-point Padé) approximants. We consider three families of continued fractions (Stieltjes fractions, real J-fractions and non-negative T-fractions) whose denominators are orthogonal polynomials or Laurent polynomials. Orthogonality of these denominators plays an important role in the analysis of errors due to numerical roundoff and truncation of infinite sequences of approximants. From the rigorous error bounds described one can determine the exact number of significant decimal digits contained in the approximation of a given function value. Results from computational experiments are given to illustrate the methods.Research supported in part by the National Science Foudation under Grant No. DMS-9302584.