Asymptotics of Jacobi matrices for a family of fractal measures GÖKALP APLAN BILKENT UNIVERSITY, TYRKEY There are many results concerning asymptotics of orthogonal polynomials and Jacobi matrices associated with a measure whose essential support is a finite union of intervals on R. In the case of totally disconnected support, spectral theory of orthogonal polynomials is less complete and numerical evaluations can be seen as a useful source of information. In this talk, we discuss various properties and asymptotics of Jacobi matrices for a special family of fractal measures. We focus on numerical results and conjectures. The talk is based on a joint work with Alexander Goncharov and Ahmet Nihat Şimşek. On the median of the beta distribution DIMITRIOS ASKITIS UNIVERSITY OF COPENHAGEN The median q of the beta distribution is defined implicitly by the equation ∫ q 0 ta−1(1− t)b−1dt = 1 2 ∫ 1 0 ta−1(1− t)b−1dt. We study the monotonicity and asymptotic properties of the median as a univariate function of the parameter a, as well as of the function a log q(a), related to its logarithm. In particular, we find asymptotic expansions for a → 0 and ∞. These are related to the polygamma function and generalised Bernoulli polynomials.

Orthogonal Polynomials and Special Functions

Signals and Systems, Alexander D. Poularikas, University of Alabama in Huntsville Fourier Transforms, Kenneth B. Howell, University of Alabama in Huntsville Sine and Cosine Transforms, Pat Yip, McMaster University, Ontario The Hartley Transform, Kraig J. Olejniczak, University of Arkansas, Fayetteville Laplace Transforms, Samuel Seely (deceased), Westbrook, Connecticut The Z-Transform, Alexander D. Poularikas, University of Alabama in Huntsville Hilbert Transforms, Stefan L. Hahn, Warsaw University of Technology Radon and Abel Transforms, Stanley Deans, University of South Florida The Hankel Transform, Robert Piessens , Katholieke Universieit Leuven, Belgium Wavelet Transform, Yunlong Sheng, Laval University, Quebec The Mellin Transform, Jacqueline Bertrand, Pierre Bertrand, Universite de Paris VII, and Jean-Philippe Ovarlez, ONERA/DES, France Mixed Time-Frequency Signal Transformations, G. Faye Boudreaux-Bartels, University of Rhode Island Fractional Fourier Transforms, Dr. Mustafa Abushagur, University of Alabama in Huntsvill, Dr. Ahmed M. Almanasrah, Photronix, Malaysia The Lapped Transforms, Ricardo L. de Queiroz, Xerox Corporation The Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Appendices, Alexander D. Poularikas, University of Alabama in Huntsville Index

The transforms and applications handbook

This article attempts to describe an approach to statistical data analysis which is simultaneously parametric and nonparametric. Given a random sample X 1, …, X n of a random variable X, one would like (1) to test the parametric goodness-of-fit hypothesis H 0 that the true distribution function F is of the form F(x) = F0[(x − μ)/σ)], where F 0 is specified, and (2) when H 0 is not accepted, to estimate nonparametrically the true density-quantile function fQ(u) and score function J(u) = − (fQ)'(u). The article also introduces density-quantile functions, autoregressive density estimation, estimation of location and scale parameters by regression analysis of the sample quantile function, and quantile-box plots.

Nonparametric Statistical Data Modeling

Systems made up of arrays of sensors (seismographs, hydrophones, antennas, meteorological stations) receive geophysical data on a number of channels. Such multichannel data can be processed by means of a multichannel least-squares (Wiener) digital filter; the filter is determined by requiring that the sum of the mean-square errors between the desired outputs and the actual outputs for each channel be minimized. This requirement leads to a set of simultaneous linear equations, called the normal equations, where each element is itself a square matrix. The coefficients of the digital filter are given by the solution of these normal equations. An exact recursive procedure is derived for the numerical solution of the normal equations. In this recursive procedure advantage is taken of the fact that the matrix of the normal equations is an autocorrelation matrix, and hence is in the form of a Toeplitz matrix, namely one with equal elements along any diagonal. A set of auxiliary coefficients that are generated in the recursive procedure are the coefficients of matrix-valued polynomials which are the counterparts of the classical (scalar-valued) polynomials orthogonal on the unit circle. This set of auxiliary coefficients may also be used for a sideward recursion that corresponds to shifting the time index of the right-hand side of the normal equations. This sideward recursion is valuable in applications because it represents a relatively inexpensive way of determining the filter that incorporates the optimum time delay between the multichannel input signal and the multichannel output signal. The machine time required to solve the normal equations for a filter with m (matrix-valued) coefficients is proportional to m2 for the proposed recursive procedure, as compared with m3 for conventional techniques for the solution of simultaneous equations. Another advantage of using the recursive procedure is that it only requires computer storage space proportional to m, rather than to m2 as in the case of the conventional techniques.

Recursive solution to the multichannel filtering problem

SUMMARY

High-quality high-resolution transmission and reflection images produced using a scanning optical microscope and the split-detector technique are presented. These images exhibit differential phase contrast, the method avoiding some drawbacks of the usual Nomarski DIC arrangement. Imaging is treated theoretically and compared with the Nomarski method.

Differential phase contrast in scanning optical microscopy

A course of mathematical analysis

Polynomials orthogonal on a circle and interval

Integral Transforms and Operational Calculus.

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A course of mathematical analysis

Polynomials orthogonal on a circle and interval

Integral Transforms and Operational Calculus.