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A well-known theorem by N. Wiener characterizes the discrete part of a complex Borel measure μ ∈ M(T) on the torus group T. In this note an analoguous result is presented for orthonormal polynomial sequences (p_n)_{n ∈ ℕ 0}. For Jacobi polynomials the characterization is investigated in detail.Copyright 1995, 1999 Academic Press, Inc.

This paper describes the measurements of the normal cholangiogram as demonstrated by endoscopic retrograde cholangiopancreatography. Two groups of patients were studied. There were 49 patients who had normal livers and no evidence of biliary tract disease. In addition, there were 25 patients with known liver disease and an apparent normal extrahepatic biliary tract. The extrahepatic bile duct was measured in three regions: (1) the common hepatic duct (above the cystic duct and below the bifurcation); (2) the prepancreatic portion (below the cystic duct and above the pancreas); and (3) the interpancreatic portion (bile duct within the pancreas). These measurements ranged from 2.1 to 9.2 mm. Although the common bile ducts were generally quite uniform in caliber, there were instances where each portion was the largest diameter in an individual case. There was a slight increase in bile-duct caliber with age. This study showed somewhat smaller measurements than previously reported and may relate to case selection in this or previous studies.

The efficiency in estimating the mean of a weakly stationary process is investigated. Estimators Mnλare optimum provided the spectral density has a zero in t= 0 of order λ. Here we study the asymptotic behavior of Mnλin case the spectral density has a zero in t= 0 of order different from λ. In particular we prove that Mnλare optimum if λ is greater than this order.

Trigonometric polynomial kernels defined by Jacobi polynomials are investigated. They generalize the classical Dirichlet kernel and the Fejér kernel. The asymptotic behavior of the corresponding Fourier summation obtained is leading to optimal kernels.

A formula is developed which allows to calculate then-fold convolution power of elementary functions recursively.

The estimation of a μ-density f by a finite weighted series of μ -orthogonal polynomials is studied. The measures μ, are positive finite Borel measures with infinite support contained in [-1,1] and include large subclasses of Jacobi measures and generalized Chebyshev measures. We prove the consistency in L1(μ) while imposing only minimal or no conditions on f.

The close relationship between orthogonal polynomial sequences and polynomial hypergroups is further studied in the case of even weight function, cf. [18]. Sufficient criteria for the recurrence relation of orthogonal polynomials are given such that a polynomial hypergroup structure is determined on . If the recurrence coefficients are convergent the dual spaces are determined explicitly. The polynomial hypergroup structure is revealed and investigated for associated ultraspherical polynomials, Pollaczek polynomials, associated Pollaczek polynomials, orthogonal polynomials with constant monic recursion formula and random walk polynomials.

For many applications it is desirable to have extensions of the classical theory of weakly stationary processes to certain classes of nonstationary ones. A large family for which Fourier methods still play a major role is the harmonizable class and some related processes. The purpose of this paper is to initiate the study of a class of stochastic processes with a stationarity condition based on the notion of hypergroups.

1) Average weights per inch of height for both sexes in older population, ages 65 to 94, have been presented. This is the first study of its kind to appear. It is based on a population sample of apparently healthy persons and the sample size is considered statistically adequate. 2) There exists a definite trend towards a decrease in average weight of successive 5 year age groups after age of 65. There were 2 reasons for this: 1) individual weight loss accompanying aging due to loss of body substance, 2) high mortality rate among obese which favors survival of a larger proportion of underweight people. 3) Height and survival were not related.