Polynomials orthogonal on the semicircle,II

Generalizing previous work [2], we study complex polynomials {π _k },π _k (z)=z ^k +?, orthogonal with respect to a complex-valued inner product (f,g)=∫ ₀ ^π f(e ^iθ)g(e ^iθ)w(e ^iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ ₀ ^π w(e ^iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π _k } in the case of the Jacobi weight functionw(z)=(1?z)^α(1+z)^β, α>?1, and its special case \(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ _n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ _n (z). It has regular singular points atz=1, ?1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.     

Notes on the Feynman integral,II

In the setting of Cameron and Storvick's recent theory, our main result establishes the existence of the analytic Feynman integral for functions on v-dimensional Wiener space of the form F(X) = exp {? ∝_a^b (A(s) X(s), X(s)) ds}. Here X is a $\text{R}$^v-valued continuous function on [a, b] such that X(a) = 0 and {A(s): a ? s ? b} is a commutative family of real, symmetric, positive definite matrices such that the square roots of the eigenvalues are functions of bounded variation on [a, b]. We obtain the existence theorem just referred to without having to construct special spaces, quadratic forms, etc., to fit the particular problem of interest.     

The Generalized Radon Transform on the Plane, the Inverse Transform, and the Cavalieri Conditions

In the two-dimensional case, the generalized Radon transform takes each function supported in a disk to the values of the integrals of that function over a family of curves. We assume that the curves differ only slightly from straight lines and the network formed by these curves has the same topological structure as the network of straight lines. Thus, the generalized Radon transform specifies a function on the set of straight lines. Under these conditions, we obtain a solution of the inversion problem for the generalized Radon transform and indicate a Cavalieri condition describing the range of this transform in the space of functions on the set of straight lines.     

The epistemic,the cognitive,the human: a commentary on the mathematical working space approach

ZDM – Mathematics Education - This article is a commentary on the mathematical working space (MWS) approach and draws on the articles contained in this ZDM issue. The article is divided into...     

Effect of liquid media on the weight,hardness, volume,and deformation of polyurethane

The resistance of polyurethane to aviation fluids and water has been investigated. The deformation of polyurethane directly exposed to these fluids has been determined. It is shown that polyurethane is insoluble in aviation fluids and sensitive to the action of acids and RDV liquid; it has good resistance to water and atmospheric conditions. The deformation that develops in the first hours of exposure subsequently shows little change.Kiev Institute of Civil Aviation Engineers. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 342–344, March–April, 1969.     

Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations

We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments. It is demonstrated that these geometrically defined quantities coincide with the Verblunsky coefficients, which appear in the Szegö recursions for the matrix orthogonal polynomials. Finally, we provide an alternative proof of the Geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval [−1, 1] and the Verblunsky coefficients corresponding to matrix measures on the unit circle.     

Uniform,on the Real Line,Equiconvergence of Spectral Expansions for the Higher-Order Differential Operators

Doklady Mathematics - Result on the uniform, over the entire real line, equiconvergence of spectral expansions related to the self-adjoint extension of a general differential operation of any even...