Homogeneity in powers of subspaces of the real line

Working in ZFC, we prove that for every zero-dimensional subspace of the real line, the Tychonoff power is homogeneous ( denotes the nonnegative integers). It then follows as a corollary that is homogeneous whenever is a separable zero-dimensional metrizable space. The question of homogeneity in powers of this type was first raised by Ben Fitzpatrick, and was subsequently popularized by Gary Gruenhage and Hao-xuan Zhou.

     

Approximation of locally integrable functions on the real line

We introduce the notion of generalized-derivatives for functions locally integrable on the real axis and investigate problems of approximation of the classes of functions determined by these derivatives with the use of entire functions of exponential type. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1549–1561, November, 1999.     

Special functions as subordinated semigroups on the real line

We give examples of convolution semigroups on the positive half-line and on the real line. Such semigroups are expressed in terms of special functions which arise in classical differential equations.     

Zygmund functions on the real line and quasiconformal deformations

By a complex approach, we obtain some results concerning Zygmund functions on the real line, which corresponds to some classical results about Zygmund functions on the unit circle.     

Topologically transitive semigroup actions of real linear fractional transformations

Let , where I is a proper closed subinterval of R. Our main goal in this paper is to describe all f,g∈FI such that the action of the semigroup generated by f and g is topologically transitive on I. We also prove that this action is never weakly topologically mixing. Finally, we describe all pairs of affine functions on R whose generated semigroup action is weakly topologically mixing.     

Approximation for the measures of ergodic transformations of the real line

Summary Let be the Frobenius-Perron operator corresponding to a nonsingular point-transformation of the real lineR into itself and let for each natural numbern, P _n be the discrete analogue ofP _f. It is shown that under fairly weak restrictions on, the equationf=P _f f has an unique solutionf ₀ such thatf ₀>0 (a.e.), ¦f ₀¦=1, and that this solution can be approximated inL ¹ (R) in two different ways: (1) by the sequence wheref0, ¦f¦=1, and (2) by the sequence {s _On } of simple functions such thats _n=P_n(s _On ).     

Universality of the automorphism group of the real line

We discuss the question, whether each automorphism group (of cardinality at most ) of a linear order is embeddable into the automorphism group of the real line. We show that the answer to this question is independent of the axioms of ZFC: the answer is positive, if we assume < and Souslin's hypothesis; the answer is negative, if we assume orV=L.     

Orthogonal rational functions and quadrature on the real half line

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Padé approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Padé approximants is discussed.     

Evaluation of generalized Mittag–Leffler functions on the real line

This paper addresses the problem of the numerical computation of generalized Mittag–Leffler functions with two parameters, with applications in fractional calculus. The inversion of their Laplace transform is an effective tool in this direction; however, the choice of the integration contour is crucial. Here parabolic contours are investigated and combined with quadrature rules for the numerical integration. An in-depth error analysis is carried out to select suitable contour’s parameters, depending on the parameters of the Mittag–Leffler function, in order to achieve any fixed accuracy. We present numerical experiments to validate theoretical results and some computational issues are discussed.     

Approximation by entire functions in the mean on the real line

Approximations in a locally integral norm of functions representable as a convolution on the whole line by means of Fourier operators are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 121–125, January, 1991.     

A method of approximation by rational functions on the real line

For a given system of numbers {z _k } _k=1 ⁿ , IMz _k > 0, rational functions of order 4n — 2 are constructed which effect for a functionf(x∈C _∞) an approximation of the same order as the best approximation by proper rational functions having poles at the points {z _{k k=1} ⁿ and . Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 375–380, September, 1977. In conclusion the author thanks E. P. Dolzhenko and S. B. Stechkin, whose discussions contributed to improvements in this work.     

Weighted approximation of functions on the real line by Bernstein polynomials

The authors give error estimates, a Voronovskaya-type relation, strong converse results and saturation for the weighted approximation of functions on the real line with Freud weights by Bernstein-type operators.     

Stable real invariant semidefinite subspaces and stable factorizations of symmetric rational matrix functions

Stability of certain classes of invariant subspaces of a real matrix which is selfadjoint in a real indefinite inner product, is studied. The study is motivated by, and subsequently applied to, the problem of stability for minimal factorizations of real symmetric rational matrix functions.