Interpolation of rational functions by simple partial fractions

We study a generalized interpolation of a rational function at n nodes by a simple partial fraction of degree n and reduce the consideration to the solvability question for a special difference equation. We construct explicit interpolation formulas in the case where the equation order is equal to 1. We show that for functions A(x − a) ^m , m ? \mathbbN m \in \mathbb{N} , it is possible to reduce the consideration to a system of m + 1 independent first order equations and construct explicit interpolation formulas. Bibliography: 6 titles.     

Integral bounds for simple partial fractions

For p ≥ 2 we obtain bounds for L _p -norms of the Fourier transform of real parts of simple partial fractions. For even p our estimate is sharp. We also prove a new inequality for L _p -norms of simple partial fractions which in some cases is stronger than the corresponding inequality obtained by V. Yu. Protasov.     

A criterion for the best approximation of constants by simple partial fractions

The problem of the best uniform approximation of a real constant c by real-valued simple partial fractions R _n on a closed interval of the real axis is considered. For sufficiently small (in absolute value) c, |c| ≤ c _n , it is proved that R _n is a fraction of best approximation if, for the difference R _n ? c, there exists a Chebyshev alternance of n + 1 points on a closed interval. A criterion for best approximation in terms of alternance is stated.     

Estimates for L p -norms of simple partial fractions

We obtain estimates for L _p -norms of simple partial fractions in terms of their L _r -norms on bounded and unbounded segments of the real axis for various p > 1 and r > 1 (S. M. Nikolskii type inequalities). We adduce examples and remarks concerning sharpness of the inequalities and area of their application.     

Approximation of differentiable functions by partial sums of their Fourier series

For the classes of differentiable functions W ^r , r > 0, which include the classes of functions which have derivativesf(r) or (r)with moduli bounded by one, we obtain an asymptotic formula for the supremum of the difference between a function and the partial sums of its Fourier series. The remainder term in our formula is Cn^–r, in which C is a constant.Translated from Matematicheskii Zametki, Vol. 4, No. 3, pp. 291–300, September, 1068.     

Criterion for the appearance of singular nodes under interpolation by simple partial fractions

Under simple interpolation by simple partial fractions, the poles of the interpolation fraction may arise at some nodes irrespective of the values of the interpolated function at these nodes. Such nodes are said to be singular. In the presence of singular nodes, the interpolation problem is unsolvable. We establish two criteria for the appearance of singular nodes under an extension of interpolation tables and obtain an algebraic equation for calculating such nodes.     

Chebyshev’s alternance in the approximation of constants by simple partial fractions

Uniform approximation of real constants by simple partial fractions on a closed interval of the real axis is studied. It is proved that a simple partial fraction of best approximation of degree n for a constant is unique and coincides with this constant at n nodes lying on the interval; moreover, there is a Chebyshev alternance consisting of n + 1 points.     

Comparison of various generalizations of continued fractions

We use the Euler, Jacobi, Poincaré, and Brun matrix algorithms as well as two new algorithms to evaluate the continued fraction expansions of two vectorsL related to two Davenport cubic formsg ₁ andg ₂. The Klein polyhedra ofg ₁ andg ₂ were calculated in another paper. Here the integer convergentsP _k given by the cited algorithms are considered with respect to the Klein polyhedra. We also study the periods of these expansions. It turns out that only the Jacobi and Bryuno algorithms can be regarded as satisfactory. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 339–348, March, 1997. Translated by V. E. Nazaikinskii     

A criterion for the solvability of the multiple interpolation problem by simple partial fractions

Using reduction to polynomial interpolation, we study the multiple interpolation problem by simple partial fractions. Algebraic conditions are obtained for the solvability and the unique solvability of the problem. We introduce the notion of generalized multiple interpolation by simple partial fractions of order ≤ n. The incomplete interpolation problems (i.e., the interpolation problems with the total multiplicity of nodes strictly less than n) are considered; the unimprovable value of the total multiplicity of nodes is found for which the incomplete problem is surely solvable. We obtain an order n differential equation whose solution set coincides with the set of all simple partial fractions of order ≤ n.     

Finite Shmidt's groups and their generalizations

There are considered direct generalizations of finite Shmidt's groups, i.e., finite nonnilpotent groups, whose all proper subgroups are nilpotent. As corollaries, there are proved assertions confirming the dependence of the structure of the entire group on the presence of some system of Shmidt's groups. In particular, it is proved that a finite group is dispersive, if all its Shmidt's subgroups are upper-solvable.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 963–968, July–August, 1991.     

On biharmonic maps and their generalizations

We give a new proof of regularity of biharmonic maps from four-dimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives.Received: 6 February 2003, Accepted: 12 March 2003, Published online: 16 May 2003Mathematics Subject Classification (2000): 35J60, 35H20Pawel Strzelecki: Current address (till September 2003): Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany (email: strzelec@math.uni-bonn.de). The author is partially supported by KBN grant no. 2-PO3A-028-22;he gratefully acknowledgesthe hospitality of his colleagues from Bonn,and the generosity of Humboldt Foundation.     

k-th record times and their generalizations

A series of asymptotic results are given for the k-th record times and for their generalizations, the so-called k_n-record times. All possible limiting distributions are obtained for some generalizations of the k-th record variables. These distributions are closely related with the corresponding limit distributions of the extremal order statistics.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 115–121, 1986.     

Nets defined by hyperdistributions in affine space and their generalizations

This paper investigates special classes of distributions of hyperplane elements with which piecewise Chebyshev nets are associated.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol.8, pp. 197–223, 1977.