Approximation properties of Jacobi's algorithm

The paper improves a previous result which is a generalization of Hurwitz' theorem for Jacobi's algorithm.     

Ergodic computations with continued fractions and Jacobi's algorithm

Ergodic computational aspects of the Jacobi algorithm, a generalization to two dimensions of the continued fraction algorithm, are considered. By means of such computations the entropy of the algorithm is estimated to be 3.5. An approximation to the invariant measure of the transformation associated with the algorithm is obtained. The computations are tested by application to the continued fraction algorithm for which both entropy and the invariant measure are known.Work partly performed under the auspices of the U.S. Atomic Energy Commission while one of the authors (M.S.W.) was a faculty participant of the Associated Western Universities at Los Alamos Scientific Laboratory. The work was also supported in part by NSF grant GP-28313 to M. S. W.     

Approximation properties of multivariate wavelets

Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in provides approximation order .

     

Approximation properties of q-Baskakov operators

We establish direct estimates for the q-Baskakov operator introduced by Aral and Gupta in [2], using the second order Ditzian-Totik modulus of smoothness. Furthermore, we define and study the limit q-Baskakov operator.     

Arithmetic Consequences of Jacobi's Two-Squares Theorem

There is a well-known formula due to Jacobi for the number r₂(n) of representations of the number n as the sum of two squares. This formula implies that the numbers r₂(n) satisfy elegant arithmetic relations. Conversely, these arithmetic properties essentially imply Jacobi's formula. So it is of interest to give direct proofs of these arithmetic relations, and this we do.     

Jacobi's Two-Square Theorem and Related Identities

We show that Jacobi's two-square theorem is an almost immediate consequence of a famous identity of his, and draw combinatorial conclusions from two identities of Ramanujan.     

Approximation properties in tensor products

For distinct classes of locally convex spaces and tensor topologies α =? and α = π it is proved that \(E\hat \otimes _\alpha F\) has the approximation property if and only if E and F have this property.     

Approximation properties of the spline fit

Different topics in connection with spline fit are discussed in this paper. In particular, an example is given showing non-convergence of splines, and further some error bounds of cubic spline interpolation are proved.     

Approximation properties of theM n -operators

Aequationes mathematicae -     

Approximation properties of modified Gamma operators

In this paper we present a survey of rates of pointwise approximation of modified Gamma operators Gn for locally bounded functions and absolutely continuous functions by using some inequalities and results of probability theory with the method of Bojanic-Cheng. In the paper a kind of locally bounded functions is introduced with different growth conditions in the fields of both ends of interval (0,+∞), and it is found out that the operators have different properties compared to the Gamma operators discussed in [X.M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl. 311 (2005) 389-401]. And we obtain two main theorems. Theorem 1 gives an estimate for locally bounded functions which subsumes the approximation of functions of bounded variation as a special case. Theorem 2 gives an estimate for absolutely continuous functions which is best possible in the asymptotical sense.     

Approximation properties of fine hyperbolic graphs

In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use the techniques of Ozawa’s to prove that a fine hyperbolic graph has the metric invariant translation approximation property.     

Approximation properties of theM n -operators

Aequationes mathematicae -     

Canonical Euler-Lagrange equations and Jacobi's theorem on regular surfaces

In this article we establish conditions under which canonical variables can be defined for a variational problem defined on a geometric (compact) surface. Also, we show the form the corresponding Euler-Lagrange equations assume once we rewrite them in terms of such canonical variables. Furthermore, we prove a version of Jacobi's theorem generalizing the univariate standard version of this theorem. The main results are applied to the conformal Gauss curvature functional.     

Approximation properties of nilpotent interlacings of groups

The work is devoted to the study of properties of verbal products and interlacings of groups, in particular, the problem of approximability and finite approximability. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 49, Algebra and Geometry, 2007.     

Approximation and shape-preserving properties of q-Stancu operator

We introduce the definition of q-Stancu operator and investigate its approx- imation and shape-preserving property. With the help of the sign changes of f(x) and Ln = f ( f,q;x) the shape-preserving property of q-Stancu operator is obtained.