Convergence Of Certain Sequences Research Articles

Application of Banach contraction principle to approximate the golden number

This work is devoted to the application of selected fixed point theorems in the problems of convergence of certain sequences to the golden number. It contains the theorem about the fixed point of so-called ψ-contraction specified on the closed interval <a, b> and the local version of Banach Contraction Principle as a conclusion. It will also be used to approximate the golden number.

Uniform Convergence and Spectra of Operators in a Class of Fréchet Spaces

Well-known Banach space results (e.g., due to J. Koliha and Y. Katznelson/L. Tzafriri), which relate conditions on the spectrum of a bounded operatorTto the operator norm convergence of certain sequences of operators generated byT, are extended to the class of quojection Fréchet spaces. These results are then applied to establish various mean ergodic theorems for continuous operators acting in such Fréchet spaces and which belong to certain operator ideals, for example, compact, weakly compact, and Montel.

On the convergence of certain sequences of rational approximants to meromorphic functions in several variables

In a previous paper, the author introduced a new class of multivariate rational interpolants, which are called Optimal Padé-type Approximants (OPTA). There, for this class of rational interpolants, which extends classical univariate Padé Approximants, a direct extension of the “de Montessus de Ballore's Theorem” for meromorphic functions in several variables is established. In the univariate case, this theorem ensures uniform convergence of a row of Pade Approximants when the denominator degree equals the number of poles (counting multiplicities) in a certain disc. When one overshoots the number of poles when fixing the denominator degree, convergence in measure or capacity has been proved and, under certain additional restrictions, the uniform convergence of a subsequence of the row. The author tackles the latter case and studies its generalization to functions in several variables by using OPTA.

Boundaries and random walks on finitely generated infinite groups

We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of ϖΓ and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.

Quadrature on the half-line and two-point Padé approximants to Stieltjes functions. Part I. Algebraic aspects

Let f( z) be a Stieltjes function with asymptotic expansions L 0 and L ∞ at z = 0 and z = ∞, respectively. Let [ k n ] denote the two-point Padé approximant of type ( m, n) which matches k of the coefficients of the series L 0 and which takes its remaining interpolation conditions from L ∞. We discuss in this paper the algebraic aspects of this problem. We shall emphasize the relation between quadrature formulas and two-point Padé approximants and derive expressions for the error. In a subsequent paper we shall consider the convergence aspects of these approximants. For example, the positivity of the error, which is obtained here, will result in the monotonic convergence of certain sequences of two-point Padé approximants, a property which is well known in the case of classical Padé approximants.

Limit theorems for the number of crossings of a fixed plane by certain sequences of generalized dbffusion processes

We characterize the weak convergence of certain sequences of generalized diffusion processes by using a specific functional of a process, namely, the number of crossings of a fixed plane by this process.

A New Theory for Fourier Analysis. II. Further Analysis on the Space of Test Functions

This is the second in a series of papers extending the theory supporting Fourier analysis. Bounds for basic mappings on the space of test functions, the convergence of certain sequences of lest functions, and the convolution of test functions are discussed.

Convergence of algorithms of decomposition type for the eigenvalue problem

We develop the theory of convergence of a generic GR algorithm for the matrix eigenvalue problem that includes the QR, LR, SR, and other algorithms as special cases. Our formulation allows for shifts of origin and multiple GR steps. The convergence theory is based on the idea that the GR algorithm performs nested subspace iteration with a change of coordinate system at each step. Thus the convergence of the GR algorithm depends on the convergence of certain sequences of subspaces. It also depends on the quality of the coordinate transformation matrices, as measured by their condition numbers. We show that with a certain obvious shifting strategy the GR algorithm typically has a quadratic asymptotic convergence rate. For matrices possessing certain special types of structure, cubic convergence can be achieved.

Some Examples Illustrating Richardson's Improvement

(1990). Some Examples Illustrating Richardson's Improvement. The College Mathematics Journal: Vol. 21, No. 4, pp. 314-322.

On the approximation of the delta distribution in sobolev spaces of negative order

The rate of convergence of certain sequences of type δ in Sobolev spaces of negative order is investigated.

A contour integral representation of linear extrapolation methods

Extrapolation methods are well-known to be very efficient tools for the acceleration of the convergence of certain sequences of numbers or functions, cf. [2, 3, 4, 8, 9]. In this note we present a representation of linear extrapolation procedures in terms of complex contour integrals. For the proof we make use of a complete characterization of these procedures as linear functionals, which is itself of some interest and can be found, for example, in [2] or [8].

An algorithm for reconstructing convex bodies from their projections

An algorithm is described for the approximative reconstruction of a plane convex body from its projections in a finite number of directions.A priori anda posteriori error estimates are given, and the convergence of certain sequences of an approximative solution of the reconstruction problem to the exact solution is proven. Finally, it is shown that, after small modifications, the algorithm can be applied to reconstruct convex bodies from discrete projectional data. The algorithm consists in an approximation of the convex body, to be reconstructed, by recursively defined cores and envelopes, following the ideas of Kuba [6] for the reconstruction of binary patterns.

On Rates of Convergence of Certain Sequences

On Rates of Convergence of Certain Sequences

Convergence and almost convergence of certain sequences of positive linear operators

Convergence and almost convergence of certain sequences of positive linear operators

On the degree of convergence of sequences of rational functions

The writer has recently studiedt the convergence of certain sequences of rational functions of the complex variable, under the hypothesis that the poles of these functions are prescribed and satisfy certain asymptotic conditions. The rational functions are determined either by interpolation to a given analytic function, or by some extremal property of best approximation to such a function. Degree of convergence and regions of uniform convergence of the sequence of rational functions are then obtained (op. cit.). It is the object of the present paper to go more deeply than hitherto into properties of degree of convergence of sequences of rational functions, to make more precise the previous results, and especially to introduce and study the concept of maximal convergence of a sequence of rational functions with preassigned poles; this is a generalization of the corresponding concept for sequences of polynomials. The analogy between convergence properties of sequences of polynomials and convergence properties of sequences of more general rational functions is strong, but has hitherto not been sufficiently strong to justify the use of the term maximal convergence in the latter case (compare op. cit., p. 258). We show now that maximal convergence is characteristic of various sequences of rational functions determined by interpolation and by extremal properties. The present results would seem to be more or less definitive in form. 1. Introductory results. We choose as point of departure the following relatively simple but typical formulation (op. cit., ?8.3):

Note on the degree of convergence of sequences of analytic functions

It is the object of this note first to present a general theorem (Theorem 1) on maxima and superior limits of sequences of analytic functions; this is essentially a result in the theory of subharmonic functions, presumably of some interest in itself. Second, Theorem 1 is employed to establish a sharper theorem (Theorem 2) on the precise degree of convergence of general sequences of analytic functions. Finally, these general results are used to establish a quite specific result (Theorem 3) on the degree of convergence of certain sequences of analytic functions in an annular or more general region. Parts of Theorem 3 have previously been established, but in a more restricted form and by another method. The present Theorem 3 is sufficiently general to dominate results established elsewhere by the author on maximal convergence of sequences of polynomials,t on approximation by functions analytic in a given region,: on interpolation by functions analytic in a given region,? and on maximal convergence of sequences of rational functions.11 Various other applications of Theorem 3 seem to suggest themselves and will presumably appear in the future.T The following treatment is chosen rather for elements of simplicity than for ultimate generality. Some extensions present themselves at once as obvious; others are scarcely less immediate. But even in its present form, Theorem