Sequence Of Linear Operators Research Articles

Optimal Simultaneous Approximation via𝒜-Summability

We present optimal convergence results for themth derivative of a function by sequences of linear operators. The usual convergence is replaced by𝒜-summability, with𝒜being a sequence of infinite matrices with nonnegative real entries, and the operators are assumed to bem-convex. Saturation results for nonconvergent but almost convergent sequences of operators are stated as corollaries.

Admissibility versus nonuniform exponential behavior for noninvertible cocycles

We study the relation between the notions of exponential dichotomy and admissibility for a nonautonomous dynamics with discrete time. More precisely, we consider $\mathbb{Z}$-cocycles defined by a sequence of linear operators in a Banach space, and we give criteria for the existence of an exponential dichotomy in terms of the admissibility of the pairs $(\ell^p,\ell^q)$ of spaces of sequences, with $p\le q$ and $(p,q)\ne(1,\infty)$. We extend the existing results in several directions. Namely, we consider the general case of nonuniform exponential dichotomies; we consider $\mathbb{Z}$-cocycles and not only $\mathbb{N}$-cocycles; and we consider exponential dichotomies that need not be invertible in the stable direction. We also exhibit a collection of admissible pairs of spaces of sequences for any nonuniform exponential dichotomy.

Statistical approximation with a sequence of 2-Banach spaces

Let X ¯ = { X ¯ n : n ∈ N } be a sequence of 2 -Banach spaces, where X denotes an arbitrary 2 -Banach space, and let { T n } be a sequence of linear operators T n : X → X ¯ . First, a relationship is constructed between X ¯ n and X by means of T n and next the notion of statistical T -convergence is introduced. Hence, a statistical approximation theory is constructed between the elements of X ¯ n and X . Then, the properties of this statistical approximation theory are examined. On the other hand, the necessary and sufficient conditions for statistical T -convergence of the elements of X ¯ n to an element of X are investigated, and the methods of the determination of the statistical convergence velocity are examined. Finally, we define the statistical approximation and statistical stability conditions of linear operators, and give an application of our results.

Divergence of general operators on sets of measure zero

We consider sequences of linear operators $U_n$ with a localization property. It is proved that for any set $E$ of measure zero there exists a set $G$ for which $U_n{\mathbb I}_G(x)$ diverges at each point $x\in E$. This result is a generalization of ana

Korovkin-type theorem for sequences of operators preserving shape

In the paper we present Korovkin-type theorem concerning conditions of convergence sequences of linear operators preserving shape.

A Grobman–Hartman theorem for general nonuniform exponential dichotomies

For a nonautonomous dynamics with discrete time given by a sequence of linear operators A m , we establish a version of the Grobman–Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates e c ρ ( n ) , determined by a sequence ρ ( n ) . For all sufficiently small Lipschitz perturbations A m + f m we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators A m . We also show that all conjugacies are Hölder continuous. We note that the usual exponential behavior is included as a very special case when ρ ( n ) = n , but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ ( n ) = log n .

Korovkin-type theorems and applications

Abstract Let {T n} be a sequence of linear operators on C[0,1], satisfying that {T n (e i)} converge in C[0,1] (not necessarily to e i) for i = 0,1,2, where e i = t i. We prove Korovkin-type theorem and give quantitative results on C 2[0,1] and C[0,1] for such sequences. Furthermore, we define King’s type q-Bernstein operator and give quantitative results for the approximation properties of such operators.

Approximation processes for resolvent operators

We introduce general sequences of linear operators obtained from classical approximation processes which are useful in the approximation of the resolvent operators of the generators of suitable C0-semigroups. The main aim is the representation of the resolvent operators in terms of classical approximation operators.

Local saturation of conservative operators

We state a result about the local saturation of sequences of linear operators that preserve the sign of the k-th derivative of the functions. We apply it to the well known approximation operators of Bernstein, Szasz–Mirakjan, Meyer–Konig and Zeller, and Bleimann, Butzer and Hahn.

Approximation of *Weak-to-Norm Continuous Mappings

The purpose of this paper is to study the approximation of vector-valued mappings defined on a subset of a normed space. We investigate Korovkin-type conditions useful to recognize if a given sequence of linear operators is a so-called approximation process. First, we give a sufficient condition for this sequence to approximate the class of bounded, uniformly continuous functions. Then we present some sufficient and necessary conditions guaranteeing the approximation within the class of unbounded, *weak-to-norm continuous mappings. We also derive some estimates of the rate of convergence. We apply concrete approximation processes to derive representation formulae for semigroups of bounded linear operators.

Accelerating infinite products

Slowly convergent infinite products $$\prod\nolimits_{n - 1}^\infty {b_n }$$ are considered, where $$\left\{ {b_n } \right\}$$ is a sequence of numbers, or a sequence of linear operators. Using an asymptotic expansion for the “remainder” of the infinite product a method for convergence acceleration is suggested. The method is in the spirit of the d-transformation for series. It is very simple and efficient for some classes of sequences $$\left\{ {b_n } \right\}$$ . For complicated sequences $$\left\{ {b_n } \right\}$$ it involves the solution of some linear systems, but it is still effective.

Divergence Almost Everywhere of a PointwiseComparison of Two Sequences of Linear Operators

This paper deals with a negative result concerning a pointwisecomparison of two quite general sequences of linear operatorson the space of continuous functions on the torus or a segment

Multiresolution Analysis and Multivariate Approximation of Smooth Signals in CB(Rd

In this paper we derive rates of approximation for a class of linear operators on \(C_B({\bf R}^d)\) associated with a multiresolution analysis \(\{ V_n\}_{n\in{\bf Z}}.\) We show that for a uniformly bounded sequence of linear operators \(\{T_n\}_{n\in{\bf Z}}\) satisfying \(T_n f\equiv f\) on the subspace \(V_n\cap C_B({\bf R}^d),\) a lower bound for the approximation order is determined by the number of vanishing moments of a prewavelet set. We consider applications to extensions of generalized projection operators as well as to sampling series.

Qualitative Korovkin-Type Theorems for RF-Convergence

In this paper we study sequences of linear operators which are "almost positive" outside sets of small Jordan measure. For them, we prove Korovkin-type theorems in terms of a modification of the R-convergence used previously by W. Dickmeis, K. Mevissen, R. J. Nessel, and E. Van Wickeren and the test families of functions which the author introduced in a previous paper.

Expanding direction of the period doubling operator

We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a “Perron-Frobenius type operator,” to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point.

On the synthesis of radar signals using the woodward ambiguity function

Optimum radar signal synthesis is one of finding a signal (the complex envelope of the radar transmitter waveform), whose ambiguity function has a magnitude closest, in the mean square sense, to a nonnegative function G(t,f), of two variables, which function is chosen keeping in mind the given radar environment. Sussman (1962) has sought a solution to this problem in a finite-dimensional subspace of the space of square-integrable functions, defined over the time interval −∞ < t < ∞. In this paper, however, the optimum signal is sought in the class L2(−T, T) of all square-integrable functions vanishing outside a finite interval, thereby removing the finite-dimensional restriction imposed by Sussman, in his work, on the class of admissible signals. This optimization problem is shown to lead to an eigenvalue problem for a nonlinear integral operator on L2(−T, T). Next, an iteration scheme is presented whereby this nonlinear operator is approximated by a sequence of linear operators. These linear operators are recursively constructed utilizing a sequence of functions Gsn (n = 0, 1, 2, …), obtained at each stage by assigning to the G-function the phase of the ambiguity function corresponding to an eigenfunction sn associated with the largest eigenvalue of the linear operator at the previous stage. It is shown that any cluster point of the sequence of solutions of the linear eigenvalue problems thus generated furnishes at least a suboptimal solution to the basic problem of the synthesis of radar signals.

Comparison of the orders of approximation of the function classesW r H r

The first result in this direction has been obtained by Jackson [i], who has constructed a sequence of polynomial operators approximating the functions of class Lip 1 with order O (n-~), and then has proved that these operators approximate functions from any class H~ with order O (o (I/n)). For the proof of this result he has obtained and used an estimate of the order of approximation of the functions of class H~ by functions whose graphs represent polygonal lines, inscribed in the graphs of the functions /(X) ~ H~ (i.e., by functions (x) E Lip i). Thus, one has also proved that the problem of the approximation of various classes of functions by the same sequence of linear operators and the problem of the approximation of a class of functions by another are closely related with each other.

Stability of spectral properties for sequences of abstract operators of the type of Sturm-Liouville

We examine sequences of linear operators which are an abstract generalization of the classical operators of Sturm-Liouville. We give some hypotheses which ensure the convergence of the eigenvalues and the norm-convergence of eigenvectors of sequences of such operators. A possible application to problems of time evolution is also briefly mentioned.

Some Good Sequences of Interpolatory Polynomials: Addendum

In 1974 [2] we used the n + 2 zeros of (1 - x2)Pn(α, β)(x), α, β &gt; — 1, where Pn(α, β)(x) denotes Jacobi polynomials, to construct a sequence of linear operators {An(α, β)(f, x)} which has the following properties:(i) An(α, β)(f, x) is a linear polynomial operator mapping C[— 1, 1] into polynomials of degree ≦ n(1 + c), (c &gt; 0 fixed but arbitrary)

Descent methods in smooth, rotund spaces with applications to approximation in Lp

The purpose of this paper is to adapt some of the ideas used for approximation on finite-dimensional subspaces of L2 to the more general setting of smooth, rotund spaces. A general principle for construction of approximation methods is obtained. As an application we give a unified treatment of some of the Ln-approximation methods found in the literature, both for, 1 < p < 2 and 2 < p < 03. By employing sequences of LPI approximations with p, + 1 or p, + co these methods can be used for L1 or L” approximations. Section 2 contains background material on smoothness, rotundity, and orthogonality along with presentation of some properties of the best approximation operator. One of the L2 ideas which can be generalized is the Gram-Schmidt orthogonalization method. An imitation of the Gram-Schmidt process results in the method of “alternation” with B-operators which is given in Section 3. Another property of approximation in L2 is that the best approximation operator is given by a contractive linear projection. In Section 4 we obtain a theorem which gives techniques for obtaining best approximations using sequences of linear operators. These are based on using the derivatives of the norm functional to define B-operators. One of these involves the second derivative of the norm-functional and is a sequence of “weighted” La-projections. We show the connection between this and the derivative of the best approximation operator. We also consider “mixed” methods which combine alternation with sequences of linear operators. In Section 5 we show how the methods apply to LP and comment briefly on some numerical experiments.