Nonuniform hyperbolicity and one-sided admissibility

For a nonautonomous dynamics defined by a sequence of linear operators, we introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in lp spaces, both for the space of perturbations and the space of solutions. This allows unifying the notions of uniform and nonuniform exponential behavior. Moreover, we consider the general case of a noninvertible dynamics. As a nontrivial application we show that the conditional stability of a nonuniform exponential dichotomy persists under sufficiently small linear perturbations.

The algorithm of application of the method of steepest descent to the solution of problems of structural mechanics and solid mechanics, described by nonlinear differential equations. For application of this method to nonlinear operators are described by a sequence of linear operators in incremental form, unlimited and complex linear operator, in line with the idea of L. V. Kantorovich is limited to a simple linear unbounded operator.

For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a uniform exponential dichotomy and we characterizeit completely in terms of the admissibility of a large class of function spaces. We apply those results to show that structural stability of a diffeomorphism is equivalent to a very general type of Lipschitz shadowing property. Our results extend those in [37] in various directions.

Let X and Y be vector lattices and Ti: X → Y a sequence of linear operators which are sequentially continuous with respect to relative uniform convergence. If {Tjx} is relatively uniformly convergent to Tx for each x ∈ X, under appropriate assumptions on the spaces, we show that the linear operator T is also continuous and that the {Ti} are order equicontinuous in a certain sense. We also establish an order version of the Uniform Boundedness Principle.

On the synthesis of radar signals using the woodward ambiguity function

In this paper we give an elementary proof of the approximate continuous partition of unit on a compact Hausdorff-space using results developed in [2]. With the aid of this result we give a proof of the well-known localizability theorem for A-modules in C[T,E]. Using a similar but less elementary method we derive a localizability theorem in the case of approximation of elements in C[T,E] by certain sequences of linear operators.

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.

The notion of a saturation class, due to Favard, can be explained as follows. (For a more general discussion and for the determination of the saturation classes for trigonometric approximation, compare the article of Sunouchi [4].) We try to approximate continuous functions f (x) by a given sequence of linear operators L n (f, x). Sometimes the following phenomena occur. There is a subclass K of the space C of all continuous functions for which the degree of approximation of f by L n (f) is optimal. A still better approximation happens only for functions f of a very narrow subclass K o of K, for which L n (f) = f. In this situation, K is called the saturation class of the operators L n .

We construct $C^k$ invariant center manifolds for differentialequations $u'=A(t)u+f(t,u)$ obtained fromsufficiently small perturbations of a nonuniformexponential trichotomy. We emphasize that our results are optimal,in the sense that the invariant manifolds are as regular as thevector field. In addition, we can also consider linearperturbations with the same method.

This is a survey (without proofs except for verifying a few new facts) of the slowest possible rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. A sequence of linear operators (L n ) is said to converge to a linear operator Larbitrarily slowly (resp., almost arbitrarily slowly) provided that (L n ) converges to L pointwise, and for each sequence of real numbers (ϕ(n)) converging to 0, there exists a point x = x ϕ such that \(\|{L}_{n}(x) - L(x)\| \geq \phi (n)\) for all n (resp., for infinitely many n). Two main “lethargy” theorems are prominent in this study, and they have numerous applications. The first lethargy theorem (Theorem 11.16) characterizes almost arbitrarily slow convergence. Applications of this lethargy theorem include the fact that a large class of polynomial operators (e.g., Bernstein, Hermite–Fejer, Landau, Fejer, and Jackson operators) all converge almost arbitrarily slowly to the identity operator. Also all the classical quadrature rules (e.g., the composite Trapezoidal Rule, composite Simpson’s Rule, and Gaussian quadrature) converge almost arbitrarily slowly to the integration functional. The second lethargy theorem (Theorem 11.21) gives useful sufficient conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, there is a “dichotomy” theorem (Theorem 11.27) which states that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. Some applications of the dichotomy theorem include generalizations and sharpening of (1) the von Neumann-Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., “almost” randomly) ordered projections, and (3) a theorem of Xu and Zikatanov.

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.

A sequence ${T_n}_{n=1}^{infty}$ of bounded linear operators on a separable infinite dimensional Hilbert space $mathcal{H}$ is called subspace-diskcyclic with respect to the closed subspace $Msubseteq mathcal{H},$ if there exists a vector $xin mathcal{H}$ such that the disk-scaled orbit ${alpha T_n x: nin mathbb{N}, alpha inmathbb{C}, | alpha | leq 1}cap M$ is dense in $M$. The goal of this paper is the studying of subspace diskcyclic sequence of operators like as the well known results in a single operator case. In the first section of this paper, we study some conditions that imply the diskcyclicity of ${T_n}_{n=1}^{infty}$. In the second section, we survey some conditions and subspace-diskcyclicity criterion (analogue the results obtained by some authors in cite{MR1111569, MR2261697, MR2720700}) which are sufficient for the sequence ${T_n}_{n=1}^{infty}$ to be subspace-diskcyclic(subspace-hypercyclic).

In this paper, we define a new sequence of linear positive operators of integral type to approximate functions in the space,. First, we study the basic convergence theorem in simultaneous approximation and then study Voronovskaja-type asymptotic formula. Then, we estimate an error occurs by this approximation in the terms of the modulus of continuity. Next, we give numerical examples to approximate three test functions in the space by the sequence. Finally, we compare the results with the classical sequence of Szãsz operators  on the interval . It turns out that, the sequence  gives better than the results of the sequence  for the two test functions using in the numerical examples.

For a two-sided sequence of compact linear operators acting on a Banach space, we consider the notion of spectrum defined in terms of the existence of exponential dichotomies under homotheties of the dynamics. This can be seen as a natural generalization of the spectrum of a matrix—the set of its eigenvalues. We give a characterization of all possible spectra and explicit examples of sequences for which the spectrum takes a form not occurring in finite-dimensional spaces. We also consider the case of a one-sided sequence of compact linear operators.

An extension based on qR-integral for a sequence of operators