Abstract: Introduction of q-calculus.- q-Discrete operators and their results.- q-Integral operators.- q-Bernstein type integral operators.- q-Summation-integral operators.- Statistical convergence of q-operators.- q-Complex operators.

Abstract: For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability.

Abstract: Part I. Hilbert spaces: 1. Introduction 2. Spaces Part II. Operators: 3. Introduction to operators 4. Bounded operators 5. Compact operators 6. Integral operators Part III. Applications: 7. Signals and systems analysis on the 2-sphere 8. Signal concentration and joint spatio-spectral analysis 9. Convolution on 2-sphere 10. Reproducing kernel Hilbert spaces.

Abstract: As applications of atomic decomposition results of Hardy spaces with variable exponents, we shall prove the boundedness of commutators and the fractional integral operators as well as the Hardy operators. There are many ways to prove such boundedness. For example, the boundedness of commutators can be proved by the sharp maximal inequalities. But here, we propose a different method based upon our atomic decomposition.

Abstract: In this paper, we consider a certain King type operators which includes general families of Sz´ asz-Mirakjan, Baskakov, Post-Widder and Stancu operators. By introducing two parameter family of Lipschitz type space, which provides global approximationfortheabovementionedoperators,weobtaintherateofconvergence of this class. Furthermore, we give local approximation results by using the first and the second modulus of continuity.

Abstract: We analyze a class of partial differential equations that arise as "backwards Kolmogorov operators" in infinite population limits of the Wright-Fisher models in population genetics and in mathematical finance. These are degenerate elliptic operators defined on manifolds with corners. The classical example is the Kimura diffusion operator, which acts on functions defined on the simplex in R^n. We introduce anisotropic Holder spaces, and prove existence, uniqueness and regularity results for the heat and resolvent equations defined by this class of operators. This suffices to prove that the C^0-graph closure generates a strongly continuous semigroup, and that the associated Martingale problem has a unique solution. We give a detailed description of the nullspace of the forward Kolmogorov operator.

Abstract: Some elementary inequalities providing upper bounds for the difference of the norm and the numerical radius of a bounded linear operator on Hilbert spaces under appropriate conditions are given.

Abstract: We derive a new generalization of Prony?s method to reconstruct M-sparse expansions of (generalized) eigenfunctions of linear operators from only suitable values in a deterministic way. The proposed method covers the well-known reconstruction methods for M-sparse sums of exponentials as well as for the interpolation of M-sparse polynomials by using special linear operators in . Further, we can derive new reconstruction formulas for M-sparse expansions of orthogonal polynomials using the Sturm?Liouville operator. The method is also applied to the recovery of M-sparse vectors in finite-dimensional vector spaces.

Abstract: Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of differential operators. For instance, we show that this class is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking "square roots", and under the direct sum construction. Symmetric hyperbolic systems are studied in detail.

Abstract: First published in Proceedings of the American Mathematical Society, volume 141, published by the American Mathematical Society. Also available electronically from http://www.ams.org/journals/proc/2013-141-10/S0002-9939-2013-11689-8/home.html

Abstract: We extend results of Caffarelli–Silvestre and Stinga–Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in particular to infinitesimal generators of bounded C0 semigroups and operators with purely imaginary symbol. We give integral representations to the extension problem in terms of solutions to the heat equation and the wave equation.

Abstract: We study statistical approximation properties of -Bernstein-Shurer operators and establish some direct theorems. Furthermore, we compute error estimation and show graphically the convergence for a function by operators and give its algorithm.

Abstract: The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples.

Abstract: In this paper, we study the convergence of paths for continuous pseudocontractions in a real Banach space. As an application, we consider the problem of finding zeros of m-accretive operators based on an iterative algorithm with errors. Strong convergence theorems for zeros of m-accretive operators are established in a real Banach space.

Abstract: We investigate Schrodinger operators with \delta- and \delta'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schrodinger operators with \delta- and \delta'-interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrodinger operators and, in particular, it allows to transform known results for Schrodinger operators with \delta-interactions to Schrodinger operators with \delta'-interactions.

Abstract: We have developed and implemented a general formalism for fast numerical solution of time-independent linear partial differential equations as well as integral equations through the application of numerically separable integral operators in d ≥ 1 dimensions using the non-standard (NS) form. The proposed formalism is universal, compact and oriented towards the practical implementation into a working code using multiwavelets. The formalism is applied to the case of Poisson and bound-state Helmholtz operators in d = 3. Our algorithms are fully adaptive in the sense that the grid supporting each function is obtained on the fly while the function is being computed. In particular, when the function g = O f is obtained by applying an integral operator O, the corresponding grid is not obtained by transferring the grid from the input function f. This aspect has significant implications that will be discussed in the numerical section. The operator kernels are represented in a separated form with finite but arbitra...

Abstract: These Proceedings include 42 of the 49 invited conference papers, three papers sub- mitted subsequently, and a report devoted to new and unsolved problems based on two special problem sessions and as augmented by later communications from the participants. In addition, there are four short accounts that emphasize the personality of the scholars to whom the proceedings are dedicated. Due to the large number of contributors, the length of the papers had to be restricted. This volume is again devoted to recent significant results obtained in approximation theory, harmonic analysis, functional analysis, and operator theory. The papers solicited include in addition survey articles that not only describe fundamental advances in their subfields, but many also emphasize basic interconnections between the various research areas. They tend to reflect the range of interests of the organizers and of their immediate colleagues and collaborators. The papers have been grouped according to subject matter into ten chapters. Chap- ter I, on operator theory, is devoted to certain classes of operators such as contraction, hyponormal, and accretive operators, as well as to suboperators and semi groups of operators. Chapter II, on functional analysis, contains papers on function spaces, algebras, ideals, and generalized functions. Chapter III, on abstract approximation, is concerned with the comparison of approximation processes, the gliding hump method, certain inter- polation spaces, and n-widths.

Abstract: Magnitude is a numerical invariant of finite metric spaces, recently introduced by Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended to infinite metric spaces in several a priori distinct ways. This paper develops the theory of a class of metric spaces, positive definite metric spaces, for which magnitude is more tractable than in general. Positive definiteness is a generalization of the classical property of negative type for a metric space, which is known to hold for many interesting classes of spaces. It is proved that all the proposed definitions of magnitude coincide for compact positive definite metric spaces and further results are proved about the behavior of magnitude as a function of such spaces. Finally, some facts about the magnitude of compact subsets of $$\ell _p^n$$ for $$p \le 2$$ are proved, generalizing results of Leinster for $$p=1,2$$ using properties of these spaces which are somewhat stronger than positive definiteness.

Abstract: This article deals with linear operators T on a complex Hilbert space ℋ, which are bounded with respect to the seminorm induced by a positive operator A on ℋ. The A-adjoint and A 1/2-adjoint of T are considered to obtain some ergodic conditions for T with respect to A. These operators are also employed to investigate the class of orthogonally mean ergodic operators as well as that of A-power bounded operators. Some classes of orthogonally mean ergodic or A-ergodic operators, which come from the theory of generalized Toeplitz operators are considered. In particular, we give an example of an A-ergodic operator (with an injective A) which is not Cesaro ergodic, such that T * is not a quasiaffine transform of an orthogonally mean ergodic operator.

Abstract: We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones.

Abstract: Many recent papers have been devoted to the ideal of p-compact operators, denoted by Kp. In this note, we will revisit their main results by very simple proofs that are based on the general theory of operator ideals as developed in the author’s monograph more than 30 years ago. In several cases the outcome is even better. Moreover, the ideal of all operators with p-summing duals is identified as the maximal hull of Kp. As a byproduct, we show that Kmax p is stable under the formation of projective tensor products. Operator ideals play an important role in the study of Banach spaces and their operators. The aim of this note is to demonstrate the well-known interplay of the basic operations on operator ideals. This approach allows us to replace several long reasonings by one-line-proofs. Our methods are applied to the ideal Kp formed by the p-compact operators. The most significant observation says that this ideal is surjective and regular. Moreover, we show that the maximal hull of Kp consists of all operators whose dual is p-summing. For unexplained notation and terminology in the text we refer to [15]; auxiliary results, adopted from [15], are just marked by triplets, such as (4.7.16), which means [15, 4.7.16]. For simplicity, we mostly deal with operator ideals A, B, . . . only, but not with the underlying norms A, B, . . . . It should be noted, however, that (in our context) A=B always includes A=B. According to [19, p. 20], a subset K of a Banach space E is called relatively p-compact if there exists a sequence (xn) in E such that K ⊆ { ∞ ∑

Abstract: We develop techniques to study the correlation functions of “large operators” whose bare dimension grows parametrically with N, in SO(N) gauge theory. We build the operators from a single complex matrix. For these operators, the large N limit of correlation functions is not captured by summing only the planar diagrams. By employing group representation theory we are able to define local operators which generalize the Schur polynomials of the theory with gauge group U(N). We compute the two point function of our operators exactly in the free field limit showing that they diagonalize the two point function. We explain how these results can be used to obtain the exact free field answers for correlators of operators in the trace basis.

Abstract: In the present paper we introduce a q-analogue of the Bernstein-Kantorovich operators and investigate their approximation properties. We study local and global approximation properties and Voronovskaja type theorem for the q-Bernstein-Kantorovich operators in case 0 < q < 1.