Abstract: We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators The principal innovation in these algorithms is that they are block-iterative in the sense that, at each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in established methods Flexible strategies are used to select the blocks of operators activated at each iteration In addition, we allow lags in operator processing, permitting asynchronous implementation The decomposition phase of each iteration of our methods is to generate points in the graphs of the selected monotone operators, in order to construct a half-space containing the Kuhn–Tucker set associated with the system The coordination phase of each iteration involves a projection onto this half-space We present two related methods: the first method provides weakly convergent primal and dual sequences under general conditions, while the second is a variant in which strong convergence is guaranteed without additional assumptions Neither algorithm requires prior knowledge of bounds on the linear operators involved or the inversion of linear operators Our algorithmic framework unifies and significantly extends the approaches taken in earlier work on primal-dual projective splitting methods

Abstract: The present paper deals with the Stancu-type generalization of (p, q)-Baskakov–Durrmeyer operators. We investigate local approximation, weighted approximation properties of new operators and present the rate of convergence by means of suitable modulus of continuity. At the end of the paper, we introduce a new modification of (p, q)-Baskakov–Durrmeyer–Stancu operators with King approach.

Abstract: In the present paper, we introduce Kantorovich modifications of (p, q)-Bernstein operators using a new (p, q) -integral. We first estimate the moments and central moments. We obtain uniform convergence of new operators, rate of convergence in terms of classical modulus of continuity and second order modulus of continuity. We also investigate the rate of convergence of new operators for functions belonging to Lipschitz class and finally, we give an upper bound for the error of approximation via modulus of continuity of the derivative of approximating function.

Abstract: This paper presents a new approach to improve the order of approximation of the Bernstein operators. Three new operators of the Bernstein-type with the degree of approximations one, two, and three are obtained. Also, some theoretical results concerning the rate of convergence of the new operators are proved. Finally, some applications of the obtained operators such as approximation of functions and some new quadrature rules are introduced and the theoretical results are verified numerically.

Abstract: In this paper, we discuss convergence of the $$q-$$ derivatives of a new sequence of positive linear operators. We also find degree of approximation in terms of modulus of smoothness of the $$q-$$ derivatives of the corresponding functions.

Abstract: A new representation for a regular solution of the perturbed Bessel equation of the form is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to . For the coefficients of the series, explicit direct formulas are obtained in terms of the systems of recursive integrals arising in the spectral parameter power series (SPPS) method, as well as convenient for numerical computation recurrent integration formulas. The result is based on application of several ideas from the classical transmutation (transformation) operator theory, recently discovered mapping properties of the transmutation operators involved and a Fourier–Legendre series expansion of the transmutation kernel. For convergence rate estimates, asymptotic formulas, a Paley–Wiener theorem, and some results from constructive approximation theory were used. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able...

Abstract: The class of matrix optimization problems (MOPs) has been recognized in recent years to be a powerful tool to model many important applications involving structured low rank matrices within and beyond the optimization community. This trend can be credited to some extent to the exciting developments in emerging fields such as compressed sensing. The Lowner operator, which generates a matrix valued function via applying a single-variable function to each of the singular values of a matrix, has played an important role for a long time in solving matrix optimization problems. However, the classical theory developed for the Lowner operator has become inadequate in these recent applications. The main objective of this paper is to provide necessary theoretical foundations from the perspectives of designing efficient numerical methods for solving MOPs. We achieve this goal by introducing and conducting a thorough study on a new class of matrix valued functions, coined as spectral operators of matrices. Several fundamental properties of spectral operators, including the well-definedness, continuity, directional differentiability and Frechet-differentiability are systematically studied.

Abstract: The Paulsen problem is a basic open problem in operator theory: Given vectors u1, …, un ∈ ℝd that are є-nearly satisfying the Parseval’s condition and the equal norm condition, is it close to a set of vectors v1, …, vn ∈ ℝd that exactly satisfy the Parseval’s condition and the equal norm condition? Given u1, …, un, the squared distance (to the set of exact solutions) is defined as infv ∑i=1n || ui − vi ||22 where the infimum is over the set of exact solutions. Previous results show that the squared distance of any є-nearly solution is at most O(poly(d,n,є)) and there are є-nearly solutions with squared distance at least Ω(d є). The fundamental open question is whether the squared distance can be independent of the number of vectors n. We answer this question affirmatively by proving that the squared distance of any є-nearly solution is O(d13/2 є). Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any є-nearly solution is O(d2 n є). Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an є-nearly solution is O(d5/2 є) when n is large enough and є is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyzing the operator scaling algorithm.

Abstract: In the present article, we study modified form of Szasz–Mirakyan–Kantorovich operators, which reproduce constant and $$e^{-x}$$ functions. We discuss a uniform convergence result along with a quantitative estimate for the modified operators.

Abstract: In the present paper we introduce (p, q) variant of Szasz–Mirakyan–Baskakov operators, we estimate moments and establish some approximation results which include weighted approximation and a direct estimate in terms of modulus of continuity. We have used linear approximating method namely Steklov mean.

Abstract: In this paper, we introduce and study new concepts of almost L-weakly and almost M-weakly compact operators.

Abstract: Boundary value problems for Sturm-Liouville operators with potentials from the class $W_2^{-1}$ on a star-shaped graph are considered. We assume that the potentials are known on all the edges of the graph except two, and show that the potentials on the remaining edges can be constructed by fractional parts of two spectra. A uniqueness theorem is proved, and an algorithm for the constructive solution of the partial inverse problem is provided. The main ingredient of the proofs is the Riesz-basis property of specially constructed systems of functions.

Abstract: The current article deals with the study of Baskakov–Szasz–Mirakyan operators which reproduces constant and exponential functions. We discuss a uniform estimate and establish a quantitative result for the modified operators.

Abstract: A tuple of commuting operators $$(S_1,\dots ,S_{n-1},P)$$ for which the closed symmetrized polydisc $$\Gamma _n$$ is a spectral set is called a $$\Gamma _n$$ -contraction. We show that every $$\Gamma _n$$ -contraction admits a decomposition into a $$\Gamma _n$$ -unitary and a completely non-unitary $$\Gamma _n$$ -contraction. This decomposition is an analogue to the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $$\Gamma _n$$ and $$\Gamma _n$$ -contractions.

Abstract: Nonlinear dynamical systems are ubiquitous in science and engineering, yet analysis and prediction of these systems remains a challenge. Koopman operator theory circumvents some of these issues by considering the dynamics in the space of observable functions on the state, in which the dynamics are intrinsically linear and thus amenable to standard techniques from numerical analysis and linear algebra. However, practical issues remain with this approach, as the space of observables is infinite-dimensional and selecting a subspace of functions in which to accurately represent the system is a nontrivial task. In this work we consider time-delay observables to represent nonlinear dynamics in the Koopman operator framework. We prove the surprising result that Koopman operators for different systems admit universal (system-independent) representations in these coordinates, and give analytic expressions for these representations. In addition, we show that for certain systems a restricted class of these observables form an optimal finite-dimensional basis for representing the Koopman operator, and that the analytic representation of the Koopman operator in these coordinates coincides with results computed by the dynamic mode decomposition. We provide numerical examples to complement our results. In addition to being theoretically interesting, these results have implications for a number of linearization algorithms for dynamical systems.

Abstract: In this paper, we introduce and study vector valued multiplier spaces with the help of the sequence of continuous linear operators between normed spaces and Cesaro convergence. Also, we obtain a new version of the Orlicz–Pettis Theorem by means of Cesaro summability.

Abstract: We analyze eigenvalues emerging from thresholds of the essential spectrum of one-dimensional Dirac operators perturbed by complex and non-symmetric potentials In the general non-self-adjoint setting, we establish the existence and asymptotics of weakly coupled eigenvalues and Lieb–Thirring inequalities As physical applications, we investigate the damped wave equation and armchair graphene nanoribbons

Abstract: In the present paper, we discuss the pseudo-fractional calculus, including two fields of fractional calculus and pseudo-analysis. We also provide pseudo-fractional integral/derivative operators on a semiring \(([a,b],\oplus ,\odot )\). Then, some basic properties of these operators are proposed such as the Leibniz rule, chain rule and g-Laplace transform formulas.

Abstract: Matrix valued truncated Toeplitz operators act on vector-valued model spaces. They represent a generalization of block Toeplitz matrices. A characterization of these operators analogue to the scalar case is obtained, as well as the determination of the symbols that produce the zero operator.

Abstract: Extrapolation results in weighted grand Lebesgue spaces defined with respect to product measure $$\mu \times u $$ on $$X\times Y$$ , where $$(X, d, \mu )$$ and $$(Y, \rho , u )$$ are spaces of homogeneous type, are obtained. As applications of the derived results we prove new one-weight estimates for multiple integral operators such as strong maximal, Calderon–Zygmund and fractional integral operators with product kernels in these spaces.

Abstract: We characterize those operators that satisfy the properties of monotonicity, permutation invariance, positive homogeneity, and translation invariance. As these operators do not necessarily satisfy comonotonic additivity, their class is larger than that of ordered weighted averaging (OWA) operators. We give a representation theorem for these operators, which shows, nonetheless, that this more general class can be constructed directly from that of OWA operators. In addition, we characterize the special classes consisting of operators that are either subadditive or superadditive. We suggest applications to the evaluation of complex systems.

Abstract: We establish an analog for bilinear operators of the compactness interpolation result for bounded linear operators proved by Cwikel and Cobos, Kuhn and Schonbek. We work with the assumption that $$T:(A_0+A_1) \times (B_0+B_1) \longrightarrow E_0+E_1$$ is bounded, but we also study the case when this does not hold. Applications are given to compactness of convolution operators and compactness of commutators of bilinear Calderon–Zygmund operators.

Abstract: This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based on nuclear algebra of white noise operators acting on spaces of entire functions with $$\theta $$ -exponential growth of minimal type. First, we use extended techniques of rotation invariance operators, the commutation relations with respect to the QWN-derivatives and the QWN-conservation operator. Second, we employ the new concept of QWN-convolution operators. As application, we study and characterize the powers of the QWN-Gross Laplacian. As for their associated Cauchy problem it is solved using a QWN-convolution and Wick calculus.

Abstract: In this paper, using power series method we obtain a Korovkin type theorem for double sequences of real valued functions defined on a compact subset of $$\mathbb {R}^{2}$$ (the real two-dimensional space). We also present an example that satisfies our theorem. Finally, we calculate the rate of convergence.

Abstract: In the current article, we study Szasz–Mirakyan–Durrmeyer operators which reproduces constant and e2ax,a>0 functions. We discuss a uniform estimate and estimate a quantitative asymptotic formula fo...