Semi-classical linear functionals of class three: the symmetric case

In this paper, we obtain all the symmetric semi-classical linear functionals of class three taking into account the irreducible expression of the corresponding Pearson equation. We focus our attention on their integral representations. Thus, some linear functionals very well known in the literature, associated with perturbations of semi-classical linear functionals of class one at most, appear as well as new linear functionals which have not been studied.     

Weight Monte Carlo algorithms for estimation and parametric analysis of the solution to the kinetic coagulation equation

The Smoluchowski equation with linear coagulation coefficients depending on two parameters is considered. We construct weight algorithms for estimating various linear functionals in an ensemble that is governed by the equation under study. The algorithms constructed allow us to estimate the functionals for various parameters, as well as parametric derivatives by using the same set of trajectories. Moreover, we construct the value algorithms and analyze their efficiency for estimating the total monomer concentration, as well as the total monomer and dimer concentration in the ensemble. The computational cost is considerably reduced via the approximate value simulation of the time between interactions combined with the value simulation of the interacting pair number.     

Characterization of the D w -Laguerre-Hahn Functionals

We give some characterization theorems for the D w -Laguerre-Hahn linear functionals and we extend the concept of the class of the usual Laguerre-Hahn functionals to the D w -Laguerre-Hahn functionals, recovering the classic results when w tends to zero. Moreover, we show that some transformations carried out on the D w -Laguerre-Hahn linear functionals lead to new D w -Laguerre-Hahn linear functionals. Finally, we analyze the class of the resulting functionals and we give some applications relative to the first associated Charlier, Meixner, Krawtchouk and Hahn orthogonal polynomials.     

Modifications of quasi-definite linear functionals via addition of delta and derivatives of delta Dirac functions

We consider the general theory of the modifications of quasi-definite linear functionals by adding discrete measures. We analyze the existence of the corresponding orthogonal polynomial sequences with respect to such linear functionals. The three-term recurrence relation, lowering and raising operators as well as the second order linear differential equation that the sequences of monic orthogonal polynomials satisfy when the linear functional is semiclassical are also established. A relevant example is considered in details.     

Tractability of Tensor Product Linear Operators

This paper deals with the worst case setting for approximating multivariate tensor product linear operators defined over Hilbert spaces. Approximations are obtained by using a number of linear functionals from a given class of information. We consider the three classes of information: the class of all linear functionals, the Fourier class of inner products with respect to given orthonormal elements, and the standard class of function values. We wish to determine which problems are tractable and which are strongly tractable. The complete analysis is provided for approximating operators of rank two or more. The problem of approximating linear functionals is fully analyzed in the first two classes of information. For the third class of standard information we show that the possibilities are very rich. We prove that tractability of linear functionals depends on the given space of functions. For some spaces all nontrivial normed linear functionals are intractable, whereas for other spaces all linear functionals are tractable. In “typical” function spaces, some linear functionals are tractable and some others are not.     

Variational calculus with sums of elementary tensors of fixed rank

In this article we introduce a calculus of variations for sums of elementary tensors and apply it to functionals of practical interest. The survey provides all necessary ingredients for applying minimization methods in a general setting. The important cases of target functionals which are linear and quadratic with respect to the tensor product are discussed, and combinations of these functionals are presented in detail. As an example, we consider the solution of a linear system in structured tensor format. Moreover, we discuss the solution of an eigenvalue problem with sums of elementary tensors. This example can be viewed as a prototype of a constrained minimization problem. For the numerical treatment, we suggest a method which has the same order of complexity as the popular alternating least square algorithm and demonstrate the rate of convergence in numerical tests.     

Christoffel Transforms and Hermitian Linear Functionals

In this contribution we are focused on some spectral transformations of Hermitian linear functionals. They are the analogues of the Christoffel transform for linear functionals, i. e. for Jacobi matrices which has been deeply studied in the past. We consider Hermitian linear functionals associated with a probability measure supported on the unit circle. In such a case we compare the Hessenberg matrices associated with such a probability measure and its Christoffel transform. In this way, almost unitary matrices appear. We obtain the deviation to the unit matrix both for principal submatrices and the complete matrices respectively.     

Transform formulae for linear functionals of affine processes and their bridges on positive semidefinite matrices

In this paper, we derive transform formulae for linear functionals of affine processes and their bridges whose state space is the set of positive semidefinite d×d$d\times d$ matrices. Particularly, we investigate the relationship between such transforms and certain integral equations. Our findings extend and unify the well known results of Cuchiero et al. (2011) [5] and Pitman and Yor (1982) [19], who analysed affine processes on positive semidefinite matrices and transforms of linear functionals of squared Bessel processes, respectively. We are, then, able to derive analytic expressions for Laplace transforms of some functionals of Wishart bridges.     

On the properties of McShane’s functional and their applications

In this paper we derive some remarkable properties of McShane’s functional, defined by means of positive isotonic linear functionals. These properties are then applied to weighted generalized means. A series of consequences among additive and multiplicative type mean inequalities is given, as well as a special consideration of Hölder’s inequality, in view of the new results.     

Informativeness of Linear Functionals

In this paper, we study the informativeness of linear functionals in reconstruction problems and obtain exact orders of the informativeness of linear functionals in the Besov and Sobolev classes W and SW.     

Dividierte Differenzen und Monotonie von Quadraturformeln

Summary As a generalisation of divided differences we consider linear functionals vanishing for polynomials of given degree and with discrete support. It is shown that functionals of that type may be uniquely represented by a linear combination of divided differences. On the basis of this representation theorem we introduce the concept of positivity and definiteness of functions and linear functionals. Next we show that in many cases positivity follows from the number of sign changes of the coefficients of the given linear functional. These results may be applied to the problems of nonexistence of Newton-Côtes and Gegenbauer quadrature formulas with positive weights and to the monotony problem of Gauss and Newton Côtes quadrature.

     

Asymptotic behaviour of the energy for electromagnetic systems with memory

Some linear evolution problems arising in the theory of hereditary electromagnetism are considered here. Making use of suitable Liapunov functionals, existence of solutions as well as asymptotic behaviour, are determined for rigid conductors with electric memory. In particular, we show the polynomially decay of the solutions, when the memory kernel decays exponentially or polynomially. Copyright © 2004 John Wiley & Sons, Ltd.     

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A unified approach to parallel space decomposition methods

We consider (relaxed) additive and multiplicative iterative space decomposition methods for the minimization of sufficiently smooth functionals without constraints. We develop a general framework which unites existing approaches from both parallel optimization and finite elements. Specifically this work unifies earlier research on the parallel variable distribution method in minimization, space decomposition methods for convex functionals, algebraic Schwarz methods for linear systems and splitting methods for linear least squares. We develop a general convergence theory within this framework, which provides several new results as well as including known convergence results.     

Triangular expansions

We use umbral methods to obtain several general expansion theorems for linear operators and linear functionals. We show in particular that every linear operator admits Newton-type expansions. These expansions are prototypes of numerous classical ones as well as new ones, some of which are powerful enough for use in numerical approximation.     

Sure wins, separating probabilities and the representation of linear functionals

We discuss conditions under which a convex cone K⊂RΩ admits a finitely additive probability m such that supk∈Km(k)?0. Based on these, we characterise those linear functionals that are representable as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions.     

Application of He’s variational iteration method to Helmholtz equation

In this article, we implement a new analytical technique, He’s variational iteration method for solving the linear Helmholtz partial differential equation. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via the variational theory. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the boundary/initial conditions. The results compare well with those obtained by the Adomian’s decomposition method.     

Asymptotics for statistical functionals of long-memory sequences

We present two general results that can be used to obtain asymptotic properties for statistical functionals based on linear long-memory sequences. As examples for the first one we consider L- and V-statistics, in particular tail-dependent L-statistics as well as V-statistics with unbounded kernels. As an example for the second result we consider degenerate V-statistics. To prove these results we also establish a weak convergence result for empirical processes of linear long-memory sequences, which improves earlier ones.     

Deviation inequalities for quadratic Wiener functionals and moderate deviations for parameter estimators

We study deviation inequalities for some quadratic Wiener functionals and moderate deviations for parameter estimators in a linear stochastic differential equation model. Firstly, we give some estimates for Laplace integrals of the quadratic Wiener functionals by calculating the eigenvalues of the associated Hilbert-Schmidt operators. Then applying the estimates, we establish deviation inequalities for the quadratic functionals and moderate deviation principles for the parameter estimators.     

Variational Characterizations of Weak Solutions to the Dirichlet Problem for Mixed‐Type Equations

For linear and nonlinear second‐order partial differential equations of mixed elliptic‐hyperbolic type, we prove that weak solutions to the Dirichlet problem are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functional restricted to suitable infinite‐dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights associated to the linear operator. This spectral theory has been recently developed by the authors, which in turn exploits weak well‐posedness results obtained by Morawetz and the authors. © 2015 Wiley Periodicals, Inc.