General Linear Operator Research Articles

Asymptotic properties of powers of linear positive operators which preserve [formula omitted

In this paper we prove Korovkin type theorem for iterates of general positive linear operators T : C [ 0 , 1 ] → C [ 0 , 1 ] which preserve e 2 and derive quantitative estimates in terms of moduli of smoothness. The results can be applied to several well-known operators; we present here the Bernstein, the q -Bernstein, the genuine Bernstein–Durrmeyer and the genuine q -Bernstein–Durrmeyer operators.

Approximation of integrable functions by general linear operators of their Fourier series at the Lebesgue points

Pointwise estimates of the deviation T n,A,B f(⋅)−f(⋅) in terms of moduli of continuity \(\bar{w}_{\cdot}f\) and w ⋅ f are proved. Analog results on norm approximation with remarks and corollaries are also given. In the results essentially weaker conditions than those in [2, Theorem 1, p. 437] are used.

Approximation of functions from Lp(ω)βby general linear operators of their Fourier series

We show the general and precise conditions on the functions and modulus of continuity as well as on the entries of matrices generating the summability means and give the rates of approximation of functions from the generalized integral Lipschitz classes b

An iterative method for Tikhonov regularization with a general linear regularization operator

Tikhonov regularization is one of the most popular approaches to solve discrete ill-posed problems with error-contaminated data. A regularization operator and a suitable value of a regularization parameter have to be chosen. This paper describes an iterative method, based on Golub-Kahan bidiagonalization, for solving large-scale Tikhonov minimization problems with a linear regularization operator of general form. The regularization parameter is determined by the discrepancy principle. Computed examples illustrate the performance of the method.

Toeplitz operators with L 1 symbols on Bergman spaces in the unit ball of

This paper contains new results on L1 symbols of bounded (resp. compact) Toeplitz operators on Bergman spaces in the unit disk of the complex plane, ℂ. Our results improve and extend earlier results of N. Zorboska, J. Miao and D. Zheng. We also extend these results to the unit ball of Cn. Part of our study can be done for more general linear operators on Bergman spaces.

Real analytic parameter dependence of solutions of differential equations

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation P(D)u=f , i.e., the question if for every family (f_\lambda)\subseteq \mathscr{D}'(\Omega) of distributions depending in a real analytic way on \lambda\in U , U a real analytic manifold, there is a family of solutions (u_\lambda)\subseteq \mathscr D'(\Omega) also depending analytically on \lambda such that P(D)u_\lambda=f_\lambda\qquad \text{for every $\lambda\in U$}, where \Omega\subseteq \mathbb{R}^ d an open set. For general surjective variable coefficients operators or operators acting on currents over a smooth manifold we give a solution in terms of an abstract “Hadamard three circle property” for the kernel of the operator. The obtained condition is evaluated giving the full solution (usually in terms of the symbol) for operators with constant coefficients and open (convex) \Omega\subseteq\mathbb{R}^d if P(D) is of one of the following types: 1) elliptic, 2) hypoelliptic, 3) homogeneous, 4) of two variables, 5) of order two or 6) if P(D) is the system of Cauchy–Riemann equations. An analogous problem is solved for convolution operators of one variable. In all enumerated cases, it follows that the solution is in the affirmative if and only if P(D) has a linear continuous right inverse which shows a striking difference comparing with analogous smooth or holomorphic parameter dependence problems. The paper contains the whole theory working also for operators on Beurling ultradistributions \mathscr{D}'_{(\omega)} . We prove a characterization of surjectivity of tensor products of general surjective linear operators on a wide class of spaces containing most of the natural spaces of classical analysis.

An iterative algorithm for solving a finite-dimensional linear operator equation [formula omitted] with applications

This paper proposes an iterative algorithm for solving a general finite-dimensional linear operator equation T ( x ) = f and demonstrates that it will get the exact solution within a finite number of iteration steps. This algorithm unifies all the iterative methods in Huang (2008) [3], Peng (2005) [7] and Peng and Peng (2006) [8] and provides an iterative method for solving an inverse problem related to Hermitian-generalized Hamiltonian matrices [2,12].

Relations of 3D directional derivatives and expressions of typical differential operators

Relations of the 3D multi-directional derivatives are studied in this paper. These relations are applied to a general second-order linear elliptical operator and the corresponding expression are obtained. These relations and expressions play important roles in the meshless finite point method.

Time distributed-order diffusion-wave equation. I. Volterra-type equation

A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order diffusion-wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.

First-order symmetries of the Dirac equation in a curved background: a unified dynamical symmetry condition**Dedicated to Professor T Dereli on the occasion of his 60th birthday.

It has been shown that, for all dimensions and signatures, the most general first-order linear symmetry operators for the Dirac equation including interaction with Maxwell field in a curved background are given in terms of Killing–Yano (KY) forms. As a general gauge invariant condition it is found that among all KY forms of the underlying (pseudo) Riemannian manifold, only those which Clifford commute with the Maxwell field take part in the symmetry operator. It is also proved that associated with each KY form taking part in the symmetry operator, one can define a quadratic function of velocities which is a geodesic invariant as well as a constant of motion for the classical trajectory. Some geometrical and physical implications of the existence of KY forms are also elucidated.

Identifying functions determined by linear functional operators

The new identifying problem is formulated for general linear functional operators \( \mathcal{P} \)F = ΣcjF ○ aj which significantly generalizes in particular the well-known Ulam stability problem. The results obtained can be very useful when processing experimental data of any kind as they enable to determine with high precision the structure of a compactly supported Banach-valued function F by using a rather restricted information concerning \( \mathcal{P} \)F.

Multivariate stochastic Korovkin theory given quantitatively

We introduce and study very general multivariate stochastic positive linear operators induced by general multivariate positive linear operators that are acting on multivariate continuous functions. These are acting on the space of real differentiable multivariate time stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we produce related multidimensional stochastic Shisha–Mond type inequalities of L q -type 1 ≤ q < ∞ and corresponding multidimensional stochastic Korovkin type theorems. These are regarding the stochastic q -mean convergence of a sequence of multivariate stochastic positive linear operators to the stochastic unit operator for various cases. All convergences are produced with rates and are given via the stochastic inequalities involving the maximum of the multivariate stochastic moduli of continuity of the n th order partial derivatives of the engaged stochastic process, n ≥ 0 . The astonishing fact here is that basic real Korovkin test functions assumptions are enough for the conclusions of our multidimensional stochastic Korovkin theory. We give an application.

A spectral approach to ill-posed problems for wave equations

In this article, we consider the problem of finding a solution to ill-posed problems for abstract wave equations in a Hilbert space, of the form $$\frac{{\rm d}^{2}u}{{\rm d}t^{2}}\left( t\right) + Au \left(t\right) = 0,\quad t \in \left(0, T\right),\quad u \left(0\right) = 0,\quad u \left(T\right) = u_{0},$$ when A is a general linear selfadjoint operator. We study issues like existence, uniqueness and continuance dependance of data and stability for this problem. Under precise constraint conditions on T, we make such problems well posed and in effect, generalize known results about these equations.

More on determination of jumps

We generalize some old and new results on the determination of jumps of a periodic, Lebesgue integrable function f at each point of discontinuity of first kind in terms of the partial sums of the conjugate series to the Fourier series of f in [1], and in terms of the Abel-Poisson means in [2], to some more general linear operators which satisfy some certain conditions. The linear operators in discussion include the Fejer means, de la Vallee-Poussin means, and Bernstein-Rogosinski sums.

Spectral theory for general nonautonomous/random dispersal evolution operators

We investigate the spectral theory of the following general nonautonomous evolution equation ∂ t u ( t , x ) = A ( u ( t , ⋅ ) ) ( x ) + h ( t , x ) u ( t , x ) , x ∈ D , where D is a bounded subset of R N which can be a smooth domain or a discrete set, A is a general linear dispersal operator (for example a Laplacian operator, an integral operator with positive kernel or a cooperative discrete operator) and h ( t , x ) is a smooth function on R × D ¯ . We first study the influence of time dependence on the principal spectrum of dispersal equations and show that the principal Lyapunov exponent of a time-dependent dispersal equation is always greater than or equal to that of the time-averaged one. Several results about the principal eigenvalue of time-periodic parabolic equations are extended to general time-periodic dispersal ones. Finally, the investigation is generalized to random time-dependent dispersal equations.

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.

Robust generalized cross-validation for choosing the regularization parameter

Let fλ be the regularized solution for the problem of estimating a function or vector f0 from noisy data yi = Lif0 + εi, i = 1, …, n, where Li are linear functionals. A prominent method for the selection of the crucial regularization parameter λ is generalized cross-validation (GCV). It is known that GCV has good asymptotic properties as n → ∞ but it may not be reliable for small or medium sized n, sometimes giving an estimate that is far too small. We propose a new robust GCV method (RGCV) which chooses λ to be the minimizer of γV(λ) + (1 − γ)F(λ), where V(λ) is the GCV function, F(λ) is an approximate average measure of the influence of each data point on fλ and γ ∊ (0, 1) is a robustness parameter. We show that for any n, RGCV is less likely than GCV to choose a very small value of λ, resulting in a more robust method. We also show that RGCV has good asymptotic properties as n → ∞ for general linear operator equations with uncorrelated errors. The function EF(λ) approximates the risk ER(λ) for values of λ that are asymptotically a bit smaller than the minimizer of ER(λ) (where V(λ) may not approximate well). The ‘expected’ RGCV estimate is asymptotically optimal as n → ∞ with respect to the ‘robust risk’ γER(λ) + (1 − γ)v(λ), where v(λ) is the variance component of the risk, and it has the optimal decay rate with respect to ER(λ) and stronger error criteria. The GCV and RGCV methods are compared in numerical simulations for the problem of estimating the second derivative from noisy data. The results for RGCV with n = 51 are consistent with the asymptotic results, and, for a large range of γ values, RGCV is more reliable and accurate than GCV.

On the modified beta approximating operators of second kind

We shall define a general linear operator from which we obtain as particular case the modified beta second kind operator\[(T_{p,q}f)(x)=\frac{1}{B(p,q)}\int_0^\infty\tfrac{u^{p-1}}{(1+u)^{p+q}}f\left(\frac{B(p,q)}{B(p+a,q-a)}u^ax\right){\rm d}u.\] We consider here only the case \(a=1\). We obtain several positive linear operators as a particular case of this modified beta second kind operator.

Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators

By making use of a general linear operator I p λ(a,c) , the authors introduce several new subclasses of multivalent functions and investigate various inclusion relationships and argument properties associated with these subclasses. Some interesting applications involving such and other families of linear operators are also considered. The results presented here include a number of known results as their special cases.

On the modified beta approximating operators of first kind

We define a general linear operator from which we obtain as special case the modified beta first kind operator\[(B_{p,q}f)(x)={B(p,q)}\int_0^1 t^{p-1}(1-t)^{q-1}f\left({B(p,q)}{B(p+a,q)}t^a x\right){\rm d}t.\]We consider here only the cases \(a=1\) and \(a=-1\).We obtain several positive linear operators as particular cases of this modified beta first kind operator.