The Eigenspace Spectral Regularization Method for Solving Discrete Ill-Posed Systems

This paper shows that discrete linear equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, TST matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. Gauss least square method (GLSM), QR factorization method (QRFM), Cholesky decomposition method (CDM), and singular value decomposition (SVDM) failed to regularize these ill-posed problems. This paper introduces the eigenspace spectral regularization method (ESRM), which solves ill-posed discrete equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, and banded and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularizes such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator κ K = K − 1 K = 1 . Thus, the condition number of ESRM is bounded by unity, unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.

Poisson like matrix operator and its application in p-summable space

The incomplete gamma function Γ(a, u) is defined by Γ ( a , u ) = ∫ u ∞ t a − 1 e − t d t , $$\Gamma(a,u)=\int\limits_{u}^{\infty}t^{a-1}\textrm{e}^{-t}\textrm{d} t,$$ where u > 0. Using the incomplete gamma function, we define a new Poisson like regular matrix P ( μ ) = ( p n k μ ) $\mathfrak{P}(\mu)=(p^{\mu}_{nk})$ given by p n k μ = n ! Γ ( n + 1 , μ ) e − μ μ k k ! ( 0 ≤ k ≤ n ) , 0 ( k > n ) , $$p^{\mu}_{nk}= \begin{cases} \dfrac{n!}{\Gamma(n+1,\mu)}\dfrac{\textrm{e}^{-\mu}\mu^k}{k!} \quad &(0\leq k\leq n), \\[1ex] 0\quad & (k>n), \end{cases}$$ where μ > 0 is fixed. We introduce the sequence space ℓ p ( P ( μ ) ) $\ell_p(\mathfrak{P}(\mu))$ for 1 ≤ p ≤ ∞ and some topological properties, inclusion relations and generalized duals of the newly defined space are discussed. Also we characterize certain matrix classes and compact operators related to the space ℓ p ( P ( μ ) ) $\ell_p(\mathfrak{P}(\mu))$ . We obtain Gurarii’s modulus of convexity and investigate some geometric properties of the new space. Finally, spectrum of the operator P ( μ ) $\mathfrak{P}(\mu)$ on sequence space c 0 has been investigated.

The Riemann boundary value problem related to the time-harmonic Maxwell equations

In this paper, we first give the definition of Teodorescu operator related to the $\mathcal{N}$ N matrix operator and discuss a series of properties of this operator, such as uniform boundedness, Hölder continuity and so on. Then we propose the Riemann boundary value problem related to the $\mathcal{N}$ N matrix operator. Finally, using the intimate relationship of the corresponding Cauchy-type integral between the $\mathcal{N}$ N matrix operator and the time-harmonic Maxwell equations, we investigate the Riemann boundary value problem related to the time-harmonic Maxwell equations and obtain the integral representation of the solution.

Banach spaces of absolutely k-summable series

In this paper, we present a general Banach space of absolutely k-summable series using a triangle matrix operator and prove that this is a BK-space isometrically isomorphic to the space ℓ k {\ell_{k}} . We also establish the α - {\alpha-} , β - {\beta-} , γ-duals and base of the new space. Finally, we qualify some matrix and compact operators on the new space making use of the Hausdorff measure of noncompactness. Our results include, as particular cases, a number of well-known results.

On the Spectrum of Hilbert Matrix Operator

The Hilbert matrix $$\begin{aligned} {\mathcal {H}}_\lambda =\left( \frac{1}{n+m+\lambda }\right) _{n,m=0}^{\infty }, \quad \lambda \ne 0,-1,-2, \ldots \, \end{aligned}$$ H λ = 1 n + m + λ n , m = 0 ∞ , λ ≠ 0 , - 1 , - 2 , … generates a bounded linear operator in the Hardy spaces $$H^p$$ H p and in the $$l^p$$ l p -spaces. The aim of this paper is to study the spectrum of this operator in the spaces mentioned. In a sense, the presented investigation continues earlier works of various authors. More information concerning the history of the topic can be found in the introduction.

The Eigenspace Spectral Regularization Method for solving Discrete Ill-Posed Systems

In this paper, it is shown that discrete equations with Hilb ert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. These ill-posed problems cannot be regularized by Gauss Least Square Method (GLSM), QR Factorization Method (QRFM), Cholesky Decomposition Method (CDM) and Singular Value Decomposition (SVDM). To overcome the limitations of these methods of regularization, an Eigenspace Spectral Regularization Method (ESRM) is introduced which solves ill-p os ed discrete equations with Hilb ert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularize such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator κ (K) = ||K − 1K|| = 1. Thus, the condition number of ESRM is bounded by unity unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.