The Spectral Approximation of Linear Operators with Applications to the Computation of Eigenelements of Differential and Integral Operators

SIAM Review ◽  
1981 ◽  
Vol 23 (4) ◽  
pp. 495-522 ◽  
Author(s):  
Françoise Chatelin
Keyword(s):  
Integral Operators ◽  
Linear Operators ◽  
Spectral Approximation

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

scholarly journals The Convergence of the Legendre–Galerkin Spectral Method for Constructing Atmospheric Acoustic Normal Modes

2020 ◽  
Vol 28 (03) ◽  
pp. 2050002
Author(s):  
Richard B. Evans
Keyword(s):  
Legendre Polynomials ◽  
Normal Modes ◽  
Sobolev Norm ◽  
Linear Operators ◽  
Basis Sets ◽  
Spectral Approximation ◽  
Spectral Approximations ◽  
Potential Applications ◽  
Galerkin Spectral Method ◽  
Theoretical Results

The asymptotic rate of convergence of the Legendre–Galerkin spectral approximation to an atmospheric acoustic eigenvalue problem is established, as the dimension of the approximating subspace approaches infinity. Convergence is in the [Formula: see text] Sobolev norm and is based on the existing theory [F. Chatelin, Spectral Approximations of Linear Operators (SIAM, 2011)]. The assumption is made that the eigenvalues are simple. Numerical results that help interpret the theory are presented. Eigenvalues corresponding to acoustic modes with smaller [Formula: see text] norms are especially accurately approximated, even with lower dimensioned basis sets of Legendre polynomials. The deficiencies in the potential applications of the theoretical results are noted in connection with the numerical examples.

scholarly journals Spectral approximation theorems for bounded linear operators

1973 ◽  
Vol 8 (2) ◽  
pp. 279-287 ◽  
Author(s):  
W.S. Lo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Eigenvalue Problem ◽  
Linear Operators ◽  
Spectral Approximation ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Approximation Theorems ◽  
Generalized Eigenvectors

In this paper we present some approximation theorems for the eigenvalue problem of a compact linear operator defined on a Banach space. In particular we examine: criteria for the existence and convergence of approximate eigenvectors and generalized eigenvectors; relations between the dimensions of the eigenmanifolds and generalized eigenmanifolds of the operator and those of the approximate operators.

scholarly journals Some Properties of Fractional Calculus and Linear Operators Associated with Certain Subclass of Multivalent Functions

2009 ◽  
Vol 2009 ◽  
pp. 1-16
Author(s):  
Sh. Khosravianarab ◽  
S. R. Kulkarni ◽  
O. P. Ahuja
Keyword(s):  
Fractional Calculus ◽  
Integral Operators ◽  
Linear Operators ◽  
New Approaches ◽  
Special Cases

We investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivatives, and some well-known integral operators. In special cases, the results presented in this paper provide new approaches to several previously known results.

Spectral Approximation of Linear Operators (Françoise Chatelin)

SIAM Review ◽  
1985 ◽  
Vol 27 (3) ◽  
pp. 459-459
Author(s):  
P. M. Anselone
Keyword(s):  
Linear Operators ◽  
Spectral Approximation
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