Approximate singular values of the fractional difference and summation operators

We obtain sharp bounds on the singular values of the fractional difference and summation operators on R n . These bounds allow us to establish that the convergence rates of the distributions of the singular values of these operators are O(1/ n). Since the fractional difference operator of order α is...

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Bibliographic Details
Published in:Linear algebra and its applications 2006-07, Vol.416 (2), p.677-687
Main Author: Burman, Prabir
Format: Article
Language:English
Subjects:
Algebra
Eigenvalues
Exact sciences and technology
Fisher–Hartwig symbol
Fractional difference
Fractional summation
Linear and multilinear algebra, matrix theory
Mathematics
Sciences and techniques of general use
Singular values
Toeplitz matrix
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Summary:We obtain sharp bounds on the singular values of the fractional difference and summation operators on R n . These bounds allow us to establish that the convergence rates of the distributions of the singular values of these operators are O(1/ n). Since the fractional difference operator of order α is associated with the Toeplitz matrix with Fisher–Hartwig symbol (2 − 2 cos u) α , α > 0, we are able to obtain similar bounds on the eigenvalues of this Toeplitz matrix and a similar result on the convergence rate of the distribution of its eigenvalues.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2005.12.014