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Generalized sampling and infinitedimensional compressed sensing
"... We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demo ..."
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Cited by 33 (20 self)
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We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finitedimensional techniques are illsuited for solving a number of important problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. The main conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. The key to these developments is the introduction of two new concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize the fundamentally infinitedimensional reconstruction problem.
Geometry of higher order relative spectra and projection methods
 Journal of Operator Theory
"... Abstract. Let H be a densely defined linear operator acting on a Hilbert space H, let P be the orthogonal projection onto a closed linear subspace L and let n ∈ N. The nth order spectrum Specn(H,L) of H relative to L is the set of z ∈ C such that the restriction to L of the operator P (H − zI)nP is ..."
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Cited by 26 (1 self)
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Abstract. Let H be a densely defined linear operator acting on a Hilbert space H, let P be the orthogonal projection onto a closed linear subspace L and let n ∈ N. The nth order spectrum Specn(H,L) of H relative to L is the set of z ∈ C such that the restriction to L of the operator P (H − zI)nP is not invertible within the subspace L. We study restrictions which may be placed on this set under given assumptions on Spec(H) and the behaviour of Specn(H,L) as L increases towards H.
Which eigenvalues are found by the Lanczos method
 SIAM J. Matrix Anal. Appl
"... Abstract. When discussing the convergence properties of the Lanczos iteration method for the real symmetric eigenvalue problem, Trefethen and Bau noted that the Lanczos method tends to find eigenvalues in regions that have too little charge when compared to an equilibrium distribution. In this paper ..."
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Cited by 25 (5 self)
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Abstract. When discussing the convergence properties of the Lanczos iteration method for the real symmetric eigenvalue problem, Trefethen and Bau noted that the Lanczos method tends to find eigenvalues in regions that have too little charge when compared to an equilibrium distribution. In this paper a quantitative version of this rule of thumbis presented. We describe, in an asymptotic sense, the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of eigenvalues and on the ratio between the size of the matrix and the number of iterations, and it is characterized by an extremal problem in potential theory which was first considered by Rakhmanov. We give examples showing the connection with the equilibrium distribution. Key words. Ritz values, equilibrium distribution, potential theory
C*Algebras in Numerical Analysis
 IRISH MATH. SOC. BULLETIN
, 2000
"... These are the notes for two lectures I gave at the Belfast Functional Analysis Day 1999. The purpose of these notes is to give an idea of how C∗algebra techniques can be successfully employed in order to solve some concrete problems of Numerical Analysis. I focus my attention on several questions c ..."
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Cited by 13 (2 self)
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These are the notes for two lectures I gave at the Belfast Functional Analysis Day 1999. The purpose of these notes is to give an idea of how C∗algebra techniques can be successfully employed in order to solve some concrete problems of Numerical Analysis. I focus my attention on several questions concerning the asymptotic behavior of large Toeplitz matrices. This limitation ignores the potential and the triumphs of C∗algebra methods in connection with large classes of other operators and plenty of different approximation methods, but it allows me to demonstrate the essence of the C∗algebra approach and to illustrate it with nevertheless nontrivial examples.
Convergence analysis of the finite section method and Banach algebras of matrices
 OPER. THEORY
, 2008
"... The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted `pspaces. Our approach uses recent results f ..."
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Cited by 11 (0 self)
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The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted `pspaces. Our approach uses recent results from the theory of Banach algebras of matrices with offdiagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable to large classes of unstructured nonhermitian matrices as well as to least squares problems.
Spectral Approximation of Multiplication Operators
 New York J. Math
, 1995
"... . A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of the Hilbert space. For an operator that is not compact such approximations cannot converge in the norm topology on the space of operators. Multiplication operators on spaces of L 2 fun ..."
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Cited by 10 (0 self)
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. A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of the Hilbert space. For an operator that is not compact such approximations cannot converge in the norm topology on the space of operators. Multiplication operators on spaces of L 2 functions are never compact; for them we consider how well the eigenvalues of the matrices approximate the spectrum of the multiplication operator, which is the essential range of the multiplier. The choice of the orthonormal basis strongly affects the convergence. Toeplitz matrices arise when using the Fourier basis of exponentials exp(ik`). We also consider the basis of Legendre polynomials and the basis of Walsh functions. Contents 1. Introduction 75 2. Multiplication operators 78 2.1. Toeplitz Matrices 78 2.2. Matrices Associated to Legendre Polynomials 78 2.3. WalshToeplitz Matrices 79 3. Spectral Convergence 81 4. Spectral convergence for Toeplitz matrices 83 5. Spectral Convergence wit...
Quantitative estimates for the finite section method
"... Abstract. The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. We present quantitative estimates for the rate of the convergence of the finite section method on weighted ℓ pspaces. Our approach uses recent results from the theory of ..."
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Cited by 10 (1 self)
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Abstract. The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. We present quantitative estimates for the rate of the convergence of the finite section method on weighted ℓ pspaces. Our approach uses recent results from the theory of Banach algebras of matrices with offdiagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable to large classes of nonhermitian matrices. An example from digital communication illustrates the practical usefulness of the proposed theoretical framework.
Decay properties for functions of matrices over C∗algebras
, 2013
"... We extend previous results on the exponential offdiagonal decay of the entries of analytic functions of banded and sparse matrices to the case where the matrix entries are elements of a C∗algebra. ..."
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Cited by 3 (1 self)
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We extend previous results on the exponential offdiagonal decay of the entries of analytic functions of banded and sparse matrices to the case where the matrix entries are elements of a C∗algebra.
OF THE ALMOST MATHIEU OPERATOR
, 2006
"... Abstract. Let Aθ be the rotation C*algebra for angle θ. For θ = p/q with p and q relatively prime, Aθ is the subC*algebra of Mq(C(T2)) generated by a pair of unitaries u and v satisfying uv = e2πiθvu. Let hθ,λ = u+u−1 +λ/2(v+ v−1) be the almost Mathieu operator. By proving an identity of rational ..."
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Cited by 3 (2 self)
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Abstract. Let Aθ be the rotation C*algebra for angle θ. For θ = p/q with p and q relatively prime, Aθ is the subC*algebra of Mq(C(T2)) generated by a pair of unitaries u and v satisfying uv = e2πiθvu. Let hθ,λ = u+u−1 +λ/2(v+ v−1) be the almost Mathieu operator. By proving an identity of rational functions we show that for q even, the constant term in the characteristic polynomial of hθ,λ is (−1) q/2 (1 + (λ/2) q) − (z q 1 1.