Results 1  10
of
93
The Classical Moment Problem as a SelfAdjoint Finite Difference Operator
, 1998
"... This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong r ..."
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Cited by 89 (7 self)
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This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong resolvent convergence of finite matrix approximations to a Jacobi matrix. As a bonus of this, we obtain new results on the convergence of certain Pade approximants for series of Hamburger.
Filter Bank Frame Expansions with Erasures
, 2002
"... We study frames for robust transmission over the Internet. In our previous work, we used quantized finitedimensional frames to achieve resilience to packet losses; here, we allow the input to be a sequence in ` 2 (Z) and focus on a filterbank implementation of the system. We present results in par ..."
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Cited by 50 (4 self)
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We study frames for robust transmission over the Internet. In our previous work, we used quantized finitedimensional frames to achieve resilience to packet losses; here, we allow the input to be a sequence in ` 2 (Z) and focus on a filterbank implementation of the system. We present results in parallel, R N or C N versus ` 2 (Z), and show that uniform tight frames, as well as newly introduced strongly uniform tight frames, provide the best performance.
Generalized gradients: priors on minimization flows
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2007
"... This paper tackles an important aspect of the variational problem underlying active contours: optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literatu ..."
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Cited by 23 (3 self)
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This paper tackles an important aspect of the variational problem underlying active contours: optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literature. Most authors, explicitely or implicitely, assume that the space of admissible deformations is ruled by the canonical L 2 inner product. The classical gradient flows reported in the literature are relative to this particular choice. Here, we investigate the relevance of using (i) other inner products, yielding other gradient descents, and (ii) other minimizing flows not deriving from any inner product. In particular, we show how to induce different degrees of spatial consistency into the minimizing flow, in order to decrease the probability of getting trapped into irrelevant local minima. We report numerical experiments indicating that the sensitivity of the active contours method to initial conditions, which seriously limits its applicability and efficiency, is alleviated by our applicationspecific spatially coherent minimizing flows. We show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors.
Gibbs sampling, exponential families and orthogonal polynomials
 Statistical Sciences
, 2008
"... Abstract. We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical ort ..."
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Cited by 19 (6 self)
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Abstract. We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions. Key words and phrases: Gibbs sampler, running time analyses, exponential families, conjugate priors, location families, orthogonal polynomials, singular value decomposition. 1.
Tight frames and their symmetries
 Constr. Approx
, 2005
"... The aim of this paper is to investigate symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to g ..."
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Cited by 18 (6 self)
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The aim of this paper is to investigate symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finitedimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.
Graph subspaces and the spectral shift function
 2003), 449 – 503; math.SP/0105142 v3
"... Abstract. We extend the concept of Lifshits–Krein spectral shift function associated with a pair of selfadjoint operators to the case of pairs of (admissible) operators that are similar to selfadjoint operators. An operator H is called admissible if: (i) there is a bounded operator V with a bounde ..."
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Cited by 14 (10 self)
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Abstract. We extend the concept of Lifshits–Krein spectral shift function associated with a pair of selfadjoint operators to the case of pairs of (admissible) operators that are similar to selfadjoint operators. An operator H is called admissible if: (i) there is a bounded operator V with a bounded inverse such that H = V −1 ̂HV for some selfadjoint operator ̂H; (ii) the operators H and ̂ H are resolvent comparable, i.e., the difference of the resolvents of H and ̂ H is a trace class operator (for nonreal values of the spectral parameter); (iii) tr(V R − RV) = 0 whenever R is bounded and the commutator V R − RV is a trace class operator. The spectral shift function ξ(λ, H, A) associated with the pair of resolvent comparable admissible operators (H, A) is introduced then by the equality ξ(λ, H, A) = ξ(λ, ̂ H, Â) where ξ(λ, ̂ H, Â) denotes the Lifshits– Krein spectral shift function associated with the pair ( ̂H, Â) of selfadjoint operators. Our main result is the following. Let H0 and H1 be separable Hilbert spaces, A0 a selfadjoint operator in H0, A1 a selfadjoint operator in H1, and Bij a bounded operator from Hj to Hi, i = 0, 1, j = 1 − i, and
Assche W. Criterion for the resolvent set of nonsymmetric tridiagonal operators
, 1995
"... Abstract. We study nonsymmetric tridiagonal operators acting in the Hilbert space ℓ 2 and describe the spectrum and the resolvent set of such operators in terms of a continued fraction related to the resolvent. In this way we establish a connection between Padé approximants and spectral properties o ..."
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Cited by 10 (3 self)
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Abstract. We study nonsymmetric tridiagonal operators acting in the Hilbert space ℓ 2 and describe the spectrum and the resolvent set of such operators in terms of a continued fraction related to the resolvent. In this way we establish a connection between Padé approximants and spectral properties of nonsymmetric tridiagonal operators. 1.
The many proofs of an identity on the norm of oblique projections
 Numer. Algorithms
"... Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P 2 = P, which is neither null nor the identity, it holds that ‖P ‖ = ‖I − P ‖. This useful equality, while not widelyknown, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler o ..."
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Cited by 9 (1 self)
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Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P 2 = P, which is neither null nor the identity, it holds that ‖P ‖ = ‖I − P ‖. This useful equality, while not widelyknown, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler ones are presented.
Central limit theorem for linear eigenvalue statistics of random matrices with . . .
, 2009
"... We consider n × n real symmetric and Hermitian Wigner random matrices n −1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n −1 X ∗ X with independent entries of m × n matrix X. Assuming first that the 4th cumulant (excess) κ4 of entries of W and X ..."
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Cited by 9 (0 self)
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We consider n × n real symmetric and Hermitian Wigner random matrices n −1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n −1 X ∗ X with independent entries of m × n matrix X. Assuming first that the 4th cumulant (excess) κ4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, ∞) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5). This is done by using a simple “interpolation trick ” from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially C 5 test function. Here the variance of statistics contains an additional term proportional to κ4. The proofs of all limit theorems follow essentially the same scheme.