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Spectral clustering properties of block multilevel Hankel matrices

Spectral clustering properties of block multilevel Hankel matrices,10.1016/S0024-3795(99)00251-7,Linear Algebra and Its Applications,Dario Fasino,Paol

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Spectral clustering properties of block multilevel Hankel matrices   (Citations: 11)
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By means of recent results concerning spectral distributions of Toeplitz matrices, we show that the singular values of a sequence of block p-level Hankel matrices Hn(μ), generated by a p-variate, matrix-valued measure μ whose singular part is finitely supported, are always clustered at zero, thus extending a result known when p=1 and μ is real valued and Lipschitz continuous. The theorems hold for both eigenvalues and singular values; in the case of singular values, we allow the involved matrices to be rectangular.
Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL , vol. 306, no. 1, pp. 155-163, 2000
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    • ...Moreover, from [8] we know that the Hankel sequence {Hn(f)} is distributed as the zero function over Q...
    • ...x = cos(s), the matrix ˜ Mn[ ˜ φ] can be written as Tn( ˜ f) plus Hn( ˜ f). Moreover, w ∈ L 1 [−1,1] and therefore ˜ f ∈ L 1 (Q), Q = (−π, π). Finally by Theorem 2.2 (which holds for L 1 functions) and by [8], we know that...
    • ...Moreover, by [8], k Hn( ˜ f)k 1 = o(n) and therefore by Item 6. k Opt(Hn( ˜ f))k 1 = o(n)...
    • ...Once we arrive here, the rest is a straightforward generalization of the univariate case since the result on Hankel matrices (see [8]) are directly stated in an arbitrary number of dimensions...
    • ...Once we arrive here, the rest is a generalization of the multivariate case since the result on Hankel matrices (see [8]) are directly stated in an arbitrary number of dimensions with blocks of fixed dimension...

    Stefano Serra Capizzano. The spectral approximation of multiplication operators via asymptotic ...

    • ...Part (a) of the followingtheorem is essentially a reformulationof Theorem 3 in [5]...

    William F. Trench. Absolute equal distribution of families of finite sets

    • ...Tyrtyshnikov and others (see, e. g, [5, 6, 11, 12, 13, 16, 17, 18, 19, 20, 22]) have used Theorem 1 or variations of it to obtain significant results on the asymptotic distribution of eigenvalues and singular values of structured matrices...
    • ...Fasino and Tilli [6] have obtained distributional results in which the assumptions on the perturbation matrices are in terms of the trace norm...

    William F. Trench. Absolute equal distribution of the spectra of Hermitian matrices

    • ...Proof. As shown by Fasino and Tilli [6], [15],...
    • ...Finally, Fasino and Tilli [6], [15] proved that always...

    Albrecht Bottcher, et al. THE NORM OF THE PRODUCT OF A LARGE MATRIX AND A RANDOM VECTOR

    • ...Moreover, from [8] we know that the Hankel sequence {Hn(f)} is distributed as the zero function over Q...
    • ...x = cos(s), the matrix ˜ Mn[ ˜ φ] can be written as Tn( ˜ f) plus Hn( ˜ f). Moreover, w ∈ L 1 [−1,1] and therefore ˜ f ∈ L 1 (Q), Q = (−π, π). Finally by Theorem 2.2 (which holds for L 1 functions) and by [8], we know that...
    • ...Moreover, by [8], k Hn( ˜ f)k 1 = o(n) and therefore by Item 6. k Opt(Hn( ˜ f))k 1 = o(n)...
    • ...Once we arrive here, the rest is a straightforward generalization of the univariate case since the result on Hankel matrices (see [8]) are directly stated in an arbitrary number of dimensions...
    • ...Once we arrive here, the rest is a generalization of the multivariate case since the result on Hankel matrices (see [8]) are directly stated in an arbitrary number of dimensions with blocks of fixed dimension...

    Stefano Serra-Capizzano. The spectral approximation of multiplication operators via asymptotic ...

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