Pizzetti Formulae for Stiefel Manifolds and Applications

Coulembier, Kevin; Kieburg, Mario

doi:10.1007/s11005-015-0774-x

Cited by 11 publications

(15 citation statements)

References 46 publications

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“…Then the question for the generic eigenvalues of a truncation T HT * (T * is the Hermitian conjugate of T ) is deeply related to the fact which representations of U(n) are contained in a certain representation of U(m). In particular group integrals and coset integrals like integrals over Stiefel manifolds are deeply related to representation theory, see [35,9,10,38,19]. Though our results are more general they can be partially interpreted in this framework.…”

Section: )mentioning

confidence: 63%

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Singular Value Statistics of Matrix Products with Truncated Unitary Matrices

Kieburg

Kuijlaars

Stivigny

2015

Int Math Res Notices

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121

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We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix with a random matrix. We show that the structure of polynomial ensembles and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint singular value density of a product of truncated unitary matrices and its corresponding correlation kernel which can be written as a double contour integral. This leads to hard edge scaling limits that also include new finite rank perturbations of the Meijer G-kernels found for products of complex Ginibre random matrices.

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Section: )mentioning

confidence: 63%

“…In this way we integrate in (5.2) over the larger space M(m, n). See also the discussion in [19]. The complex matrix M can be partitioned into two blocks…”

Section: Preliminariesmentioning

confidence: 99%

Singular Value Statistics of Matrix Products with Truncated Unitary Matrices

Kieburg

Kuijlaars

Stivigny

2015

Int Math Res Notices

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121

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“…Proof. The first statement is trivial, the second follows immediately from a repeated application of relation (10).…”

Section: Description Of the Modulementioning

confidence: 91%

“…Remark 7.1. In [10], the authors also obtained a Pizzetti formula for integration on the Stiefel manifold. Altough their formula allows for easier and faster computation, our formula is more clear from a conceptual perspective.…”

Section: Fischer Inner Product and Orthogonalitymentioning

confidence: 99%

The harmonic transvector algebra in two vector variables

2017

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The decomposition of polynomials of one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the Howe duality theorem and leads to the study of spherical harmonics. The aim of the present paper is to describe a decomposition of polynomials in two vector variables and to obtain projection operators on each of the irreducible components. To do so, a particular transvector algebra will be used as a new dual partner for the orthogonal group leading to a generalisation of the classical Howe duality. The results are subsequently used to obtain explicit projection operators and formulas for integration of polynomials over the associated Stiefel manifold.

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“…the supersphere (see [7]). A Pizzetti formula for Stiefel manifolds was developed in [5]. We state here the result for the first case St (1) .…”

Section: Plane Wave Formulasmentioning

confidence: 95%

Plane wave formulas for spherical, complex and symplectic harmonics

Bie

Sommen

Wutzig

2017

Journal of Approximation Theory

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This paper is concerned with spherical harmonics, and two refinements thereof: complex harmonics and symplectic harmonics. The reproducing kernels of the spherical and complex harmonics are explicitly given in terms of Gegenbauer or Jacobi polynomials. In the first part of the paper we determine the reproducing kernel for the space of symplectic harmonics, which is again expressible as a Jacobi polynomial of a suitable argument. In the second part we find plane wave formulas for the reproducing kernels of the three types of harmonics, expressing them as suitable integrals over Stiefel manifolds. This is achieved using Pizzetti formulas that express the integrals in terms of differential operators

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Pizzetti Formulae for Stiefel Manifolds and Applications

Cited by 11 publications

References 46 publications

Singular Value Statistics of Matrix Products with Truncated Unitary Matrices

Singular Value Statistics of Matrix Products with Truncated Unitary Matrices

The harmonic transvector algebra in two vector variables

Plane wave formulas for spherical, complex and symplectic harmonics

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