Time and Band Limiting for Matrix Valued Functions, an Example

Grünbaum, F. Alberto; Pacharoni, I.; Zurrián, Ignacio

doi:10.3842/sigma.2015.044

Cited by 11 publications

(19 citation statements)

References 46 publications

(45 reference statements)

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“…In the first example we extend results previously obtained in [18,19]; in the second one we verify a result that was conjectured in [4]; in the third example we exploit the power of our construction to give a commuting differential operator for a case where the commuting operator problem was not studied before; the last example is included to indicate that bispectrality may not always guarantee the existence of a commuting differential operator.…”

Section: Introductionsupporting

confidence: 69%

“…Remark 3.9. From Corollary 3.5 it is clear that L breaks into two blocks, an upper-left block of size (N + 1) × (N + 1) yielding a matrix such as the one displayed in [18] and a lower-right block which is semi-infinite.…”

Section: The Symmetric Bispectral Problemmentioning

confidence: 99%

“…On the other hand, the matrices L 1 , L 2 , L 3 given in [18] are in the span of {F −1 T 1 F, F −1 T 2 F, I}. Furthermore, L 1 and L 2 scalar multiples of F −1 T 2 F and F −1 T 1 F, respectively, and…”

Section: Matrix Gegenbauer Weightmentioning

confidence: 99%

“…A much more recent look at the relation between these two topics involves matrix valued orthogonal polynomials, a subject started by M. G. Krein, see [26,27]. The papers where this relation has been explored recently are [1,18,19,3,4].…”

Section: Introductionmentioning

confidence: 99%

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Bispectrality and Time–Band Limiting: Matrix-valued Polynomials

Grünbaum

Pacharoni

Zurrián

2018

International Mathematics Research Notices

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The subject of time-band-limiting, originating in signal processing, is dominated by the miracle that a naturally appearing integral operator admits a commuting differential one allowing for a numerically efficient way to compute its eigenfunctions. Bispectrality is an effort to dig into the reasons behind this miracle and goes back to joint work with H. Duistermaat. This search has revealed unexpected connections with several parts of mathematics, including integrable systems.Here we consider a matrix valued version of bispectrality and give a general condition under which we can display a constructive and simple way to obtain the commuting differential operator. Furthermore, we build an operator that commutes with both the time-limiting operator and the band-limiting operators.

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Section: Introductionsupporting

confidence: 69%

Section: The Symmetric Bispectral Problemmentioning

confidence: 99%

Section: Matrix Gegenbauer Weightmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bispectrality and Time–Band Limiting: Matrix-valued Polynomials

Grünbaum

Pacharoni

Zurrián

2018

International Mathematics Research Notices

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“…Namely, W (x) = W p,n = (1 − x 2 ) n 2 −1 p x 2 + n − p −nx −nx (n − p)x 2 + p , x ∈ [−1, 1], for real parameters 0 < p < n/2. This was the example employed in the study of the first case of the time and band limiting problem for matrix-valued functions (see [GPZ15]). This algebra is of interest by itself.…”

Section: Introductionmentioning

confidence: 99%

The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

Zurrián¹

2016

Int Math Res Notices

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Abstract. In this paper we study in detail algebraic properties of the algebra D(W ) of differential operators associated to a matrix weight of Gegenbauer type. We prove that two second order operators generate the algebra, indeed D(W ) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W ) is a finitely-generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra D(W ) and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra D(W ) as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this paper represents a good step in this direction.

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Matrix Gegenbauer Polynomials: The $$2\times 2$$ 2 × 2 Fundamental Cases

Pacharoni¹,

Zurrián²

2015

Constr Approx

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In this paper, we exhibit explicitly a sequence of 2 × 2 matrix valued orthogonal polynomials with respect to a weight Wp,n, for any pair of real numbers p and n such that 0 < p < n. The entries of these polynomiales are expressed in terms of the Gegenbauer polynomials C λ k . Also the corresponding three-term recursion relations are given and we make some studies of the algebra of differential operators associated with the weight Wp,n.2010 Mathematics Subject Classification. 22E45 -33C45 -33C47.

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Time and Band Limiting for Matrix Valued Functions, an Example

Cited by 11 publications

References 46 publications

Bispectrality and Time–Band Limiting: Matrix-valued Polynomials

Bispectrality and Time–Band Limiting: Matrix-valued Polynomials

The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

Matrix Gegenbauer Polynomials: The $$2\times 2$$ 2 × 2 Fundamental Cases

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