Rigidity for Matrix-Valued Hardy Functions

Kasahara, Yukio; Inoue, Akihiko; Pourahmadi, Mohsen

doi:10.1007/s00020-015-2265-y

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…Recall the unitary part of the product ℎ ℎ from (2.10). By [15,Theorem 4.1], ℎ ℎ is rigid if and only if 2 ∩ 2 = {0}, which is the same as to say that the direct sum 2 + 2 is dense in 2 . The following fact (cf.…”

Section: Wiener's Lemma a Function In ℳ Is Invertible Therein If It mentioning

confidence: 96%

“…Indeed, if

g_{L}^{- 1}

and

g_{R}^{- 1}

lie in

H_{scriptM}^{2}

, then, for any function

f = f_{L} f_{R}

H_{scriptM}^{1}

sharing with g the same unitary part,

{(f_{L}^{*})}^{- 1} f_{R} = {(g_{L}^{*})}^{- 1} g_{R}

implies

c \equiv {((), g_{L}^{- 1}, f_{L})}^{*} = f_{R} g_{R}^{- 1} \in M = {((), H_{M}^{1})}^{*} \cap H_{M}^{1}, det c \neq 0,

showing that

f = g_{L} k g_{R}

for

k = c^{*} c > 0

, as required. See Kasahara–Inoue–Pourahmadi for a general concept of

scriptM

‐valued rigid functions and its applications to

scriptV

‐valued stationary processes.…”

Section: Baxter's Theoremmentioning

confidence: 99%

“…showing that = for = * > , as required. See Kasahara-Inoue-Pourahmadi [15] for a general concept of ℳ-valued rigid functions and its applications to -valued stationary processes.…”

Section: Introductionmentioning

confidence: 99%

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Matricial Baxter's theorem with a Nehari sequence

Kasahara

Bingham

2018

Mathematische Nachrichten

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In the theory of orthogonal polynomials, (non‐trivial) probability measures on the unit circle are parametrized by the Verblunsky coefficients. Baxter's theorem asserts that such a measure is absolutely continuous and has positive density with summable Fourier coefficients if and only if its Verblusnky coefficients are summable. This note presents a version of Baxter's theorem in the matrix case from a viewpoint of the Nehari problem.

show abstract

Section: Wiener's Lemma a Function In ℳ Is Invertible Therein If It mentioning

confidence: 96%

“…Indeed, if

g_{L}^{- 1}

and

g_{R}^{- 1}

lie in

H_{scriptM}^{2}

, then, for any function

f = f_{L} f_{R}

H_{scriptM}^{1}

sharing with g the same unitary part,

{(f_{L}^{*})}^{- 1} f_{R} = {(g_{L}^{*})}^{- 1} g_{R}

implies

c \equiv {((), g_{L}^{- 1}, f_{L})}^{*} = f_{R} g_{R}^{- 1} \in M = {((), H_{M}^{1})}^{*} \cap H_{M}^{1}, det c \neq 0,

showing that

f = g_{L} k g_{R}

for

k = c^{*} c > 0

, as required. See Kasahara–Inoue–Pourahmadi for a general concept of

scriptM

‐valued rigid functions and its applications to

scriptV

‐valued stationary processes.…”

Section: Baxter's Theoremmentioning

confidence: 99%

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Matricial Baxter's theorem with a Nehari sequence

Kasahara

Bingham

2018

Mathematische Nachrichten

Self Cite

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show abstract

“…In fact, by [27, Theorem 3.5], (IPF) and (CND) are equivalent under (A). The condition (CND) is also closely related to the rigidity for matrix-valued Hardy functions (see [29]). It should be noticed that if {X k } satisfies (IPF), then so does the time-reversed process {X k }, and that the same holds for (M) and (CND).…”

Section: (M)mentioning

confidence: 99%

Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes

Inoue¹,

Kasahara²,

Pourahmadi³

2018

Bernoulli

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For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the finite predictor coefficients, prediction error covariances, and the PACF as well as for proving a multivariate version of Baxter's inequality for a multivariate FARIMA process with a common fractional differencing order for all components of the process.2010 Mathematics Subject Classification. Primary 60G25; secondary 62M20, 62M10.

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“…which goes back to Levinson-McKean [15]. See Kasahara-Inoue-Pourahmadi [14] for a general concept of M -valued rigid functions and its application to V -valued stationary processes. Let γ = (γ 1 , γ 2 , .…”

Section: Introductionmentioning

confidence: 99%