Rational Functions Associated with the White Noise Space and Related Topics

Alpay, Daniel; Levanony, David

doi:10.1007/s11118-008-9094-4

Cited by 14 publications

(12 citation statements)

References 32 publications

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“…The structure of H 0 has been studied in [4,Section 7]. Recall now that the function t → X σ (t) is continuous, and even uniformly continuous, from R into W, and hence from R into H p for any p ∈ N 0 since…”

Section: Proof By (53) We Havementioning

confidence: 99%

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A class of Gaussian processes with fractional spectral measures

Alpay

Jørgensen

Levanony

2011

Journal of Functional Analysis

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We study a family of stationary increment Gaussian processes, indexed by time. These processes are determined by certain measures σ (generalized spectral measures), and our focus here is on the case when the measure σ is a singular measure. We characterize the processes arising from σ when σ is in one of the classes of affine selfsimilar measures. Our analysis makes use of Kondratiev white noise spaces. With the use of a priori estimates and the Wick calculus, we extend and sharpen (see Theorem 7.1) earlier computations of Ito stochastic integration developed for the special case of stationary increment processes having absolutely continuous measures. We further obtain an associated Ito formula (see Theorem 8.1).

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Section: Proof By (53) We Havementioning

confidence: 99%

“…We will prove this as an application of Kolmogorov's test for the existence of a continuous version [34, p. 14]. Because {X σ (t)} is Gaussian, there exists K independent of t, s such that E X σ (t) − X σ (s) 4 = K E X σ (t) − X σ (s) 2 2 .…”

Section: )mentioning

confidence: 99%

A class of Gaussian processes with fractional spectral measures

Alpay

Jørgensen

Levanony

2011

Journal of Functional Analysis

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“…This H (G, c) turns out to be a reproducing kernel Hilbert space (RKHS), but not in the traditional sense of the notion of RKHS, see [7,8]. It turns out that the reproducing kernel in H (G, c) is a function depending on a pair of points in G (0) ; not just one point, which is the traditional notion of RKHS; see e.g., [17].…”

Section: The Graph Laplacianmentioning

confidence: 99%

“…8) where · op on the RHS in (7.8) refers to the H E → H E operator norm. Then (B, · B ) is a Banach algebra containing the functions δ x in (7.6) for all x ∈ G (0) .…”

Section: Sampling and Interpolationmentioning

confidence: 99%

A sampling theory for infinite weighted graphs

Jørgensen¹

2011

OpMath

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Abstract. We prove two sampling theorems for infinite (countable discrete) weighted graphs G; one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing G, and there are Hilbert spaces of functions on X that allow interpolation by sampling values of the functions restricted only on the vertices in G. We sample functions on X from their discrete values picked in the vertex-subset G. We prove two theorems that allow for such realistic ambient spaces X for a fixed graph G, and for interpolation kernels in function Hilbert spaces on X, sampling only from points in the subset of vertices in G. A continuum is often not apparent at the outset from the given graph G. We will solve this problem with the use of ideas from stochastic integration.

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“…The reader might want to look at a similar table in [3], where the analogies between the white noise space case and the hyper-holomorphic case are presented.…”

Section: The White Noise Space Setting and A Tablementioning

confidence: 99%