Stochastic mappings and random distribution fields III. Module propagators and uniformly bounded linear stationarity

Gaşpar, Păstorel; Popa, Lorena

doi:10.1016/j.jmaa.2015.10.082

Cited by 4 publications

(6 citation statements)

References 11 publications

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“…T , called the continuous transform of T , is a pure contraction 4 from H 1 into H 2 (i.e. it satisfies the strict inequality Z T x H 2 <…”

Section: Lemma 31mentioning

confidence: 99%

“…In quite recent times, the study of stochastic processes or random fields was enlarged to the framework of multivariate stochastic mappings (see [2,20]) in order to treat in a unified way also other probabilistic concepts such as stochastic measures and stochastic integrals, random distributions or random distribution fields, as well as random operators (see [3][4][5]7,8,15,[17][18][19][20][21][22]), but also in an attempt to develop in this setting a corresponding random spectral theory (see [7,8,15,18,24,25]). B Pȃstorel Gaşpar pastorel.gaspar@uav.ro 1 Department of Mathematics and Computer Science, Faculty of Exact Sciences, "Aurel Vlaicu" University, Arad, Str.…”

Section: Introductionmentioning

confidence: 99%

“…For a study of such mappings see [19,20], for p = 0, 2 and [2][3][4] for p = 2. In this paper, we restrain ourselves to the case in which is a complex separable Hilbert space G (which can differ from H ) or some dense subspace of G. We shall call these p-order random operators.…”

Section: Introductionmentioning

confidence: 99%

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On Random Normal Operators and Their Spectral Measures

Gaşpar

2018

J Theor Probab

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The main aim of this paper is to introduce and study the subclass of not necessarily continuous, normal random operators, establishing connections with other subclasses of random operators, as well as with the existing concept of random projection operatorvalued measure. Hence, after recalling some basic facts regarding random operators on a complex separable Hilbert space, theorems about transforming the class of not necessarily continuous decomposable random operators into the class of purely contractive random operators are proved. These are applied to obtain integral representations for not necessarily continuous normal or self-adjoint random operators on a Hilbert space with respect to the corresponding random projection operator-valued measures.

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“…T , called the continuous transform of T , is a pure contraction 4 from H 1 into H 2 (i.e. it satisfies the strict inequality Z T x H 2 <…”

Section: Lemma 31mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Random Normal Operators and Their Spectral Measures

Gaşpar

2018

J Theor Probab

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“…The former, detecting thresholds to ensure bounded result tends to be too restricted and conceals some of the stochastic behavior of the spectral structure. At this stage, it is worth mentioning a different approach where positive definite functions R(n) on the integers and with values continuous operators from a Banach space into its anti-dual play an important role in the theory of stochastic processes and related topics ( [17,5,19,20,23] and [4]). An important feature of Banach spaces (not always used in the above references) is that they possess the factorization property: a positive operator from a Banach space into its anti-dual can be factorized via a Hilbert space; see [9] and e.g.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Wiener filter in the white noise space

Alpay¹,

Pinhas²

2020

Opuscula Math.

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In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions.

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“…In quite recent times the study of stochastic processes or random fields was enlarged to the framework of multivariate stochastic mappings (see [20], [3]) in order to treat in a unitary way also other probabilistic concepts such as stochastic measures and stochastic integrals, random distributions or random distribution fields, as well as random operators (see [19], [17], [20], [18], [23], [21], [22], [16], [4], [5], [6], [9], [8]), but also in an attempt to develop in this setting a corresponding random spectral theory (see [16], [24], [18], [9], [8]). Specifically, in [24], in terms of measurable families of deterministic continuous linear operators, the continuous normal and Hermitian random operators, random spectral measures were defined and the random version of the spectral (integral) representation theorems were given.…”

Section: Introductionmentioning

confidence: 99%