Biorthogonal appell systems in analysis on co-nuclear spaces

Kachanovsky, N. A.; Us, G. F.

doi:10.1007/bf02465759

Cited by 4 publications

(8 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 3.1 [18,12,13]. An S-transform is a topological isomorphism between the space ′ ( )′ S a n d the algebra Hol 0 of germs of functions on S C holomorphic at zero.…”

Section: Elements Of Wick Calculus and Stochastic Equationsmentioning

confidence: 97%

Extended Stochastic Integral and Wick Calculus on Spaces of Regular Generalized Functions Connected with Gamma Measure

Kachanovskii

2005

Ukr Math J

View full text Add to dashboard Cite

show abstract

“…Theorem 3.1 [18,12,13]. An S-transform is a topological isomorphism between the space ′ ( )′ S a n d the algebra Hol 0 of germs of functions on S C holomorphic at zero.…”

Section: Elements Of Wick Calculus and Stochastic Equationsmentioning

confidence: 97%

Extended Stochastic Integral and Wick Calculus on Spaces of Regular Generalized Functions Connected with Gamma Measure

Kachanovskii

2005

Ukr Math J

View full text Add to dashboard Cite

show abstract

“…be the set of continuous polynomials on D 0 : One can show (see, e.g., [67]) that it is possible to understand P.D 0 / as the set of polynomials…”

Section: Spaces Of Test Functions Letmentioning

confidence: 99%

“…I z/ 2 Hol 0 .D C / for each z 2 D 0 C : Therefore, by using the Cauchy inequality (see, e.g., [35]) and the kernel theorem (see, e.g., [17]), one can show that, for z 2 D 0 C and from some neighborhood of 0 2 D C (depending on z ), we have .z/; f .n/ E ; f .n/ 2 D y n C ; n 2 Z C g are called generalized Appell-like polynomials (or Schefer polynomials in a different terminology). The reader can find more detailed information about generalized Appell-like polynomials, e.g., in [9,26] (one-dimensional case) or in [67,77,88,91] (infinite-dimensional case).…”

Section: Biorthogonal Approach To the Construction Of Non-gaussian Inmentioning

confidence: 99%

“…On the other hand, numerous experts study the (generally speaking, non-Gaussian) analysis on the so-called Hida (e.g., [32,34,56]), Kondrat'ev (e.g., [7,8,12,24,67,68,88,90,91,112]), and other similar spaces of test and generalized functions (and on the corresponding weighted Fock spaces). This analysis includes the theory of stochastic integration, Wick calculus, and various applications (including the theory of normally ordered whitenoise equations or, in a different terminology, of stochastic equations with Wick-type nonlinearities).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Elements of a non-gaussian analysis on the spaces of functions of infinitely many variables

Kachanovsky

2011

Ukr Math J

Self Cite

View full text Add to dashboard Cite

show abstract

“…One of generalizations of the Gaussian white noise analysis is a so-called biorthogonal analysis (see [1,2,5,23,24,36]) that developed actively in 90th of the last century. Its main idea is to use as orthogonal bases in spaces of test functions so-called generalized Appell polynomials (or their generalizations), in this case orthogonal bases in spaces of generalized functions are biorthogonal to the above-mentioned polynomials generalized functions.…”

Section: Remarkmentioning

confidence: 99%

On Wick calculus on spaces of nonregular generalized functions of Levy white noise analysis

Kachanovsky

2018

Carpathian Math. Publ.

View full text Add to dashboard Cite

Development of a theory of test and generalized functions depending on infinitely many variables is an important and actual problem, which is stipulated by requirements of physics and mathematics. One of successful approaches to building of such a theory consists in introduction of spaces of the above-mentioned functions in such a way that the dual pairing between test and generalized functions is generated by integration with respect to some probability measure. First it was the Gaussian measure, then it were realized numerous generalizations. In particular, important results can be obtained if one uses the Lévy white noise measure, the corresponding theory is called the Lévy white noise analysis.In the Gaussian case one can construct spaces of test and generalized functions and introduce some important operators (e.g., stochastic integrals and derivatives) on these spaces by means of a so-called chaotic representation property (CRP): roughly speaking, any square integrable random variable can be decomposed in a series of repeated Itô's stochastic integrals from nonrandom functions. In the Lévy analysis there is no the CRP, but there are different generalizations of this property.In this paper we deal with one of the most useful and challenging generalizations of the CRP in the Lévy analysis, which is proposed by E. W. Lytvynov, and with corresponding spaces of nonregular generalized functions. The goal of the paper is to introduce a natural product (a Wick product) on these spaces, and to study some related topics. Main results are theorems about properties of the Wick product and of Wick versions of holomorphic functions. In particular, we prove that an operator of stochastic differentiation satisfies the Leibniz rule with respect to the Wick multiplication. In addition we show that the Wick products and the Wick versions of holomorphic functions, defined on the spaces of regular and nonregular generalized functions, constructed by means of Lytvynov's generalization of the CRP, coincide on intersections of these spaces.Our research is a contribution in a further development of the Lévy white noise analysis.

show abstract

Biorthogonal appell systems in analysis on co-nuclear spaces

Cited by 4 publications

References 3 publications

Extended Stochastic Integral and Wick Calculus on Spaces of Regular Generalized Functions Connected with Gamma Measure

Extended Stochastic Integral and Wick Calculus on Spaces of Regular Generalized Functions Connected with Gamma Measure

Elements of a non-gaussian analysis on the spaces of functions of infinitely many variables

On Wick calculus on spaces of nonregular generalized functions of Levy white noise analysis

Contact Info

Product

Resources

About