Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables
We present a review of some results of the non-Gaussian analysis in the biorthogonal approach and consider elements of the analysis associated with the generalized Meixner measure. The main objects of our interest are stochastic integrals, operators of stochastic differentiation, elements of theWick...
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irk-123456789-1662872020-02-19T01:27:23Z Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables Kachanovsky, N.A. Статті We present a review of some results of the non-Gaussian analysis in the biorthogonal approach and consider elements of the analysis associated with the generalized Meixner measure. The main objects of our interest are stochastic integrals, operators of stochastic differentiation, elements of theWick calculus, and related topics. Наведено огляд деяких результатів негауссівського аналізу при біортогональному підході та розглянуто елементи аналізу, пов'язаного з узагальненою мірою Майкснера. Основними об'єктами, що розглядаються, є стохастичні інтеграли, оператори стохастичного диференціювання, елементи віківського числення та споріднені питання. 2010 Article Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables / N.A. Kachanovsky // Український математичний журнал. — 2010. — Т. 62, № 9. — С. 1220–1246. — Бібліогр.: 118 назв. — рос. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166287 512.662.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Kachanovsky, N.A. Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables Український математичний журнал |
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We present a review of some results of the non-Gaussian analysis in the biorthogonal approach and consider elements of the analysis associated with the generalized Meixner measure. The main objects of our interest are stochastic integrals, operators of stochastic differentiation, elements of theWick calculus, and related topics. |
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Kachanovsky, N.A. |
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Kachanovsky, N.A. |
| author_sort |
Kachanovsky, N.A. |
| title |
Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables |
| title_short |
Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables |
| title_full |
Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables |
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Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables |
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Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables |
| title_sort |
elements of a non-gaussian analysis on the spaces of functions of infinitely many variables |
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Інститут математики НАН України |
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2010 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/166287 |
| citation_txt |
Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables / N.A. Kachanovsky // Український математичний журнал. — 2010. — Т. 62, № 9. — С. 1220–1246. — Бібліогр.: 118 назв. — рос. |
| series |
Український математичний журнал |
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AT kachanovskyna elementsofanongaussiananalysisonthespacesoffunctionsofinfinitelymanyvariables |
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2025-07-14T21:07:03Z |
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2025-07-14T21:07:03Z |
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1837657986726625280 |
| fulltext |
УДК 512.662.5
N. A. Kachanovsky (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES
OF FUNCTIONS OF INFINITELY MANY VARIABLES
We present a review of some results of a non-Gaussian analysis under the biorthogonal approach and consider
elements of an analysis associated with the generalized Meixner measure. The main objects of our interest are
stochastic integrals, operators of stochastic differentiation, elements of a Wick calculus, and related topics.
Наведено огляд деяких результатiв негауссiвського аналiзу при бiортогональному пiдходi та розглянуто
елементи аналiзу, пов’язаного з узагальненою мiрою Майкснера. Основними об’єктами, що розгля-
даються, є стохастичнi iнтеграли, оператори стохастичного диференцiювання, елементи вiкiвського
числення та спорiдненi питання.
Introduction. The development of modern mathematical branches of science, in parti-
cular, of mathematical physics and stochastic analysis, requires to construct an extensive
theory of generalized functions of infinitely many variables. One direction of this theory
that consists in the study of test and generalized function spaces as infinite tensor products
of one-dimensional spaces was triggered by Yu. M. Berezansky, Yu. G. Kondratiev,
and Yu. S. Samoilenko (see [21, 85, 86]). Afterwards, independently of one another,
Yu. G. Kondratiev [81 – 83] (see also [87]) and T. Hida (see [54, 55]) constructed a
detailed theory of generalized functions of infinitely many variables with special spaces
of test and generalized functions such that the pairing between elements of these spaces
is generated by integration with respect to the Gaussian measure.
On the other hand, approximately at the same time, Yu. M. Berezansky and disciples
were constructing a theory of generalized functions (see [14, 21, 92]) related to product
measures. This theory is less detailed than the “Gaussian” one, but is more general
because deals with a dual pairing generated by integration with respect to a non-Gaussian
measure.
With regard for these researches, it is natural to try to construct a theory of generalized
functions of infinitely many variables, which deals with a general, as far as possible,
dual pairing, but is similar by detalization of results to the ”Gaussian” theory (for this
theory we use the term a non-Gaussian infinite-dimensional analysis). The first works
in this direction are the papers of Y. Ito and I. Kubo [59, 60], in which some results of
the Gaussian analysis are transfered to the case where the Gaussian measure is replaced
by the Poissonian measure. The posterior development of the above-mentioned theory
ocured in different directions. For example, in 1991, Yu. M. Berezansky [15] offered
to construct orthogonal decompositions of “base” spaces (these spaces generate dual
pairings) using families of commuting self-adjoint operators. Typical examples of works
in this direction are [18, 95]. Another direction is based on the idea of Yu. G. Kondratiev
to use biorthogonal systems [30] (that consist of generalized Appell polynomials and
(generalized) functions dual to these polynomials) as orthogonal bases in spaces of test
and generalized functions. This idea was realized by Yu. G. Kondratiev and his colleagues
first for the case of so-called smooth twice analytic measures [7, 8, 90], later for more
general analytic nondegenerate measures [88, 91]. After that different researches and
generalizations were executed by many specialists; in particular, by G. F. Us [116], by
Yu. M. Berezansky and Yu. G. Kondratiev [16], by Yu. M. Berezansky [12], by the author,
see, e.g., [67 – 69, 74, 75], by V. A. Tesko [112], by Yu. M. Berezansky and V. A. Tesko
c© N. A. KACHANOVSKY, 2010
1220 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1221
[22 – 24], by E. Yablonsky [118] etc. Moreover, it was ascertained that insignificant
modifications of a biorthogonal approach to the construction of a non-Gaussian analysis
give a possibility to extend an area of possible applications. For example, it is possible
to use results of this approach in order to construct elements of a so-called coloured
noise analysis, see, e.g., [65, 78, 117].
Let us say several words about nonclassical examples of application of the “bi-
orthogonal analysis”. In the end of 1990-es, in the papers [84, 89], the infinite-dimensional
analog of the so-called gamma-measure was constructed and investigated. One can show
(see, e.g., [76]) that this measure satisfies all requirements of the “biorthogonal analysis”.
Moreover, the results of the “gamma analysis” that are closely related to a structure and
properties of the gamma-measure and cannot be obtained from the ”biorthogonal theory”
(for example, the construction of the extended stochastic integral on the space of square-
integrable functions), are naturally coordinated with general results of the “biorthogonal
analysis”. This example is not unique. In 2002, Yu. M. Berezansky [13] constructed and
studied an infinite-dimensional analog of the Pascal measure; in 2003, E. W. Lytvynov
[96, 97] considered infinite-dimensional analogs of so-called Meixner-type measures
(to this type belong the Gaussian, Poissonian, Pascal, Meixner measures and gamma-
measures, see [98]) and constructed elements of the corresponding analysis basingn on
the so-called Jacobi fields theory, see, e.g., [19]; in 2005, I. V. Rodionova [108] consi-
dered a wide class of infinite-dimensional Meixner-type measures (these measures are
realized as a so-called generalized Meixner measure, which subject to parameters can be
Gaussian, Poissonian, gamma-measure etc.); her research consists in the generalization
of results presented in [97]. It is worth noting that the white noise in [108] is not a
Lévy white noise, generally speaking (not time-homogeneous). In the investigations of
[96, 97, 108], an important role belongs to the so-called extended Fock space [84, 20],
which naturally arises in the “Meixner analysis” and, in fact, is the interacting Fock space
[3, 58]. All mentioned measures satisfy all requirements of the “biorthogonal analysis”;
and specific results related to properties of these measures are naturally coordinated with
the results of the ”biorthogonal analysis”. Moreover, in order to obtain these specific
results it is convenient to use a “tooling” that is developed in the “biorthogonal analysis”.
During recent years, an analysis associated with the Gamma, Pascal and Meixner
measures and the corresponding white noises (including stochastic integration theory)
became the object of investigation of many authors. In particular, in [1], Lévy processes
on the Lie algebra sl(2,R) were investigated, components of these processes are classical
Lévy processes on R corresponding to Meixner classes; in [4], the stochastic integral
was introduced and studied for a wide class of stochastic processes and it was proved
in [2] that the results of [4] can be applied in the “Meixner analysis”; in [52, 53], the
stochastic integration theory with applications was constructed for Meixner processes
and its generalizations; in [113 – 115], some properties of gamma-processes were studied;
in [5], all Meixner classes within a quantum white noise context were considered from
a general point of view.
On the other hand, many specialists study a (non-Gaussian, generally speaking)
analysis on the so-called Hida (e.g., [32, 34, 56]), Kondratiev (e.g., [7, 8, 12, 24, 67, 68,
88, 90, 91, 112]) and another similar spaces of test and generalized functions (and on the
corresponding weighted Fock spaces). Such an analysis includes a stochastic integration
theory, a Wick calculus and different applications (including a theory of normally ordered
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1222 N. A. KACHANOVSKY
white noise equations or, in another terminology, of stochastic equations with Wick-type
nonlinearities). Thereupon we refer, in particular, to the papers [8, 27 – 29, 32, 34,
61, 62, 90, 93, 102 – 106]. One of tasks in these investigations consists in the study of
properties of different operators (including stochastic integrals and stochastic derivatives)
and operations (e.g., of a Wick multiplication) subject to the particular spaces under
consideration. For example, in [32 – 34], stochastic integrals with respect to a wide class
of Lévy processes on Hida spaces are studied and the corresponding Wick calculus
is developed; the constructions in these works are based on the so-called power jump
processes [101].
So, the problem of development and improvement of the biorthogonal approach to
the construction of a non-Gaussian analysis (in particular, the construction and study
of stochastic integrals and operators of stochastic differentiation, of a Wick calculus;
improvement of the “tooling” etc.), just as the problem of construction and detailed
study of nonclassical examples (in particular, of the ”Meixner analysis”), are natural and
relevant.
In this paper we review some results of the “biorthogonal analysis” and of the
analysis associated with the generalized Meixner measure [108]. In particular, from a
general point of view, we consider stochastic integrals, stochastic derivatives, elements
of a Wick calculus, and related topics. Note that these objects were not considered in
details in the existing surveys [22 – 24]. As for the recent survey [80], the investigation
of in that paper only stochastic integrals in a very specific context is presented there.
The present paper is organized in the following manner. In the first section, we deal
with the “biorthogonal analysis”. Namely, we introduce so-called generalized Appell-like
polynomials (these polynomials form orthogonal bases in test function spaces), construct
test function spaces and describe their properties, introduce pseudodifferential operators
on the test function spaces and consider some properties of these operators, introduce a
probability measure µ that satisfies certain conditions and consider generalized function
spaces that are dual spaces of the test function spaces with respect to L2(µ), derscribe
natural orthogonal bases in the generalized function spaces; consider elements of a
Wick calculus (an S-transform, a Wick multiplication, Wick versions of holomorphic
functions) on the generalized function spaces; introduce and study an analog of the
extended stochastic integral on the generalized function spaces, in particular, consider
the interconnection between the “extended stochastic integration” and the Wick calculus,
consider an example of an equation with Wick-type nonlinearities; introduce and study
operators of stochastic differentiation on the generalized function spaces.
The second section is devoted to an analysis that is associated with the generalized
Meixner measure µ (more exactly, in this section, we observe results that are associated
with peculiarities of µ and can not be obtained from the general “biorthogonal theory”,
and describe the interconnection between these results and results of the ”biorthogonal
analysis”). Namely, following [108], we give a definition of the generalized Meixner
measure, consider the space L2(µ) and construct a natural orthogonal basis in this
space; consider a so-called nonregular rigging of L2(µ) (this rigging is the main ”chain
of spaces” in the “biorthogonal analysis”), describe natural orthogonal bases in spaces
of this rigging and consider the interconnection between these bases and the bases
that are constructed in the framework of the “biorthogonal analysis”, introduce a so-
called parametrized regular rigging of L2(µ) and describe natural orthogonal bases
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1223
in spaces of this rigging, consider elements of a Wick calculus on the parametrized
generalized function spaces, introduce and study the extended stochastic integral with
respect to the so-called Meixner process on the parametrized generalized function spaces
(in particular, on the space of square integrable with respect to µ functions), consider
the interconnection between the extended stochastic integration and the Wick calculus,
consider an example of an equation with Wick-type nonlinearities, introduce and study
stochastic derivatives and operators of stochastic differentiation on the parametrized
generalized function spaces.
1. On a biorthogonal approach to construction of a non-Gaussian infinite-
dimensional analysis. 1.1. Generalized Appell-like polynomials. Let R+ := [0,+∞),
T := {τ = (τ1, τ2)}, where τ1 ∈ N, τ2 : R+ → [1,+∞) are infinite differentiable
functions; {Hτ}τ∈T are the Sobolev spaces on R+ of order τ1 weighted by τ2, i.e., the
scalar product in Hτ has the form
(f, g)Hτ =
∫
R+
(
f(u)g(u) +
τ1∑
k=1
f (k)(u)g(k)(u)
)
τ2(u) du.
We consider the chain
D′ = ind lim
τ ′∈T
H−τ ′ ⊃ H−τ ⊃ H := L2(R+, du) ⊃ Hτ ⊃ D := pr lim
τ ′∈T
Hτ ′ , (1.1)
where L2(R+, du) is the space of real-valued functions on R+ square integrable with
respect to the Lebesgue measuse, pr lim and ind lim denote projective and inductive
limits with the coresponding topologies, respectively (e.g., [17]), H−τ and D′ are the
spaces dual of Hτ and D with respect to the “zero space”H, respectively. One can show
(e.g., as in [25]) that D is the Schwartz space of infinite-differentiable functions on R+
with compact supports.
Denote by the subindex C complexifications of spaces, by Hol0(DC) and
Hol0(DC,DC) algebras of (germs of) functions on DC holomorphic for 0 ∈ DC wi-
th values in C and DC respectively (see, e.g., [35] for more details), by ⊗̂ a symmetric
tensor product. Together with chain (1.1), we consider the chains
D′C
b⊗n = ind lim
τ ′∈T
Hb⊗n
−τ ′,C ⊃ H
b⊗n
−τ,C ⊃ H
b⊗n
C ⊃ Hb⊗n
τ,C ⊃ D
b⊗n
C := pr lim
τ ′∈T
Hb⊗n
τ ′,C, (1.2)
n ∈ N, D′C
b⊗0 = Hb⊗0
−τ,C = Hb⊗0
C = Hb⊗0
τ,C = Db⊗0
C := C. Denote the norms in the spaces
Hb⊗n
−τ,C, H
b⊗n
C and Hb⊗n
τ,C by | · |−τ , | · |0 and | · |τ , respectively, the dual pairings between
elements of negative and positive spaces of chains (1.2) by 〈·, ·〉 ; these pairings are
generated by the scalar products in Hb⊗n
C .
Assume that γ ∈ Hol0(DC), γ(0) = 1, α ∈ Hol0(DC,DC), α(0) = 0. Let there
exist the function α−1 ∈ Hol0(DC,DC) inverse to α and let χ : C → C be an entire
function such that χ(0) = 1 and in the decomposition
χ(u) =
∞∑
n=0
χn
n!
un, (1.3)
χn 6= 0 for all n ∈ Z+. We consider a function χγ,α(λ; z) := γ(λ)χ(〈z, α(λ)〉). It is
easy to see that χγ,α(·; z) ∈ Hol0(DC) for each z ∈ D′C, therefore, using the Cauchy
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1224 N. A. KACHANOVSKY
inequality (e.g., [35]) and the kernel theorem (e.g., [17]), one can show, that for z ∈ D′C
and λ from some (depending on z) neighborhood of 0 ∈ DC, we have
χγ,α(λ; z) =
∞∑
n=0
1
n!
〈
Pχ,γ,αn (z), λ⊗n
〉
,
where Pχ,γ,αn (z) ∈ D′C
b⊗n (moreover, if z ∈ H−τ,C, then Pχ,γ,αn (z) ∈ Hb⊗n
−τ,C). The
polynomials {
〈
Pχ,γ,αn (z), f (n)
〉
, f (n) ∈ Db⊗n
C , n ∈ Z+} are called generalized Appell-
like polynomials (or Schefer polynomials in another terminology). The interested reader
can find more detailed information about generalized Appell-like polynomials in, e.g.,
[9, 26] (one-dimensional case), [67, 77, 88, 91] (infinite-dimensional case).
1.2. Test function spaces. Let
P(D′) =
{
N∑
n=0
〈
x⊗n, g(n)
〉
: x ∈ D′, g(n) ∈ Db⊗n
C , N ∈ Z+
}
be the set of continuous polynomials on D′. One can show (e.g., [67]) that it is possible
to understand P(D′) as the set of polynomials
f(x) =
Nf∑
n=0
〈
Pχ,γ,αn (x), f (n)
〉
, x ∈ D′, f (n) ∈ Db⊗n
C , Nf ∈ Z+. (1.4)
Let us introduce the family of Hilbert norms ‖ · ‖τ,q,χ,γ,α, τ ∈ T, q ∈ Z+, on P(D′) by
setting, for f of form (1.4), the norm
‖f‖2τ,q,χ,γ,α :=
Nf∑
n=0
(n!)22qn
∣∣f (n)
∣∣2
τ
. (1.5)
By (Hτ )q,χ,γ,α denote a Hilbert space that is the closure of P(D′) with respect
to norm (1.5). Let also (Hτ )χ,γ,α := pr limq∈Z+
(Hτ )q,χ,γ,α, (D)χ,γ,α:=
:= pr limτ∈T,q∈Z+
(Hτ )q,χ,γ,α. The spaces (Hτ )q,χ,γ,α, (Hτ )χ,γ,α, (D)χ,γ,α are called
Kondratiev test function spaces. It is easy to see that f ∈ (Hτ )q,χ,γ,α if and only if f
can be presented in the form
f =
∞∑
n=0
〈
Pχ,γ,αn , f (n)
〉
, f (n) ∈ Hb⊗n
τ,C, (1.6)
with
‖f‖2(Hτ )q,χ,γ,α =
∞∑
n=0
(n!)22qn|f (n)|2τ <∞.
The interested reader can find a more detailed information about Kondratiev spaces in,
e.g., [7, 12, 22 – 24, 67, 68, 88, 90, 91, 112]; here, we note only that
1) for q ∈ Z+ sufficiently large, (Hτ )q,χ,γ,α are functional spaces, their elements are
restrictions onH−τ of entire functions onH−τ,C, and for each function f ∈ (Hτ )q,χ,γ,α
decomposition (1.6) is unique;
2) the spaces (Hτ )χ,γ,α and (D)χ,γ,α do not depend on γ and α, therefore below
we denote these spaces by (Hτ )χ and (D)χ, respectively.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1225
1.3. Pseudodifferential operators. Let ν ∈ Hol0(DC). On P(D′), we define a
pseudodifferential operator ν(Dχ) :=
∑∞
n=0
1
n!
〈
νn, D
⊗n
χ
〉
, where νn ∈ D′C
b⊗n
, n ∈
Z+, are the kernels from the decomposition ν(λ) =
∑∞
n=0
1
n!
〈
νn, λ
⊗n〉 ;
〈
νn, D
⊗n
χ
〉 〈
x⊗m, f (m)
〉
:= 1{m≥n}
m!χm−n
(m− n)!χm
〈
x⊗m−n⊗̂νn, f (m)
〉
,
where χn ∈ C (n ∈ Z+) are the coefficients from decomposition (1.3) for χ; here and
below, 1A is the indicator of an event A. One can show [67, 77] that
1) for each νn ∈ D′C
b⊗n
,〈
νn, α
−1(Dχ)⊗n
〉 〈
Pχ,γ,αm (x), f (m)
〉
= 1{m≥n}
m!
(m− n)!
〈
Pχ,γ,αm−n (x)⊗̂νn, f (m)
〉
;
(1.7)
2) ν(α−1(Dχ)) can be continued to a linear isometric operator acting from (Hτ )q,χ,γ,α
to (Hτ )q,χ,γν,α: for f ∈ (Hτ )q,χ,γ,α of form (1.6), we have
ν(α−1(Dχ))f =
∞∑
n=0
〈
Pχ,γν,αn , f (n)
〉
.
1.4. Generalized function spaces. Let µ be a probability measure on (D′,F(D′)),
here and below, F denotes the σ-algebra generated by cylindrical sets. Denote (L2)µ :=
:= L2(D′, µ) the space of functions F : D′ → C square integrable with respect to µ.
Let ‖ · ‖µ be the norm in this space. In addition, we accept that µ satisfies the following
assumptions:
1) there exists τ ∈ T such that µ(H−τ ) = 1;
2) there exists K > 1 such that, for each n ∈ Z+, |χn|‖| · |n−τ‖µ ≤ n!Kn;
3) the set P(D′) of continuous polynomials on D′ is dense in (L2)µ;
4) there exists a nonempty open set O ⊆ H−τ such that µ is positive on nonempty
open subsets of O.
Let χ, γ, α and µ satisfy all conditions mentioned above. If we exclude from T
some indexes (see [73] for more details), then it is posiible to show [79] that, for
q ∈ Z+ sufficiently large, the spaces (Hτ )q,χ,γ,α are embedded in (L2)µ, and these
embeddings are dense and continuous. Moreover, this result holds true if, inconvenient
for verification, condition 4 on µ is substituted by the following one:
4′) the polynomials
{〈
Pχ,γ,αn , f (n)
〉}
are orthogonal in (L2)µ and if
∥∥〈Pχ,γ,αn ,
f (n)〉
∥∥
µ
= 0, then |f (n)|τ = 0.
We note also that if χ = exp, then condition 2 is equivalent to holomorphy at zero
of the Laplace transform of µ (e.g., [91]), and it follows from this holomorphy that
condition 3 is satisfied (e.g., [109]).
We can now consider the chain
(D)′χ,µ ⊃ (H−τ )χ,µ ⊃ (H−τ )−q,χ,γ,α,µ ⊃ (L2)µ ⊃ (Hτ )q,χ,γ,α ⊃ (Hτ )χ ⊃ (D)χ,
(1.8)
where (H−τ )−q,χ,γ,α,µ, (H−τ )χ,µ and (D)′χ,µ are the spaces dual of (Hτ )q,χ,γ,α, (Hτ )χ
and (D)χ with respect to (L2)µ, correspondingly. It will be convenient to understand
(H−τ )χ,µ and (D)′χ,µ as topological spaces with the inductive limit topologies. The
negative spaces of chain (1.8) are called Kondratiev generalized functions spaces.
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1226 N. A. KACHANOVSKY
Let us describe natural orthogonal bases in the spaces (H−τ )−q,χ,γ,α,µ. Let 〈〈·, ·〉〉µ
denote the scalar product in (L2)µ (we preserve this notation for dual pairings generated
by this scalar product); let
γ̃(λ) :=
1
〈〈1, χ(〈·, α(λ)〉)〉〉µ
. (1.9)
One can show [67] that γ̃ ∈ Hol0(DC), therefore
γ̃
γ
∈ Hol0(DC) and one can consider
the decomposition
γ̃(λ)
γ(λ)
=
∞∑
n=0
1
n!
〈
ρn, λ
⊗n〉 , ρn ∈ H
b⊗n
−τ,C.
We set
Qχ,γ,αµ,m (F (m); ·) :=
:=
∞∑
k=m
1
(k −m)!
(〈
F (m)⊗̂ρk−m, α−1(Dχ)⊗k
〉∗
1
)
(·), F (m) ∈ Hb⊗m
−τ,C,
where
〈
F (m)⊗̂ρk−m, α−1(Dχ)⊗k
〉∗
is the operator adjoint to pseudodifferential ope-
rator (1.7). It follows from results of [67] that the system of generalized functions{
Qχ,γ,αµ,m (F (m)), F (m) ∈ Hb⊗m
−τ,C, m ∈ Z+
}
form orthogonal bases in (H−τ )−q,χ,γ,α,µ:
F ∈ (H−τ )−q,χ,γ,α,µ if and only if there exists a sequence of kernels
{
F (m) ∈
∈ Hb⊗m
−τ,C
}∞
m=0
such that
F =
∞∑
m=0
Qχ,γ,αµ,m (F (m)) (1.10)
with
‖F‖2−τ,−q,χ,γ,α := ‖F‖2(H−τ )−q,χ,γ,α,µ =
∞∑
m=0
2−qm|F (m)|2−τ <∞. (1.11)
Moreover, the biorthogonality relation〈〈
Qχ,γ,αµ,m (F (m)),
〈
Pχ,γ,αn , f (n)
〉〉〉
µ
= δmnn!〈F (n), f (n)〉
is satisfied, here, δmn is the Kroneker symbol.
Another (based on properties of Fock spaces) approach to the construction of
orthogonal bases in generalized function spaces is described in [73].
1.5. Elements of a Wick calculus. Now we consider elements of a so-called Wick
calculus on the generalized function spaces. For F ∈ (D)′χ,µ, we introduce a so-called
S-transform by setting
(Sχ,γ,α,µF )(λ) := 〈〈F, χγ,α(λ; ·)〉〉µ =
∞∑
m=0
〈F (m), λ⊗m〉, (1.12)
where F (m) ∈ D′C
b⊗m
, m ∈ Z+, are the kernels from decomposition (1.10) for F ;
λ ∈ DC. It is easy to show that the series in the right-hand side of (1.12) converges on
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ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1227
some (depending on F ) neighborhood of 0 ∈ DC. Further, one can show [67] that the
S-transform is a one-to-one mapping between (D)′χ,µ and Hol0(DC).
For F,G ∈ (D)′χ,µ and a function h : C → C, holomorphic for (Sχ,γ,α,µF )(0) =
F (0) (see (1.10)) we define a Wick product F♦χ,γ,αµ G ∈ (D)′χ,µ and a Wick version
h♦χ,γ,αµ (F ) ∈ (D)′χ,µ by setting
F♦χ,γ,αµ G := S−1
χ,γ,α,µ(Sχ,γ,α,µF · Sχ,γ,α,µG),
h♦χ,γ,αµ (F ) := S−1
χ,γ,α,µh(Sχ,γ,α,µF ).
Note that if γ(λ) has form (1.9), then Sχ,γ,α,µ1 ≡ 1, and therefore,
h♦χ,γ,αµ (F ) =
∞∑
n=0
hn(F − F (0))♦χ,γ,αµ n,
where F♦χ,γ,αµ n := F♦χ,γ,αµ . . .♦χ,γ,αµ F︸ ︷︷ ︸
n
= S−1
χ,γ,α,µ
[
(Sχ,γ,α,µF )n
]
, F♦χ,γ,αµ 0 := 1; and
the coefficients hn from the decomposition
h(u) =
∞∑
n=0
hn(u− F (0))n (1.13)
belong to C.
Let us consider the “coordinate form” of the Wick product and Wick versions of
holomorphic functions. It is easy to calculate with the use of (1.12) that, for F,G ∈
(D)′χ,µ,
F♦χ,γ,αµ G =
∞∑
m=0
Qχ,γ,αµ,m
(
m∑
k=0
F (k)⊗̂G(m−k)
)
, (1.14)
where F (k), G(k) ∈ D′C
b⊗k are the kernels from decompositions (1.10) for F and G. If
we apply the induction to this formula, then we obtain
F1♦
χ,γ,α
µ . . .♦χ,γ,αµ Fn =
∞∑
m=0
Qχ,γ,αµ,m
∑
k1,...,kn∈Z+,
k1+···+kn=m
F
(k1)
1 ⊗̂ . . . ⊗̂F (kn)
n
, (1.15)
where F1, . . . , Fn ∈ (D)′χ,µ, F
(k)
1 , . . . , F
(k)
n ∈ D′C
b⊗k are the kernels from decomposi-
tions (1.10) for F1, . . . , Fn, correspondingly. Further, substituting (1.12) in (1.13) and
applying S−1
χ,γ,α,µ, we obtain
h♦χ,γ,αµ (F ) = Qχ,γ,αµ,0 (h0) +
∞∑
m=1
Qχ,γ,αµ,m
m∑
n=1
hn
∑
k1,...,kn∈N,
k1+···+kn=m
F (k1)⊗̂ . . . ⊗̂F (kn)
.
(1.16)
In particular, note that,
Qχ,γ,αµ,n (F (n))♦χ,γ,αµ Qχ,γ,αµ,m (G(m)) = Qχ,γ,αµ,n+m(F (n)⊗̂G(m)).
Consider the space (H−τ )χ,µ ⊂ (D)′χ,µ instead of (D)′χ,µ. The following statement
[70] is true.
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1228 N. A. KACHANOVSKY
Theorem 1.1. Let F,G ∈ (H−τ )χ,µ and h : C→ C be a function holomorphic at
(Sχ,γ,α,µF )(0). Then F♦χ,γ,αµ G ∈ (H−τ )χ,µ, h♦χ,γ,αµ (F ) ∈ (H−τ )χ,µ, and coordinate
formulas (1.14) – (1.16) hold true. Moreover, the Wick product is continuous in the
topology of (H−τ )χ,µ : for any F1, . . . , Fn ∈ (H−τ )χ,µ,∥∥F1♦
χ,γ,α
µ . . .♦χ,γ,αµ Fn
∥∥
−τ,−q,χ,γ,α ≤
≤ c(n− 1)‖F1‖−τ,−(q−1),χ,γ,α . . . ‖Fn‖−τ,−(q−1),χ,γ,α
(see (1.11)), where c(n) :=
√
maxm∈Z+ [2−m(m+ 1)n]; q ∈ N is such that F1, . . . , Fn ∈
(H−τ )−(q−1),χ,γ,α,µ.
1.6. An analog of the extended stochastic integral. Let us consider now an analog
of the extended (Skorohod-type, see [57, 63, 64, 110]) stochastic integral in the “bi-
orthogonal analysis” (the reader can find a more detailed presentation in [74], see also
[73]). In order to explain the idea of our construction, first we briefly recall the constructi-
on of the stochastic integral in the Gaussian analysis.
Let µ be the standard Gaussian measure on (D′,F(D′)), i.e., a probability measure
with the Laplace transform
lµ(λ) =
∫
D′
e〈x,λ〉µ(dx) = e〈λ,λ〉/2.
As is well known (e.g., [54]), one can consider as an orthogonal basis in (L2)µ
((L2)µ-limits of) the generalized Hermite polynomials
〈
Hn, F
(n)
〉
; these polynomi-
als are generalized Appell-like polynomials with χ = exp, γ(λ) = exp
{
−1
2
〈λ, λ〉
}
,
α(λ) = λ.
Let now F ∈ (L2)µ ⊗HC. Then F can be presented in the form
F (·) =
∞∑
n=0
〈Hn, F
(n)
· 〉, F (n)
· ∈ Hb⊗n
C ⊗HC;
and if, in addition, F is integrable in the extended (Skorohod) sense, i.e.,
∞∑
n=0
(n+ 1)!
∣∣F̂ (n)
∣∣2
0
<∞,
where, for each n ∈ Z+, F̂
(n) ∈ Hb⊗n+1
C is the projection of F
(n)
· onto Hb⊗n+1
C , then
the extended stochastic integral of F with respect to a Wiener process W has the form
(e.g., [56]) ∫ ∞
0
F (u)d̂Wu =
∞∑
n=0
〈
Hn+1, F̂
(n)
〉
. (1.17)
Remark. The described construction of the extended stochastic integral is not uni-
que. For example, it was shown in [31] that (in the Gaussian case) the extended stochastic
integral can be interpreted as the logarithmic derivative of the Gaussian measure in the
line of a vector field. By analogy, i.e., using an integration by parts formula, one can
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ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1229
construct stochastic integrals in non-Gaussian cases, see, e.g., [10, 99, 100, 107]. Note
that, for groups of nonlinear transformations, the definitions of the logarithmic derivative
and of the extended stochastic integral were given in [111]. A very general construction
of (the analog of) the extended stochastic integral (that is based on a formal differential
rule and on the corresponding integration by parts formula) is presented in [47], see also
[36 – 46, 48 – 50].
Return now to the “biorthogonal analysis”. Let F ∈ (D)′χ,µ ⊗HC. Then
F (·) =
∞∑
m=0
Qχ,γ,α
µ,m (F (m)
· ), F (m)
· ∈ D′C
b⊗m ⊗HC.
By analogy with (1.17), for any t1, t2 ∈ [0,+∞], t1 < t2, one can define an analog of
the extended stochastic integral by setting
Iχ,γ,α,µ[t1,t2)
(F ) :=
∞∑
m=0
Qχ,γ,αµ,m+1
(
F̂
(m)
[t1,t2)
)
∈ (D)′χ,µ,
where F̂
(m)
[t1,t2)
∈ D′C
b⊗m+1 (m ∈ Z+) are the projections of F
(m)
· 1[t1,t2)(·) onto
D′C
b⊗m+1
. It is easy to show [74] that, for all t1, t2 ∈ [0,+∞], t1 < t2, I
χ,γ,α,µ
[t1,t2)
: (D)′χ,µ⊗
HC → (D)′χ,µ is a linear continuous operator.
The forthcoming statement follows from results of [74].
Theorem. For all t1, t2 ∈ [0,+∞], t1 < t2, and F ∈ (D)′χ,µ ⊗HC, the relation
Iχ,γ,α,µ[t1,t2)
(F ) =
t2∫
t1
F (u)♦χ,γ,αµ Qχ,γ,αµ,1 (δu)du
is satisfied, where δu is the Dirac delta-function, and the integral on the right-hand side
is a Pettis integral.
By analogy with the classical Gaussian analysis, as an example, we consider a simple
“integral equation with a Wick-type nonlinearity”. Let
Xt = X0 + r
t∫
0
Xu♦
χ,γ,α
µ (N −Xu)du+ vIχ,γ,α,µ[0,t)
(
X·♦
χ,γ,α
µ (N −X·)
)
, (1.18)
where X0 ∈ (D)′χ,µ (correspondingly (H−τ )χ,µ), N, r, v ∈ R, N > 0, r > 0,
(Sχ,γ,α,µX0)(0) > 0. Note that, in the Gaussian analysis, (1.18) is the so-called
“population growth equation” (see, e.g., [94]). Applying to (1.18) the S-transform, solvi-
ng the obtained equation and applying the inverse S-transform, we obtain the solution
Xt = N
[
S−1
χ,γ,α,µ1 +
(
NX
♦χ,γ,αµ (−1)
0 − S−1
χ,γ,α,µ1
)
♦χ,γ,αµ exp♦χ,γ,αµ
{
−N
(
rtS−1
χ,γ,α,µ1 + vQχ,γ,αµ,1 (1[0,t)
)}]♦χ,γ,αµ (−1)
∈ (D)′χ,µ
(correspondingly (H−τ )χ,µ), here Y ♦χ,γ,αµ (−1) := S−1
χ,γ,α,µ
(
1
Sχ,γ,α,µY
)
.
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1230 N. A. KACHANOVSKY
As is well known, in the Gaussian analysis, the extended stochastic integral is
the adjoint operator to the stochastic derivative. In the “biorthogonal analysis”, the
situation is quite similar; let us explain this in details. We define a stochastic derivative
∂χ,α· : (D)χ → (D)χ ⊗HC by setting, for f ∈ (D)χ,
∂χ,α· f :=
〈
δ·, α
−1(Dχ)
〉
f =
∞∑
n=0
(n+ 1)〈P χ,γ,α
n , f (n+1)(·)〉
(see (1.7)), where f (n+1)(·) ∈ Db⊗n
C ⊗HC (n ∈ Z+) are the kernels f (n+1) ∈ Db⊗n+1
C ⊂
⊂ Db⊗n
C ⊗HC from decomposition (1.6) for f (these kernels can be considered as elements
of Db⊗n
C ⊗HC). By using the results of [74, 73], we obtain the following statements.
Theorem 1.2. For all t1, t2 ∈ [0,+∞], t1 < t2, the operators Iχ,γ,α,µ[t1,t2)
: (D)′χ,µ⊗
HC → (D)′χ,µ and 1[t1,t2)(·)∂χ,α· : (D)χ → (D)χ⊗HC are adjoint one to another:(
Iχ,γ,α,µ[t1,t2)
)∗
= 1[t1,t2)(·)∂χ,α· ;
(
1[t1,t2)(·)∂χ,α·
)∗
= Iχ,γ,α,µ[t1,t2) .
In particular, the operator Iχ,γ,α,µ[t1,t2)
does not depend on γ and can be denoted by Iχ,α,µ[t1,t2)
.
Corollary 1.1. The operator Iχ,α,µ[t1,t2)
can be written in the form
Iχ,α,µ[t1,t2)
(F ) =
t2∫
t1
(∂χ,αu )†F (u)du,
where (∂χ,αu )† : (D)′χ,µ → (D)′χ,µ is the operator adjoint to ∂χ,αu , u ∈ R+; the integral
on the right-hand side is a Pettis integral.
Similar results are true if we consider operators Iχ,α,µ[t1,t2)
and ∂χ,α· on “pre-limit”
spaces.
1.7. Operators of stochastic differentiation. Unfortunatelly, the operator ∂χ,α· ,
generally speaking, cannot be continued on the generalized function spaces by a natural
way. Nevertheless, one can define and study a natural analog of this operator; the reader
can find a more detailed information in [72].
Let us now consider operators of stochastic differentiation on the generalized function
spaces. Let F (m) ∈ D′C
b⊗m
, f (n) ∈ Db⊗n
C , m > n. We define a “pairing” 〈F (m),
f (n)〉 ∈ D′C
b⊗m−n by setting, for each g(m−n) ∈ Db⊗m−n
C , the relation〈〈
F (m), f (n)
〉
, g(m−n)
〉
=
〈
F (m), f (n)⊗̂g(m−n)
〉
.
Further, for arbitrary f (n) ∈ Db⊗n
C , n ∈ Z+,we define a linear operator (D̃nχ,γ,α,µ◦)(f (n))
in (D)′χ,µ by setting, for F ∈ (D)′χ,µ, the relation
(
D̃nχ,γ,α,µF
)
(f (n)) :=
∞∑
m=0
(m+ n)!
m!
Qχ,γ,αµ,m
(〈
F (m+n), f (n)
〉)
,
where F (m) ∈ D′C
b⊗m
, m ∈ Z+, are the kernels from decomposition (1.10) for F. It
follows from results of [72] that
(
D̃nχ,γ,α,µ ◦
)
(f (n)) is a continuous operator in (D)′χ,µ,
and the following statement is true.
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ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1231
Theorem 1.3. The operator D̃nχ,γ,α,µ has the following properties:
for g(1)
1 , . . . , g
(1)
n ∈ DC and F ∈ (D)′χ,µ,(
D̃1
χ,γ,α,µ
(
. . .
(
D̃1
χ,γ,α,µ((D̃1
χ,γ,α,µ︸ ︷︷ ︸
n
F )(g(1)
1 ))
)
(g(1)
2 ) . . .
))
(g(1)
n ) =
= (D̃nχ,γ,α,µF )
(
g
(1)
1 ⊗̂ . . . ⊗̂g(1)
n
)
;
for each F ∈ (D)′χ,µ, the kernels F (m) ∈ D′C
b⊗m from decomposition (1.10) can be
presented in the form
F (m) =
1
m!
E(D̃mχ,γ,α,µF ),
i.e., for each f (m) ∈ Db⊗m
C , we have
〈
F (m), f (m)
〉
=
1
m!
E
[
(D̃mχ,γ,α,µF )(f (m))
]
, where
E denotes the expectation: E◦ = 〈〈◦, 1〉〉µ ;
the operator adjoint to D̃nχ,γ,α,µ has a form
(D̃nχ,γ,α,µg)(f (n))∗ =
∞∑
k=0
〈
Pχ,γ,αk+n , f (n)⊗̂g(k)
〉
, f (n) ∈ Db⊗n
C ,
where g(k) ∈ Db⊗k
C , k ∈ Z+, are the kernels from decomposition (1.4) for g ∈ (D)χ;
the operator D̃1
χ,γ,α,µ is a pre-image of the directional derivative of Sχ,γ,α,µ◦ under
the S-transform, i.e., for any F ∈ (D)′χ,µ and g ∈ DC,
(D̃1
χ,γ,α,µF )(g) = S−1
χ,γ,α,µDg(Sχ,γ,α,µF ),
where Dg is the directional derivative in the direction g;
the operator D̃1
χ,γ,α,µ is a differentiation with respect to the Wick product, i.e., for
all F, G ∈ (D)′χ,µ,
D̃1
χ,γ,α,µ(F♦χ,γ,αµ G) = (D̃1
χ,γ,α,µF )♦χ,γ,αµ G+ F♦χ,γ,αµ (D̃1
χ,γ,α,µG);
for any n ∈ Z+, F ∈ (D)′χ,µ, and a function h : C → C holomorphic for
(Sχ,γ,α,µF )(0) the ralations
D̃1
χ,γ,α,µF
♦χ,γ,αµ n = nF♦χ,γ,αµ n−1♦χ,γ,αµ (D̃1
χ,γ,α,µF ),
D̃1
χ,γ,α,µh
♦χ,γ,αµ (F ) = h′
♦χ,γ,αµ (F )♦χ,γ,αµ (D̃1
χ,γ,α,µF )
are satisfied; here, h′ is the usual derivative of h;
for all t1, t2 ∈ [0,+∞], t1 < t2, and F ∈ (D)′χ,µ ⊗HC,
(D̃1
χ,γ,α,µI
χ,α,µ
[t1,t2)
(F ))(◦) = Iχ,α,µ[t1,t2)
(
(D̃1
χ,γ,α,µF )(◦)
)
+
t2∫
t1
F (u) ◦ (u)du,
where the last integral on the right-hand side is a Pettis integral.
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1232 N. A. KACHANOVSKY
An information about possible applications of the operators D̃nχ,γ,α,µ and the corres-
ponding examples can be found in, e.g., [11, 72].
2. Elements of the white noise analysis associated with the generalized Meixner
measure. 2.1. The generalized Meixner measure. First, we define the generalized
Meixner measure (see [108] for more details and explanations). Let ρ, ν : R+ → C be
smooth functions such that
θ
df
= ρ− ν : R+ → R, η
df
= ρν : R+ → R+ (2.1)
and, moreover, θ and η are bounded on R+. Further, let, for each u ∈ R+, vρ(u),ν(u)(ds)
be a probability measure on (R,B(R)) (here B denotes the Borel σ-algebra) that is
defined by its Fourier transform
∫
R
eiλsvρ(u),ν(u)(ds) = exp
{
−iλ(ρ(u) + ν(u))+
+ 2
∞∑
m=1
(ρ(u)ν(u))m
m
[ ∞∑
n=2
(−iλ)n
n!
(νn−2(u) + νn−3(u)ρ(u) + · · ·+ ρn−2(u))
]m}
.
A probability measure µ on the measurable space (D′,F(D′)) with the Fourier transform
∫
D′
ei〈x,ξ〉µ(dx) = exp
{∫
R+
du
∫
R
vρ(u),ν(u)(ds)
1
s2
(
eisξ(u) − 1− isξ(u)
)}
is called the generalized Meixner measure.
Depending on parametrs ρ and ν, µ can be, in particular, the Gaussian, Poissonian,
Pascal, Meixner or gamma measure.
It was proved in [108] that the generalized Meixner measure µ is the measure of
a generalized random process (in the sense of [51]) with independent values; and the
Laplace transform lµ of µ is a holomorphic for zero function.
2.2. The square integrable function space. Let (L2) = L2(D′, µ) be the space
of complex-valued square integrable with respect to the generalized Meixner measure
µ functions on D′. Denote by 〈〈·, ·〉〉 the scalar product in (L2), this notation will be
preserved for generated by 〈〈·, ·〉〉 dual pairings. We construct now a natural orthogonal
basis in (L2). For n ∈ N denote by Pn the closure in (L2) of the set of all conti-
nuous polynomials on D′ of degree ≤ n, P0 := C, let also (L2
n) := Pn Pn−1–the
orthogonal difference in (L2), (L2
0) := C. Since µ has a holomorphic at zero Laplace
transform, the set of continuous polynomials on D′ is dense in (L2) [109], therefore
(L2) =
∞
⊕
n=0
(L2
n). For each f (n) ∈ Db⊗n
C , n ∈ Z+, we define :
〈
x⊗n, f (n)
〉
: as the
orthogonal projection of
〈
x⊗n, f (n)
〉
onto (L2
n). It follows from results of [108] that
:
〈
x⊗n, f (n)
〉
: =
〈
Pn(x), f (n)
〉
, where Pn(x) ∈ D′C
b⊗n (n ∈ Z+) are the kernels of
generalized Appell-like polynomials with χ = exp,
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ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1233
α(λ) = λ+
∞∑
n=2
λn
n
(ρn−1 + ρn−2ν + · · ·+ νn−1),
γ(λ) =
1
lµ(α(λ))
= exp{−
∫
R+
(λ2(u)
2
+
∞∑
n=3
λn(u)
n
(ρn−2(u) + ρn−3(u)ν(u) + · · ·+ νn−2(u))
)
du}.
(2.2)
We say that the polynomials
{〈
Pn, f
(n)
〉
, f (n) ∈ Db⊗n
C , n ∈ Z+
}
are called the
generalized Meixner polynomials.
Let us define a scalar product 〈·, ·〉ext on Db⊗n
C , n ∈ Z+, by setting for f (n), g(n) ∈
∈ Db⊗n
C
〈f (n), g(n)〉ext :=
1
n!
∫
D′
〈Pn, f (n)〉〈Pn, g(n)〉µ(dx).
It follows from results of [108] that
〈f (n), g(n)〉ext =
∑
k,lj ,sj∈N: j=1,...,k, l1>l2>···>lk,
l1s1+···+lksk=n
n!
ls11 . . . lskk s1! . . . sk!
×
×
∫
Rs1+···+sk
+
f (n)(u1, . . . , u1︸ ︷︷ ︸
l1
, . . . , us1 , . . . , us1︸ ︷︷ ︸
l1
, . . . , us1+···+sk , . . . , us1+···+sk︸ ︷︷ ︸
lk
)×
×g(n)(u1, . . . , u1︸ ︷︷ ︸
l1
, . . . , us1 , . . . , us1︸ ︷︷ ︸
l1
, . . . , us1+···+sk , . . . , us1+···+sk︸ ︷︷ ︸
lk
)×
×ηl1−1(u1) . . . ηl1−1(us1)η
l2−1(us1+1) . . . ηl2−1(us1+s2) . . .
. . . ηlk−1(us1+···+sk−1+1) . . . ηlk−1(us1+···+sk)
×du1 . . . dus1+···+sk .
Let | · |ext denote the norm generated by the scalar product 〈·, ·〉ext , i.e., |f (n)|ext :=
:=
√〈
f (n), f (n)
〉
ext
. Denote by H(n)
ext the closure of Db⊗n
C with respect to | · |ext.
The space H(n)
ext can be understood as an extension of Hb⊗n
C in a generalized sense: let
F (n) ∈ Hb⊗n
C , Ḟ (n) ∈ F (n) be a representative (a function) from the equivalence class
F (n) with a “zero diagonale”, i.e., Ḟ (n)(u1, . . . , un) = 0 if there exist i, j ∈ {1, . . . , n}
such that i 6= j but ui = uj . The function Ḟ (n) generates an equivalence class in H(n)
ext
that can be identified with F (n) [73].
For F (n) ∈ H(n)
ext , n ∈ Z+, we define
〈
Pn, F
(n)
〉
∈ (L2) as an (L2)-limit
〈Pn, F (n)〉 := lim
k→∞
〈Pn, f (n)
k 〉,
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1234 N. A. KACHANOVSKY
where Db⊗n
C 3 f
(n)
k → F (n) in H(n)
ext as k → ∞. The forthcoming statement easily
follows from the construction of polynomials
〈
Pn, F
(n)
〉
(see also [108]).
Theorem 2.1. A function F ∈ (L2) if and only if there exists a sequence of kernels
(
F (n) ∈ H(n)
ext
)∞
n=0
(2.3)
such that F can be presented in the form
F =
∞∑
n=0
〈Pn, F (n)〉, (2.4)
where the series converges in (L2), i.e., the (L2)-norm of F
‖F‖2(L2) =
∞∑
n=0
n!|F (n)|2ext <∞.
Moreover, the system
{ 〈
Pn, F
(n)
〉
, F (n) ∈ H(n)
ext , n ∈ Z+
}
is an orthogonal basis in
(L2) in the sense that for F, G ∈ (L2) of form (2.4)
〈〈F,G〉〉 =
∞∑
n=0
n!〈F (n), G(n)〉ext.
2.3. A nonregular rigging of (L2). One can show [73] that the generalized Meix-
ner measure satisfies conditions 1, 2, 3, 4′ of the previous section, therefore one can
consider (nonregular) chain (1.8) with the central space (L2), χ = exp, α and γ
from (2.2); and all results of the “biorthogonal analysis” hold true in the “Meixner
analysis”. Note that since the generalized Meixner polynomials are orthogonal in (L2),
natural orthogonal bases in the generalized function spaces (H−τ )−q,exp,γ,α consist of
generalized functions
〈
Pm, F
(m)
〉
:= (H−τ )−q,exp,γ,α− limk→∞〈Pm, f (m)
k 〉, m ∈ Z+,
where F (m) ∈ H(m)
−τ,C–the negative space from the chain
D′C
(m) ⊃ H(m)
−τ,C ⊃ H
(m)
ext ⊃ H
b⊗m
τ,C ⊃ D
b⊗m
C
(see [73] for more details), Db⊗m
C 3 f
(m)
k → F (m) in H(m)
−τ,C as k → ∞. The
interconnection between the generalized functions 〈Pm, ·〉 andQexp,γ,α
µ,m (see the previous
section) is given by the formulas
〈Pm, F (m)〉 = Qexp,γ,α
µ,m (UmF (m)),
where α and γ from (2.2) and the operators Um : H(m)
−τ,C → H
b⊗m
−τ,C, m ∈ Z+, are defined
as follows:
∀f (m) ∈ Hb⊗m
τ,C 〈F (m), f (m)〉ext =
〈
UmF
(m), f (m)
〉
. (2.5)
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ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1235
2.4. A parametrized regular rigging of (L2). Now we begin to observe some
results of the “Meixner analysis” that can not be obtained as consequences of results of
the “biorthogonal theory”. First we introduce a so-called (regular) parametrized rigging
of (L2). On the set P(D′) of all continuous polynomials on D′, presented as the set of
functions (1.4) with χ = exp, α and γ from (2.2) (i.e., now Pχ,γ,αn (x) = Pn(x)), we
introduce a family of Hilbert norms ‖ · ‖q,β , q ∈ Z+, β ∈ [0, 1] (in what follows, we
accept these conditions on default), by setting for f of form (1.4)
‖f‖2q,β :=
Nf∑
n=0
(n!)1+β2qn|f (n)|2ext. (2.6)
By (L2)βq denote a Hilbert space that is the closure of P(D′) with respect to norm (2.6).
Let also (L2)β := pr limq∈Z+
(L2)βq . The spaces (L2)βq , (L
2)β are called the parametri-
zed Kondratiev-type test function spaces. It is easy to see that f ∈ (L2)βq if and only if
f can be presented in the form
f =
∞∑
n=0
〈Pn, f (n)〉, f (n) ∈ H(n)
ext , (2.7)
with
‖f‖2
(L2)βq
=
∞∑
n=0
(n!)1+β2qn|f (n)|2ext <∞.
It is easy to show [73] that for arbitrary q ∈ Z+ and β ∈ [0, 1] the space (L2)βq is densely
and continuously embedded into (L2), therefore one can consider the chain
(L2)−β = ind lim
q′∈Z+
(L2)−β−q′ ⊃ (L2)−β−q ⊃ (L2) ⊃ (L2)βq ⊃ (L2)β ,
where (L2)−β−q , (L2)−β are the spaces dual of (L2)βq , (L2)β with respect to (L2)
correspondingly. The spaces (L2)−β−q , (L2)−β are called the parametrized Kondratiev-
type (regular) generalized function spaces. Note that for β = q = 0 (L2)00 = (L2)−0
−0 =
= (L2).
Since the generalized Meixner polynomials are orthogonal in (L2), these polynomials
form orthogonal bases in (L2)−β−q . More exactly, a function F ∈ (L2)−β−q if and only if
there exists sequence (2.3) such that F can be presented in form (2.4) with
‖F‖2−q,−β := ‖F‖2
(L2)−β−q
=
∞∑
n=0
(n!)1−β2−qn|F (n)|2ext <∞.
2.5. Elements of a Wick calculus. Now we consider elements of a Wick calculus
on the parametrized generalized function spaces. For F ∈ (L2)−β we define the S-
transform (SF )(λ), λ ∈ DC, as a formal series
(SF )(λ) :=
∞∑
n=0
〈F (n), λ⊗n〉ext,
where F (n) ∈ H(n)
ext , n ∈ Z+, are the kernels from decomposition (2.4) for F. It is
obvious that, in particular, (SF )(0) = F (0), S1 ≡ 1. Further, for F, G ∈ (L2)−β and
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1236 N. A. KACHANOVSKY
a holomorphic at F (0) function h : C→ C by analogy with the “biorthogonal case” we
define a Wick product F♦G and a Wick version h♦(F ) by seting formally
F♦G := S−1(SF · SG), h♦(F ) := S−1h(SF ).
A formal calculation shows that
F♦G =
∞∑
m=0
〈
Pm,
m∑
k=0
F (k) �G(m−k)
〉
,
F1♦ . . .♦Fn =
∞∑
m=0
〈
Pm,
∑
k1,...,kn∈Z+,
k1+···+kn=m
F
(k1)
1 � · · · � F (kn)
n
〉
, (2.8)
h♦(F ) = h0 +
∞∑
m=1
〈
Pm,
m∑
n=1
hn
∑
k1,...,kn∈N,
k1+···+kn=m
F (k1) � · · · � F (kn)
〉
(cf. (1.14) – (1.16)), where F (k), G(k), F
(k)
1 , . . . , F
(k)
n ∈ H(k)
ext are the kernels from
decompostions (2.4) for F,G, F1, . . . , Fn correspondingly; hn ∈ C, n ∈ Z+, are the
coefficients from the decomposition h(u) =
∑∞
n=0
hn(u − F (0))n; for F (n) ∈ H(n)
ext ,
G(m) ∈ H(m)
ext
F (n) �G(m) := U−1
n+m
(
(UnF (n))⊗̂(UmG(m))
)
∈ H(n+m)
ext , (2.9)
see (2.5). The fact that F (n) �G(m) ∈ H(n+m)
ext is proved in [73]; moreover, it is shown
therein that, roughly speaking, F (n) � G(m) is a symmetrization with respect to all
variables of a functionF
(n)(u1, . . . , un)G(m)(un+1, . . . , un+m), if ∀i∈{1,...,n},∀j∈{n+1,...,n+m} ui 6=uj ,
0, in other cases
and |F (n) �G(m)|ext ≤ |F (n)|ext|G(m)|ext.
The forthcoming theorem from results of [66, 73] follows.
Theorem 2.2. The following statements are fulfilled:
Let F, G ∈ (L2)−β . Then F♦G ∈ (L2)−β .Moreover, the Wick product is continuous
in the topology of (L2)−β : for arbitrary F1, . . . , Fn ∈ (L2)−β there exist (depending
on these elements) q, q′ ∈ Z+ (q > q′ + (1− β) log2 n+ 1) such that
‖F1♦ . . .♦Fn‖−q,−β ≤ c(n− 1)‖F1‖−q′,−β . . . ‖Fn‖−q′,−β ,
where c(n) :=
√
maxm∈Z+ [2−m(m+ 1)n].
For F ∈ (L2)−1 and a holomorphic at (SF )(0) function h : C → C h♦(F ) ∈
∈ (L2)−1.
Let h : C→ C be a holomorphic at u0 ∈ C not-polynomial function with nonnegative
coefficients hn from the decomposition h(u) =
∑∞
n=0
hn(u − u0)n. Then for each
β ∈ [0, 1) there exists F ∈ (L2)−β with (SF )(0) = u0 such that h♦(F ) 6∈ (L2)−β .
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ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1237
Let F =
∑N
m=0
〈Pm, F (m)〉, F (m) ∈ H(m)
ext ; h : C → C be a holomorphic at F (0)
function such that the coefficients hn from the decomposition h(u) =
∑∞
n=0
hn(u −
− F (0))n satisfy the estimates
|hn| ≤
Kn
nnN
1−β
2
with some K > 0. Then h♦(F ) ∈ (L2)−β .
Note that the proof of this theorem is based on “coordinate” representations (2.8).
Finally, if F, G ∈ (L2)−β ∩ (D)′exp,µ and α, γ are defined in (2.2) then F♦G =
= F♦exp,γ,α
µ G, see [73].
2.6. An extended stochastic integral. By analogy with the Gaussian analysis, on
the probability triplet (D′,F(D′), µ) we define the Meixner random processM by setting
for each u ∈ R+ Mu :=
〈
P1, 1[0,u)
〉
∈ (L2). Using results of [108] one can show that
M is a locally square integrable normal martingale (with respect to the generated by
M flow of σ-algebras) with orthogonal independent increments. Note that M is not a
Lévy process, generally speaking (not time-homogeneous). Let us construct an extended
(Skorohod-type) stochastic integral with respect to M. Let F ∈ (L2)−β−q ⊗HC. It follows
from above-posed results that F can be presented in the form
F (·) =
∞∑
m=0
〈Pm, F (m)
· 〉, F
(m)
· ∈ H(m)
ext ⊗HC, (2.10)
with
‖F‖2
(L2)−β−q⊗HC
=
∞∑
m=0
(m!)1−β2−qm|F (m)
· |2
H(m)
ext ⊗HC
<∞.
If in addition F is such that the kernels F
(m)
· belong to Hb⊗m
C ⊗HC ⊂ H(m)
ext ⊗HC (the
embedding in the generalized sense described above) then one can show [73] that F can
be presented in the form
F (·) =
∞∑
m=0
m!
∞∫
0
um∫
0
. . .
u2∫
0
F (m)
· (u1, . . . , um)dMu1 . . . dMum ,
i.e., as a series of repeated Itô stochastic integrals with respect to the Meixner process. In
this case for arbitrary t1, t2 ∈ [0,+∞], t1 < t2, one can define the extended stochastic
integral of F with respect to M on [t1, t2) as
t2∫
t1
F (u)d̂Mu :=
:=
∞∑
m=0
(m+ 1)!
∞∫
0
u∫
0
. . .
u2∫
0
F̂
(m)
[t1,t2)
(u1, . . . , um, u)dMu1 . . . dMumdMu =
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1238 N. A. KACHANOVSKY
=
∞∑
m=0
〈
Pm+1, F̂
(m)
[t1,t2)
〉
∈ (L2)−β−q−1,
where F̂ (m)
[t1,t2)
∈ Hb⊗m+1
C ⊂ H(m+1)
ext is the projection of F
(m)
· 1[t1,t2)(·) onto Hb⊗m+1
C .
In a general case such a definition can not be accepted because it is impossible to project
elements of H(m)
ext ⊗HC onto H(m+1)
ext , generally speaking. Nevertheless, the following
generalization is possible.
Lemma 2.1 ([73]). For given F
(m)
· ∈ H(m)
ext ⊗ HC, m ∈ Z+, and t1, t2 ∈
[0,+∞], t1 < t2, we construct the element F̂ (m)
[t1,t2)
∈ H(m+1)
ext by the following way. Let
Ḟ
(m)
· ∈ F (m)
· be some representative (a function) from the equivalence class F
(m)
· .
We set
˜̇F (m)
[t1,t2)(u1, . . . , um, u) :=
Ḟ
(m)
u (u1, . . . , um)1[t1,t2)(u), if u 6= u1, . . . , u 6= um,
0, in other cases,
̂̇F (m)
[t1,t2) := Pr ˜̇F (m)
[t1,t2), where Pr is the symmetrization operator. Let F̂ (m)
[t1,t2)
∈ H(m+1)
ext
be the equivalence class in H(m+1)
ext that is generated by ̂̇F (m)
[t1,t2). This class is well-
defined, does not depend on the representative Ḟ
(m)
· , and∣∣∣F̂ (m)
[t1,t2)
∣∣∣
ext
≤ |F (m)
· 1[t1,t2)(·)|H(m)
ext ⊗HC
≤ |F (m)
· |H(m)
ext ⊗HC
.
Now for F ∈ (L2)−β−q ⊗ HC of form (2.10) we can define the extended stochastic
integral on [t1, t2) with respect to M by setting [73]
t2∫
t1
F (u)d̂Mu :=
∞∑
m=0
〈
Pm+1, F̂
(m)
[t1,t2)
〉
∈ (L2)−β−q−1.
If we consider the extended stochastic integral as an operator acting from (L2)−β−q ⊗HC
to (L2)−β−q (for example, if β = q = 0 then it will be an operator acting from (L2)⊗HC
to (L2)) then for β < 1 this operator will be unbounded with the domain{
F ∈ (L2)−β−q ⊗HC :
∞∑
m=0
((m+ 1)!)1−β2−q(m+1)
∣∣∣F̂ (m)
[t1,t2)
∣∣∣2
ext
<∞
}
.
Theorem 2.3 ([73]). Let F ∈ (L2) ⊗HC be integrable on R+ by Itô with respect
to M (i.e., be adapted with respect to the generated by M flow of σ-algebras). Then
for all t1, t2 ∈ [0,+∞], t1 < t2, F is integrable on [t1, t2) by Itô and in the extended
sense, and
∫ t2
t1
F (u)d̂Mu =
∫ t2
t1
F (u)dMu (the last integral is the Itô one).
Let us consider main properties of the extended stochastic integral. First we note
that if t1, t2, t3 ∈ [0,+∞] and t1 < t2 < t3 then
∫ t2
t1
◦(u)d̂Mu +
∫ t3
t2
◦(u)d̂Mu =
=
∫ t3
t1
◦(u)d̂Mu; but applying this formula one has to keep in mind that for β < 1
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ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1239
the domain of
∫ b
a
◦(u)d̂Mu : (L2)−β−q ⊗ HC → (L2)−β−q depends on the integration
interval [a, b). Further, taking into consideration that each g(n) ∈ H(n)
ext , n ∈ N, can
be considered as g(n)(·) ∈ H(n−1)
ext ⊗ HC with |g(n)(·)|H(n−1)
ext ⊗HC
≤ |g(n)|ext [73],
we define the (Hida-type) stochastic derivative 1[t1,t2)(·)∂. : (L2)βq+1 → (L2)βq ⊗ HC,
t1, t2 ∈ [0,+∞], t1 < t2, by setting for g ∈ (L2)βq+1 of form (2.7)
1[t1,t2)(·)∂.g :=
∞∑
n=0
(n+ 1)
〈
Pn, g
(n+1)(·)1[t1,t2)(·)
〉
. (2.11)
Of course, one can understand 1[t1,t2)(·)∂. as a linear unbounded operator acting from
(L2)βq to (L2)βq ⊗HC with the domain{
g =
∞∑
n=0
〈Pn, g(n)〉 ∈ (L2)βq :
∞∑
n=0
(n!)1+β2qn(n+ 1)2
∣∣∣g(n+1)(·)1[t1,t2)(·)
∣∣∣2
H(n)
ext⊗HC
<∞
}
.
Theorem 2.4 ([73]). For all t1, t2 ∈ [0,+∞], t1 < t2, the operators
∫ t2
t1
◦(u)d̂Mu :
(L2)−β−q ⊗ HC → (L2)−β−q and 1[t1,t2)(·)∂. : (L2)βq → (L2)βq ⊗ HC are adjoint one to
another:
t2∫
t1
◦(u)d̂Mu =
(
1[t1,t2)(·)∂.
)∗; 1[t1,t2)(·)∂. =
t2∫
t1
◦d̂M
∗ .
In particular, these operators are closed.
Of course, the statement of this theorem holds true if we consider
∫ t2
t1
◦(u)d̂Mu : (L2)−β−q⊗
HC → (L2)−β−q−1 and 1[t1,t2)(·)∂. : (L2)βq+1 → (L2)βq ⊗HC.
By analogy one can consider the extended stochastic integral as a linear continuous
operator acting from (L2)−β ⊗HC to (L2)−β .
Let us consider now the interconnection between the Wick calculus and the extended
stochastic integration. Let M ′u = 〈P1, δu〉 ∈ (H−τ )−q,exp,γ,α,µ (see (1.8); now α and γ
from (2.2)) be the Meixner white noise [73, 108].
Theorem 2.5 ([73]). For all t1, t2 ∈ [0,+∞], t1 < t2, and F ∈ (L2)−β ⊗ HC
the formally defined Pettis integral
∫ t2
t1
F (u)♦M ′udu can be considered as a linear
continuous functional on (L2)β that coincides with
∫ t2
t1
F (u)d̂Mu, i.e.,
t2∫
t1
F (u)♦M ′udu =
t2∫
t1
F (u)d̂Mu ∈ (L2)−β .
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1240 N. A. KACHANOVSKY
Let us consider an example of solving of an equation with Wick-type nonlinearity.
Let
Xt = X0 +
t∫
0
Xu♦Fdu+
t∫
0
Xu♦Gd̂Mu,
whereX0, F, G ∈ (L2)−β .Applying the S-transform, solving the obtained nonstochastic
equation and applying the inverse S-transform we obtain the solution
Xt = X0♦ exp♦{Ft+G♦Mt} ∈ (L2)−1.
In order to have Xt ∈ (L2)−β , β < 1, we need additional restrictions. For example, if
F and G are polynomials and N := max{degF,degG + 1}, where deg denotes the
degree of a polynomial, then Xt ∈ (L2)−β if N ≤ 2
1− β
[66].
Finally we note that if F ∈ (L2)−β ⊗HC ∩ (D)′exp,µ⊗HC and α is defined in (2.2)
then for all t1, t2 ∈ [0,+∞], t1 < t2,
t2∫
t1
F (u)d̂Mu = Iexp,α,µ
[t1,t2)
(F ),
this result is proved in [73].
2.7. Operators of stochastic differentiation. In contrast to the general “biorthogo-
nal” case, now the operator 1[t1,t2)(·)∂., t1, t2 ∈ [0,+∞], t1 < t2, can be naturally
continued to the generalized function spaces. More exactly, for F ∈ (L2)−β−q of form (2.4)
we define 1[t1,t2)(·)∂.F ∈ (L2)−β−q−1 ⊗HC by setting (cf. (2.11))
1[t1,t2)(·)∂.F :=
∞∑
m=0
(m+ 1)
〈
Pm, F
(m+1)(·)1[t1,t2)(·)
〉
.
This operator can be naturally continued to a linear continuous operator acting from
(L2)−β to (L2)−β ⊗ HC; also one can consider 1[t1,t2)(·)∂. as a linear unbounded
operator acting from (L2)−β−q to (L2)−β−q ⊗HC with the domain consists of F ∈ (L2)−β−q
such that ‖1[t1,t2)(·)∂.F‖(L2)−β−q⊗HC
<∞, in this case 1[t1,t2)(·)∂. is closed [71].
Properties of 1[t1,t2)(·)∂. on the generalized function spaces are similar to properties
of 1[t1,t2)(·)∂. on the test function spaces. In particular, 1[t1,t2)(·)∂. is the adjoint operator
to the restriction of the extended stochastic integral on the corresponding test function
space. The interested reader can find a more detailed information in [71].
Now let us consider operators of stochastic differentiation on the generalized functi-
on spaces. For F (m) ∈ H(m)
ext and f (n) ∈ H(n)
ext , m > n, we define a “pairing”〈
F (m), f (n)
〉
ext
∈ H(m−n)
ext by setting for each g(m−n) ∈ H(m−n)
ext〈〈
F (m), f (n)
〉
ext
, g(m−n)
〉
ext
=
〈
F (m), f (n) � g(m−n)
〉
ext
(see (2.9)). One can show [71] that for arbitrary F (m) ∈ H(m)
ext and f (1) ∈ H(1)
ext = HC〈
F (m), f (1)
〉
ext
=
∫
R+
F (m)(u)f (1)(u)du,
where the integral in the right-hand side is a Pettis one.
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ELEMENTS OF A NON-GAUSSIAN ANALYSIS ON SPACES OF FUNCTIONS OF INFINITELY . . . 1241
For arbitrary f (n) ∈ H(n)
ext , n ∈ Z+, we define a linear operator (Dn◦)(f (n)) :
(L2)−β−q → (L2)−β−q−1 by setting for F ∈ (L2)−β−q
(DnF )(f (n)) :=
∞∑
m=0
(m+ n)!
m!
〈
Pm,
〈
F (m+n), f (n)
〉
ext
〉
,
where F (m) ∈ H(m)
ext , m ∈ Z+, are the kernels from decomposition (2.4) for F. It follows
from results of [71] that (Dn◦)(f (n)) is a continuous operator that can be continued to
a linear continuous operator in (L2)−β , and the following statement is fulfilled.
Theorem 2.6. The operator Dn has the following properties:
for g(1)
1 , . . . , g
(1)
n ∈ H(1)
ext = HC(
D1
(
. . . (D1((D1︸ ︷︷ ︸
n
◦)(g(1)
1 )))(g(1)
2 ) . . .
))
(g(1)
n ) = (DnF )
(
g
(1)
1 � · · · � g(1)
n
)
;
for each F ∈ (L2)−β−q the kernels F (n) ∈ H(n)
ext from decomposition (2.4) can be
presented in the form
F (n) =
1
n!
E(DnF ),
i.e., for each f (n) ∈ H(n)
ext
〈
F (n), f (n)
〉
ext
=
1
n!
E
[
(DnF )(f (n))
]
, here E denotes the
expectation;
for all F ∈ (L2)−β−q and f (1) ∈ H(1)
ext = HC∫
R+
∂uF · f (1)(u)du = (D1F )(f (1)),
here the integral in the left-hand side is a Pettis one;
the adjoint to Dn operator has a form
(Dng)(f (n))∗ =
∞∑
m=0
〈Pm+n, f
(n) � g(m)〉 = g♦〈Pn, f (n)〉 ∈ (L2)βq ,
where g ∈ (L2)βq+1, f
(n) ∈ H(n)
ext , and g(m) ∈ H(m)
ext (m ∈ Z+) are the kernels from
decomposition (2.7) for g;
for all F ∈ (L2)−β−q , g ∈ (L2)βq+1 and f (1) ∈ H(1)
ext = HC〈〈
F,
∫
R+
g · f (1)(u)d̂Mu
〉〉
=
〈〈
F, g♦
〈
P1, f
(1)
〉〉〉
=
=
〈〈
F, (D1g)(f (1))∗
〉〉
=
〈〈
(D1F )(f (1)), g
〉〉
;
the operator D1 is a differentiation with respect to the Wick product, i.e., for all F,G ∈
(L2)−β and g ∈ H(1)
ext = HC
(D1(F♦G))(g) = (D1F )(g)♦G+ F♦(D1G)(g) ∈ (L2)−β ;
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1242 N. A. KACHANOVSKY
for any n ∈ Z+, F ∈ (L2)−β , g ∈ H(1)
ext = HC, and a holomorphic at (SF )(0)
function h : C→ C
(D1F♦n)(g) = nF♦n−1♦(D1F )(g) ∈ (L2)−β ,
(D1h♦(F ))(g) = h′
♦(F )♦(D1F )(g) ∈ (L2)−1,
where h′ is the usual derivative of h;
for all t1, t2 ∈ [0,+∞], t1 < t2, g ∈ H(1)
ext = HC, and F ∈ (L2)−β ⊗HCD1
t2∫
t1
F (u)d̂Mu
(g) =
t2∫
t1
(D1F (u))(g)d̂Mu +
t2∫
t1
F (u)g(u)du ∈ (L2)−β ,
where the last integral in the right-hand side is a Pettis one.
An information about possible applications of the operators Dn and the corresponding
examples can be found in, e.g., [11, 71].
One can consider the operator (Dn◦)(f (n)), f (n) ∈ H(n)
ext , as a linear unbounded one
acting in (L2)−β−q . This operator is closed and the analog of the previous theorem holds
true (see details in [71]).
Finally we note that the operators D̃nexp,γ,α,µ (α and γ from (2.2)) and Dn, n ∈ N,
do not cioncide on the set (L2)−β ∩ (D)′exp,µ, generally speaking (if η 6= 0, see (2.1));
but, as we saw, their properties are quite similar.
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