#### Abstract

We investigate the problem of best approximations in the Hardy space of complex functions, defined on the infinite-dimensional unitary matrix group. Applying an abstract Besov-type interpolation scale and the Bernstein-Jackson inequalities, a behavior of such approximations is described. An application to best approximations in symmetric Fock spaces is shown.

#### 1. Introduction

Our goal is to investigate a best approximation problem in the quasinormed Hardy space of complex functions of infinitely many variables. The considered Hardy space is defined on the infinite-dimensional unitary matrix groups , acting over a suitable infinite-dimensional Hilbert space . Thus, this work can be seen as a continuation of [1].

Notice that the infinite-dimensional unitary group is one of the basic examples of big groups whose irreducible representations depend on infinitely many parameters. General principles of harmonic analysis on this group are developed by Olshanski [2].

The investigated Hardy space in the unitary case is antilinearly unitary isometric to a symmetric Fock space , generated by (see Theorem 2 in [1]). Therefore, we can also apply obtained results for to best approximations in the symmetric Fock space .

Now we talk briefly about the content. In the introductory Sections 2 and 3, we investigate an abstract problem for a complete quasinormed abelian group , containing a dense subgroup with a given continuous approximation scale. To solve this problem, we use an interpolating scale of special Besov-type subgroups , defined by approximation -functionals. Preliminary information about approximations with the help of -functionals in the general case of complete quasinormed abelian groups can be found in [3, 7.1] and [4].

In Theorem 5 we establish a general (one of many possible) form of the Bernstein-Jackson inequalities for the considered Besov-type scale .

The main result is in Theorem 7 that, in some sense, gives a solution of best approximation problem in the Hardy spaces for the case of Besov-type scale. We establish an analogue of the Bernstein-Jackson inequalities which sharply characterizes a behavior of best approximations for functions of infinitely many variables.

It should be noted that we consider the cases of linear and nonlinear approximations in the Hardy spaces . Recall that extensive information on nonlinear approximations by discrete scales in various Banach spaces, having wide constructive implementations, can be found, for example, in DeVore [5].

Moreover, in Theorem 8 we show an application of the Bernstein-Jackson inequalities to best linear and nonlinear approximations in symmetric Fock spaces.

#### 2. Besov-Type Approximation Scales

Following [4], we consider a complete quasinormed abelian group under addition “+” with the neutral element , where the quasinorm is determined by the following assumptions:(1) and for all nonzero ,(2) for all ,(3) for all and some fixed .

In what follows we additionally suppose that the group contains a dense subgroup with a continuous approximation scale of subsets , possessing the following properties: (i) for all , at that , (ii) for all , (iii) for all .

On the subgroup we define the quasinorm which satisfies the conditions and for all with the same constant .

In fact, if we put then via property (i) and for all and via property (iii). As a result, is a quasinorm with the constant , because the quasinorm is the same.

So, the following contracting dense embedding holds:

Let us endow the dense subgroup with the quasinorm and consider on the whole group the so-called approximation -functional (see, e.g., [3, 7.1]) with . Given the pairs and we assign in the Besov-type approximation abelian subgroups endowed with the quasinorm which is determined by the given subgroup . Notice that the function is a quasinorm on via [3, Lemma ].

*Definition 1. *One calls the scale Besov-type.

Notice that if then exactly coincides with the scale of classic Besov spaces.

For any pairs index or , using the Peetre -functional for the pair groups and , with and , we can define the interpolation abelian subgroup endowed with the quasinorm (see [3])

Lemma 2. *Let be the subgroup endowed with the quasinorm . Then the equality (up to a quasinorm equivalence)
**
with the parameters
**
holds. As a consequence, the subgroup is complete and densely embedded in .*

*Proof. *The first claim immediately implies from the known [3, Theorem ]. Now we are going to check the completeness and dense embedding. Let be endowed with the quasinorm
where and . Since and , we have
for all . Hence, the space with the quasinorm is complete. Therefore, every series with such that is convergent to an element . Using the inequality
we obtain that . Hence, is complete. Isomorphism ((13)) implies that is complete and densely embedded in . Thus, is complete and densely embedded as well.

Corollary 3. *If , , and with then
**
and there exist constants , such that
**
for all and
**
for all , where means the Peetre -functional of the groups and .*

*Proof. *Applying the reiteration property of real interpolation [3, Theorem ] for the indexes
we obtain the following equality (clearly, up to a quasinorm equivalence):
In turn, applying the interpolation degree property [3, Theorem ], we obtain
with . Now, equalities ((22)) and ((23)) for yield ((18)) with .

On the other hand, inequalities ((19)) and ((20)) are a consequence of ((18)) and well known property of the real interpolation [3, Theorem ].

Corollary 4. *If then the following continuous embedding holds:
*

*Proof. *Applying Corollary 3, we get that for every element there exists a constant such that the inequalities
hold. Hence, the embedding
is continuous. Finally using ((18)), we obtain ((24)).

#### 3. Best Approximations

Let the subgroup be endowed with the quasinorm of form ((2)). Now we are going to consider the problem of best approximation of a given element in a complete abelian group by elements of a fixed subset .

We denote the distance between and a subset with a fixed by This distance characterizes the error of the best approximation of by elements of .

To investigate this problem, we will use the scale of subgroups determined for the pairs of indexes or .

Theorem 5. *For every , there are constants and so that the following analogue of the Bernstein-Jackson inequalities holds:
*

*Proof. *By Lemma 2 the space is interpolating between and . As a consequence,
with . Hence, applying [3, Theorem ] for some constant we obtain
This inequality together with isomorphism ((13)) implies that there is a constant such that inequality ((28)) is true.

Now we will prove the second inequality. By [3, Theorem ] for some constant we have
Hence, in virtue of isomorphism ((13)) there is a constant such that
Following [3, 7.1], we introduce the functional
with and . The inequality
yields
for all .

By [3, Lemma ] for every and there exists such that
For any fixed the function is decreasing in the variable (see [3, Lemma ]). Therefore,
for every such that
for all . It follows that
As a result, we obtain
Using ((36)), we have
Substituting , we get

On the other hand, if
then , where is denoted
So, for all numbers such that
The embedding yields . Hence, the following inequality
holds. Now taking in ((43)) and using ((47)), we finally obtain the required inequality ((29)).

#### 4. Hardy Spaces of Infinitely Many Variables

We will investigate the Hardy space of complex functions on the infinite-dimensional group integrable with respect to a projective limit of probability Haar measures , determined on the corresponding -dimensional unitary matrix groups .

The measure is determined on, the so-called, the space of virtual unitary matrices , being a projective limit of , which was earlier studied by Neretin [6] and Olshanski [2]. The main feature of this measure is the fact that it is invariant under the right action over of the infinite-dimensional group

The Hardy space in the case was investigated in [1]. Let us describe the space for all in more detail.

Let be the -dimensional complex Hilbert space with the scalar product and the canonical orthonormal basis where . Consider the Hilbertian sum with the scalar product , where every coordinate is identified with its image under the natural embedding . Then the system with , forms the canonical orthonormal basis in and the canonical orthonormal basis of has the form

Let be the group of unitary -matrices with the unit . We equip every group with the probability Haar measure . The right action of the cartesian product over the group , we define as follows for all .

We write every matrix with in the block matrix form where , corresponding to the matrix partition In [6, Proposition 0.1], [2, Lemma 3.1] it was proven that the Livšic-type mapping (which is not a group homomorphism) is Borel and surjective onto which commutes with the right action of .

As is known [6, Theorem 1.6], the pullback of the probability Haar measure on under the projections is the probability Haar measure on , that is, Consider the projective limits taken with respect to the surjective Borel projections . The canonical projection such that are surjective by surjectivity of . The virtual unitary group acts isometrically over the Hilbert space by coordinate-wise way, with , for all elements and .

Following [2, 6] with the help of the Kolmogorov consistent theorem, we uniquely define a probability measure on , as the projective limit under mapping ((59)), which satisfies the equality

The right action of the infinite-dimensional unitary matrix groups over the spaces of virtual unitary matrices is defined (see [2, Definition 4.5]) as where is so large that . The measure is invariant under the right actions ((65)). Moreover, is a Radon measure (see [1, Lemma 1]).

A complex function on is called cylindrical [2, Definition 4.5] if it has the form for a certain and complex function on . Any such function is -essentially bounded.

Let denote the closure of the algebraic hull of all cylindrical -essentially bounded functions ((72)) with respect to the norm

Let be the space of -integrable complex functions on the space of virtual matrices with the finite quasinorm which is a quasinormed complete additive group (see [3, Lemma ]). Since is a probability measure, the contractive embedding holds and

As is well known [3], the constant in the triangle inequality may be chosen equal to for the case and for the case .

Let us denote Consider the system of all finite sequences and let and denote, respectively, the largest index and the set of all indexes in the sequence so that the integer vector is nonzero. Take the corresponding system of -integrable cylindrical functions of the variable , For any equality ((64)) yields Similarly, we have in the case . Let us form the systems of normalized cylindrical functions with fixed and nonfixed . Clearly (see [1]), they belong to the space .

*Definition 6. *The Hardy space on the space of virtual unitary matrices is defined as -closure of the complex linear span of (for see [1, Definition 5]).

It is essential to note that the system forms an orthonormal basis in the Hilbert space in the case [1, Theorem 1].

#### 5. Approximations in Hardy Spaces

Now, we will consider a quasinormed group , as an additive subgroup in the Hardy space endowed with the quasinorm . We will analyze three cases of approximations to a linear and nonlinear setting.(I)For the first case of linear approximation, we use the linear span in the space of all cylindrical functions of not greater than first variables, that is, It corresponds to the approximation of functions with infinite-dimensional variables by functions of fixed finite number variables.(II)For the second case of linear approximation, we use the linear combinations in of all cylindrical functions such that the number is not greater than ; that is, we choose It corresponds to the approximation of functions by polynomials of a fixed finite degree.(III)For nonlinear approximation, we use all not greater than -terms linear combinations in of cylindrical functions from ; that is, we choose where means cardinality of a set. Notice that, in contrast to linear approximation, the set is not linear. A sum of two elements in will in general need terms in its representation by .

In all three considered cases for any constant the embedding holds for all . Therefore, we choose the constant so that the Hardy space had to be a complete quasinormed additive subgroup. Namely, in what follows we put

Moreover, since runs over all finite subsequences in , the additive subgroups are total in .

We endowed the additive subgroups corresponding to each of the cases (I)–(III) of linear and nonlinear approximations, with the quasinorms of form ((2)) with a suitable (to choice of ) constant . In all considered cases for every pair index , or and an index the additive subgroups will be denoted by and endowed with the quasinorm of form ((9)).

Theorem 7. *(i) For every , the additive subgroup is complete under quasinorm ((81)) and the union is total in .**(ii) For every , there are constants , such that
**
for all and
*

*Proof. *Reasoning is based on the previous auxiliary statements. The groups are total in the space for any index by their definitions for all three cases. Therefore, claim (i) follows from Lemma 2.

To prove assertion (ii), we can apply Theorem 5. Namely, inequality ((28)) instantly implies estimation ((83)) for the distance from an element to the additive subgroup , while inequality ((29)) implies estimation ((84)).

#### 6. Applications to Symmetric Fock Spaces

Show one useful application to the theory of quantum systems. For this purpose we use the symmetric Fock spaces and their finite-dimensional subspaces.

Let stand for the complete th tensor power of a Hilbert space , endowed with the scalar product and the norm where , with for all and finite sums . Put . Use the denotation

Replacing by the subspace , we similarly define the tensor product . There is the isometric embedding If then .

If runs over all -elements permutations then the symmetric th tensor power is defined to be a codomain of the symmetrization mapping which is an orthogonal projector. Similarly, the symmetric th tensor power can be defined. Clearly, is a closed subspace in . Moreover, the following isometric embedding holds:

Consider the symmetric Fock space and its closed subspaces of the forms

As is well known (see, e.g., [7]), the system of normalized symmetric tensor elements indexed by the set , forms an orthonormal basis in the closed subspace .

Similarly, the system of symmetric tensor elements indexed by finite sequences , forms an orthonormal basis in the closed subspace .

In [1, Lemma 3 and Theorem 2] it is proved that in the antilinear isometric isomorphisms hold, where means the classic unitary Hardy space of analytic complex functions on the open unit ball (see [8, Theorem ]). It also follows that the system of cylindrical functions forms an orthogonal basis in .

On the other hand, the system of normalized symmetric tensor elements uniquely defines the corresponding system of -integrable normalized cylindrical functions ((72)), because the equalities for any hold. Moreover, their norms are completely determined by indices , namely, Thus (see [1, Theorem 2]), the above-mentioned antilinear isometries ((94)) and the isometric embedding are uniquely defined by the one-to-one correspondence of normalized basic elements

Now, let us use correspondence ((98)) to construct approximating scales in the Fock space , which are isometrically equivalent to similar scales in the Hardy space .(I)Let first be the linear span in the symmetric Fock space of all symmetric tensor elements , generated by all finite-dimensional spaces of not greater than dimensions; that is, we choose (II)For the second case of linear approximation, we will choose (III)For nonlinear approximation, we will choose

In all cases (I)–(III) we endowed the corresponding additive subgroups in the symmetric Fock space with the quasinorms

Let us denote by and endow with the quasinorms of the form ((9)) all three appropriate to the cases (I)–(III) additive subgroups for every pair index , or and the fixed index .

Using the isometric equalities ((94)) between the Hardy space and the symmetric Fock space , as well as the one-to-one correspondence ((98)) between their normalized basic elements, we conclude that the corresponding quasinormed subgroups in the spaces and are isometric. So, Theorem 7 can be rewritten in the following form.

Theorem 8. *(i) For every , the additive subgroup is complete under the quasinorm and the union is total in .**(ii) For every , there are constants , such that
**
for all and
*

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was partially supported by the Center for Innovation and Transfer of Natural Sciences and Engineering Knowledge at the University of Rzeszów.