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.

Our goal is to investigate a best approximation problem in the quasinormed Hardy space

Notice that the infinite-dimensional unitary group

The investigated Hardy space

Now we talk briefly about the content. In the introductory Sections

In Theorem

The main result is in Theorem

It should be noted that we consider the cases of linear and nonlinear approximations in the Hardy spaces

Moreover, in Theorem

Following [

In what follows we additionally suppose that the group

(i)

(ii)

(iii)

On the subgroup

In fact, if we put

So, the following contracting dense embedding holds:

Let us endow the dense subgroup

One calls the scale

Notice that if

For any pairs index

Let

The first claim immediately implies from the known [

If

Applying the reiteration property of real interpolation [

On the other hand, inequalities (

If

Applying Corollary

Let the subgroup

We denote the distance between

To investigate this problem, we will use the scale of subgroups

For every

By Lemma

Now we will prove the second inequality. By [

By [

On the other hand, if

We will investigate the Hardy space

The measure

The Hardy space

Let

Let

We write every matrix

As is known [

Following [

The right action of the infinite-dimensional unitary matrix groups

A complex function on

Let

Let

As is well known [

Let us denote

The Hardy space

It is essential to note that the system

Now, we will consider a quasinormed group

For the first case of linear approximation, we use the linear span in the space

For the second case of linear approximation, we use the linear combinations in

For nonlinear approximation, we use all not greater than

In all three considered cases for any constant

Moreover, since

We endowed the additive subgroups

(i) For every

(ii) For every

Reasoning is based on the previous auxiliary statements. The groups

To prove assertion (ii), we can apply Theorem

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

Replacing

If

Consider the symmetric Fock space

As is well known (see, e.g., [

Similarly, the system of symmetric tensor elements

In [

On the other hand, the system of normalized symmetric tensor elements

Now, let us use correspondence (

Let first

For the second case of linear approximation, we will choose

For nonlinear approximation, we will choose

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

Let us denote by

Using the isometric equalities (

(i) For every

(ii) For every

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

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