Best Approximations in Hardy Spaces on Infinite-Dimensional Unitary Matrix Groups

and Applied Analysis 3 Corollary 3. If 0 < τ < ∞, 0 < θ < 1, and α = (1−θ)α0+θα1 with α0 ̸ = α1 then


Introduction
Our goal is to investigate a best approximation problem in the quasinormed Hardy space    (0 <  ≤ ∞) 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 E. 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  = 2 is antilinearly unitary isometric to a symmetric Fock space F, generated by E (see Theorem 2 in [1]).Therefore, we can also apply obtained results for    to best approximations in the symmetric Fock space F. 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 G with a given continuous approximation scale.To solve this problem, we use an interpolating scale of special Besov-type subgroups    (G), 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    (G).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.

Besov-Type Approximation Scales
Following [4], we consider a complete quasinormed abelian group (, | ⋅ |  ) under addition "+" with the neutral element 0, where the quasinorm |⋅|  is determined by the following assumptions: (1) |0|  = 0 and ||  > 0 for all nonzero  ∈ , (2) In what follows we additionally suppose that the group  contains a dense subgroup with a continuous approximation scale of subsets G  , possessing the following properties: On the subgroup G we define the quasinorm which satisfies the conditions || = | − | and for all ,  ∈ G with the same constant  ≥ 1.
In fact, if we put then () = (−) via property (i) and for all  ∈ G  and  ∈ G  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 G with the quasinorm |⋅| and consider on the whole group  the so-called approximation -functional (see, e.g., [3, 7.1]) with  ∈ .Given the pairs {0 <  < ∞, 0 <  ≤ ∞} and {0 ≤  < ∞,  = ∞} we assign in  the Besov-type approximation abelian subgroups endowed with the quasinorm which is determined by the given subgroup G. endowed with the quasinorm (see [3]) with the parameters holds.As a consequence, the subgroup    (G) is complete and densely embedded in .
On the other hand, inequalities (19) and (20) are a consequence of (18) and well known property of the real interpolation [3, Theorem 3.11.2].

Best Approximations
Let the subgroup G 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 G  ⊂ G.
We denote the distance between  ∈  and a subset G  with a fixed  > 0 by This distance characterizes the error of the best approximation of  by elements of G  .

Hardy Spaces of Infinitely Many Variables
We will investigate the Hardy space    (0 <  ≤ ∞) 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 U, 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 U of the infinite-dimensional group The Hardy space    in the case  = 2 was investigated in [1].Let us describe the space    for all  ∈ (0, ∞] in more detail. Let C  ( ∈ N) be the -dimensional complex Hilbert space with the scalar product ⟨⋅ | ⋅⟩ C  and the canonical orthonormal basis where e  = ( ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 0, . . ., 0, 1  , 0, . . ., 0) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟  .Consider the Hilbertian sum with the scalar product where every coordinate   ∈ C  is identified with its image (0, . . ., 0,   , 0, . ..) ∈ E  under the natural embedding C   E. Then the system with e  ∼ ( + ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 0, . . ., 0, e  , 0, . ..) ∈ E  , ( = 1, . . ., ) forms the canonical orthonormal basis in E  and the canonical orthonormal basis of E has the form Let () be the group of unitary ( × )-matrices with the unit 1  .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 , V,  ∈ ().
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 U acts isometrically over the Hilbert space E by coordinate-wise way, with [  ()] (  ) ∈ E  , for all elements  ∈ U and  = (  ) ∈ E. Following [2,6] with the help of the Kolmogorov consistent theorem, we uniquely define a probability measure  on U, as the projective limit under mapping (59), which satisfies the equality The right action of the infinite-dimensional unitary matrix groups  2 (∞) over the spaces of virtual unitary matrices U is defined (see [2, Definition 4.5]) as where  is so large that  = (V, ) ∈  with fixed and nonfixed .Clearly (see [1]), they belong to the space  ∞  .
Definition 6.The Hardy space    (0 <  ≤ ∞) on the space of virtual unitary matrices U is defined as    -closure of the complex linear span of E * (for  = 2 see [1, Definition 5]).
It is essential to note that the system E * forms an orthonormal basis in the Hilbert space    in the case  = 2 [1, Theorem 1].

Approximations in Hardy Spaces
Now, we will consider a quasinormed group , as an additive subgroup in the Hardy space    (0 <  ≤ ∞) 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 {E *  :  ≤ } 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 E * 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 E * ; that is, we choose where # means cardinality of a set.Notice that, in contrast to linear approximation, the set G  is not linear.A sum of two elements in G  will in general need 2 terms in its representation by E * .
In all three considered cases for any constant  ≥ 1 the embedding G  ± G  ⊂ G (+) holds for all ,  ≥ 0. 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 ⨉ ∞ =1 Z  + , the additive subgroups ⋃ {G  :  > 0} are total in    .
We endowed the additive subgroups In all considered cases for every pair index {0 <  < ∞, 0 <  ≤ ∞} or {0 ≤  < ∞,  = ∞} and an index  ∈ (0, ∞] the additive subgroups Proof.Reasoning is based on the previous auxiliary statements.The groups G are total in the space    for any index  ∈ (0, ∞] by their definitions for all three cases.Therefore, claim (i) follows from Lemma 2.

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 ⊗  h E stand for the complete th tensor power of a Hilbert space E, endowed with the scalar product and the norm where Replacing E by the subspace E  , we similarly define the tensor product ⊗  h E  .There is the isometric embedding If  = 1 then ⊗  h C = C.If  : {1, . . ., }  → {(1), . . ., ()} runs over all elements permutations S() then the symmetric th tensor power ⊙  h E  is defined to be a codomain of the symmetrization mapping which is an orthogonal projector.Similarly, the symmetric th tensor power ⊙  h E can be defined.Clearly, ⊙  h E  is a closed subspace in ⊙  h E.Moreover, the following isometric embedding holds: Consider the symmetric Fock space F and its closed subspaces F  of the forms As is well known (see, e.g., [7]), the system of normalized symmetric tensor elements   .On the other hand, the system of normalized symmetric tensor elements uniquely defines the corresponding system E *  of integrable normalized cylindrical functions (72), because the equalities for every pair index {0 <  < ∞, 0 <  ≤ ∞} or {0 ≤  < ∞,  = ∞} and the fixed index  = 2.
Using the isometric equalities (94) between the Hardy space  2  and the symmetric Fock space F, as well as the oneto-one correspondence (98) between their normalized basic elements, we conclude that the corresponding quasinormed subgroups   ,2 (G) in the spaces  2  and F are isometric.So, Theorem 7 can be rewritten in the following form.
corresponding to each of the cases (I)-(III) of linear and nonlinear approximations, with the quasinorms          =            + inf { > 0 :  ∈ G  } (81) of form (2) with a suitable (to choice of ) constant  ≥ 1.