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

Authors: Oleh Lopushansky
Journal: Abstract and Applied Analysis
 (2014)

---

## 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.


---

## Body

## 1. Introduction

Our goal is to investigate a best approximation problem in the quasinormed Hardy spaceH
χ
p
(
0
<
p
≤
∞
) of complex functions of infinitely many variables. The considered Hardy space is defined on the infinite-dimensional unitary matrix groups U
(
∞
), 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 groupU
(
∞
) 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 spaceH
χ
p in the unitary case p
=
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 H
χ
p to best approximations in the symmetric Fock space F.Now we talk briefly about the content. In the introductory Sections2 and 3, we investigate an abstract problem for a complete quasinormed abelian group G, 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 B
τ
α
(
G
), defined by approximation E-functionals. Preliminary information about approximations with the help of E-functionals in the general case of complete quasinormed abelian groups can be found in [3, 7.1] and [4].In Theorem5 we establish a general (one of many possible) form of the Bernstein-Jackson inequalities for the considered Besov-type scale B
τ
α
(
G
).The main result is in Theorem7 that, in some sense, gives a solution of best approximation problem in the Hardy spaces H
χ
p 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 spacesH
χ
p. 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 Theorem8 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 (
G
,
|
·
|
G
) under addition “+” with the neutral element 0, where the quasinorm ·
G is determined by the following assumptions:(1)
0
G
=
0 and a
G
>
0 for all nonzero a
∈
G,
(2)
a
G
=
-
a
G for all a
∈
G,
(3)
a
+
b
G
≤
γ
(
a
G
+
b
G
) for all a
,
b
∈
G and some fixed γ
≥
1.In what follows we additionally suppose that the groupG contains a dense subgroup
(1)
G
=
⋃
G
t
:
t
>
0
with a continuous approximation scale of subsets G
t, possessing the following properties:(i)G
t
=
-
G
t for all t
≥
0, at that G
0
=
{
0
},
(ii)G
s
⊂
G
t for all 0
≤
s
<
t,
(iii)G
t
±
G
s
⊂
G
γ
(
t
+
s
) for all t
,
s
≥
0.On the subgroupG we define the quasinorm
(2)
a
=
a
G
+
inf
⁡
t
>
0
:
a
∈
G
t
which satisfies the conditions |
a
|
=
|
-
a
| and
(3)
a
+
b
≤
γ
a
+
b
for all a
,
b
∈
G with the same constant γ
≥
1.In fact, if we put(4)
r
a
=
inf
⁡
t
>
0
:
a
∈
G
t
then r
(
a
)
=
r
(
-
a
) via property (i) and
(5)
r
a
+
b
=
inf
⁡
τ
>
0
:
a
+
b
∈
G
τ
=
γ
inf
⁡
t
+
s
>
0
:
a
+
b
∈
G
γ
t
+
s
≤
γ
inf
⁡
t
+
s
>
0
:
a
∈
G
t
,
b
∈
G
s
≤
γ
max
⁡
r
a
,
r
b
≤
γ
r
a
+
r
b
for all a
∈
G
t and b
∈
G
s via property (iii). As a result, · is a quasinorm with the constant γ, because the quasinorm ·
G is the same.So, the following contracting dense embedding holds:(6)
G
↬
G
.Let us endow the dense subgroupG with the quasinorm · and consider on the whole group G the so-called approximation E-functional (see, e.g., [3, 7.1])
(7)
E
t
,
a
=
inf
⁡
a
-
a
0
G
:
a
0
∈
G
,
a
0
<
t
with a
∈
G. Given the pairs 0
<
α
<
∞
,
0
<
τ
≤
∞ and 0
≤
α
<
∞
,
τ
=
∞ we assign in G the Besov-type approximation abelian subgroups
(8)
B
τ
α
G
=
a
∈
G
:
a
B
τ
α
<
∞
endowed with the quasinorm
(9)
a
B
τ
α
=
∫
0
∞
t
α
E
t
,
a
τ
d
t
t
1
/
τ
,
τ
<
∞
,
sup
⁡
t
>
0
⁡
t
α
E
t
,
a
,
τ
=
∞
,
which is determined by the given subgroup G. Notice that the function ·
B
τ
α is a quasinorm on B
τ
α
(
G
) via [3, Lemma 7.1
.
6].Definition 1.
One calls the scaleB
τ
α
(
G
) Besov-type.Notice that ifG
=
R then B
τ
α
(
G
) exactly coincides with the scale of classic Besov spaces.For any pairs index{
0
<
ϑ
<
1,1
≤
p
≤
∞
} or {
0
<
ϑ
≤
1
,
p
=
∞
}, using the Peetre K-functional for the pair groups G and G,
(10)
K
t
,
a
;
G
,
G
=
inf
⁡
a
=
a
0
+
a
1
a
0
+
t
a
1
G
,
t
>
0
with a
0
∈
G and a
1
∈
G, we can define the interpolation abelian subgroup
(11)
G
,
G
ϑ
,
p
=
a
∈
G
:
a
G
,
G
ϑ
,
p
<
∞
endowed with the quasinorm (see [3])
(12)
a
G
,
G
ϑ
,
p
=
∫
0
∞
t
-
ϑ
K
t
,
a
;
G
,
G
p
d
t
t
1
/
p
,
p
<
∞
,
sup
⁡
0
<
t
<
∞
⁡
t
-
ϑ
K
t
,
a
;
G
,
G
,
p
=
∞
.Lemma 2.
LetB
τ
α
(
G
)
ϑ be the subgroup B
τ
α
(
G
) endowed with the quasinorm ·
B
τ
α
ϑ. Then the equality (up to a quasinorm equivalence)
(13)
B
τ
α
G
ϑ
=
G
,
G
ϑ
,
p
with the parameters
(14)
ϑ
=
1
α
+
1
,
τ
=
p
ϑ
holds. As a consequence, the subgroup B
τ
α
(
G
) is complete and densely embedded in G.Proof.
The first claim immediately implies from the known [3, Theorem 7.1
.
7]. Now we are going to check the completeness and dense embedding. Let G
+
G be endowed with the quasinorm
(15)
a
G
+
G
=
inf
⁡
a
=
a
0
+
a
1
a
0
+
a
1
G
,
where a
0
∈
G and a
1
∈
G. Since |
a
|
≥
|
a
|
G and G
+
G
=
G, we have
(16)
a
G
+
G
=
a
G
for all a
∈
G. Hence, the space G with the quasinorm ·
G
+
G is complete. Therefore, every series ∑
a
n with a
n
∈
(
G
,
G
)
ϑ
,
p such that ∑
‍
|
a
n
|
(
G
,
G
)
ϑ
,
p
<
∞ is convergent to an element a
∈
G
=
G
+
G. Using the inequality
(17)
∑
a
n
G
,
G
ϑ
,
p
≤
γ
∑
a
n
G
,
G
ϑ
,
p
,
we obtain that a
∈
(
G
,
G
)
ϑ
,
p. Hence, (
G
,
G
)
ϑ
,
p is complete. Isomorphism ((13)) implies that [
B
τ
α
(
G
)
]
ϑ is complete and densely embedded in G. Thus, B
τ
α
(
G
) is complete and densely embedded as well.Corollary 3.
If0
<
τ
<
∞, 0
<
ϑ
<
1, and α
=
(
1
-
ϑ
)
α
0
+
ϑ
α
1 with α
0
≠
α
1 then
(18)
B
τ
0
α
0
G
,
B
τ
1
α
1
G
ϑ
,
τ
=
B
τ
α
G
and there exist constants c
1, c
2 such that
(19)
a
B
τ
α
≤
c
1
a
B
τ
0
α
0
1
-
ϑ
a
B
τ
1
α
1
ϑ
for all a
∈
B
τ
0
α
0
(
G
)
∩
B
τ
1
α
1
(
G
) and
(20)
K
t
,
a
;
B
τ
0
α
0
,
B
τ
1
α
1
≤
c
2
t
ϑ
a
B
τ
α
for all a
∈
B
τ
α
(
X
), where K
t
,
a
;
B
τ
0
α
0
,
B
τ
1
α
1 means the Peetre K-functional of the groups B
τ
0
α
0
(
G
) and B
τ
1
α
1
(
G
).Proof.
Applying the reiteration property of real interpolation [3, Theorem 3.11.5] for the indexes
(21)
ϑ
=
1
α
+
1
,
τ
=
p
ϑ
,
0
<
η
<
1
,
ϑ
=
1
-
η
ϑ
0
+
η
ϑ
1
,
ϑ
i
=
1
α
i
+
1
(
i
=
0,1
) we obtain the following equality (clearly, up to a quasinorm equivalence):
(22)
B
τ
0
α
0
ϑ
0
,
B
τ
1
α
1
ϑ
1
η
,
p
=
B
τ
α
ϑ
.
In turn, applying the interpolation degree property [3, Theorem 3.11.6], we obtain
(23)
B
τ
0
α
0
ϑ
0
,
B
τ
1
α
1
ϑ
1
η
,
p
=
B
τ
0
α
0
,
B
τ
1
α
1
ϱ
,
τ
ϑ
with ϱ
=
η
ϑ
1
/
ϑ. Now, equalities ((22)) and ((23)) for α
=
(
1
-
ϱ
)
α
0
+
ϱ
α
1 yield ((18)) with ϱ
=
ϑ.
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].Corollary 4.
If0
<
τ
≤
ϱ
<
∞ then the following continuous embedding holds:
(24)
B
τ
α
G
↬
B
ϱ
α
G
.Proof.
Applying Corollary3, we get that for every element x
∈
B
τ
0
α
0
(
G
)
,
B
τ
1
α
1
(
G
)
ϑ
,
τ there exists a constant c
>
0 such that the inequalities
(25)
a
B
τ
0
α
0
,
B
τ
1
α
1
ϑ
,
ϱ
≤
sup
⁡
t
>
0
t
-
ϑ
K
t
,
a
;
·
1
-
τ
/
ϱ
∫
0
∞
t
-
ϑ
K
t
,
a
;
·
τ
d
t
t
1
/
ϱ
≤
c
a
B
τ
0
α
0
,
B
τ
1
α
1
ϑ
,
τ
hold. Hence, the embedding
(26)
B
τ
0
α
0
,
B
τ
1
α
1
ϑ
,
τ
↬
B
τ
0
α
0
,
B
τ
1
α
1
G
ϑ
,
ϱ
is continuous. Finally using ((18)), we obtain ((24)).


## 3. Best Approximations

Let the subgroupG 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 G by elements of a fixed subset G
t
⊂
G.We denote the distance betweena
∈
G and a subset G
t with a fixed t
>
0 by
(27)
δ
t
,
a
=
inf
⁡
a
-
a
0
G
:
a
0
∈
G
t
.
This distance characterizes the error of the best approximation of a by elements of G
t.To investigate this problem, we will use the scale of subgroupsB
τ
α
(
G
) determined for the pairs of indexes {
0
<
α
<
∞
,
0
<
τ
≤
∞
} or {
0
≤
α
<
∞
,
τ
=
∞
}.Theorem 5.
For everyα, τ there are constants c
1 and c
2 so that the following analogue of the Bernstein-Jackson inequalities holds:
(28)
a
B
τ
α
≤
c
1
a
α
a
G
,
a
∈
G
,
(29)
δ
t
,
a
≤
c
2
a
B
τ
α
t
α
,
a
∈
B
τ
α
G
.Proof.
By Lemma2 the space [
B
τ
α
(
G
)
]
ϑ is interpolating between G and G. As a consequence,
(30)
G
⊂
B
τ
α
G
ϑ
=
G
,
G
ϑ
,
p
⊂
G
,
ϑ
=
1
α
+
1
with τ
=
p
ϑ. Hence, applying [3, Theorem 3.11.4
(
b
)] for some constant c
(
ϑ
,
p
) we obtain
(31)
a
G
,
G
ϑ
,
p
≤
c
a
1
-
ϑ
a
G
ϑ
,
a
∈
G
.
This inequality together with isomorphism ((13)) implies that there is a constant c
1
(
α
,
τ
) such that inequality ((28)) is true.
Now we will prove the second inequality. By [3, Theorem 3.11.4
(
a
)] for some constant c
′
(
ϑ
,
p
) we have
(32)
K
t
,
a
;
G
,
G
≤
c
′
t
ϑ
a
G
,
G
ϑ
,
p
,
a
∈
G
,
G
ϑ
,
p
.
Hence, in virtue of isomorphism ((13)) there is a constant c
0
(
α
,
τ
) such that
(33)
K
t
,
a
;
G
,
G
≤
c
0
t
ϑ
a
B
τ
α
ϑ
,
a
∈
B
τ
α
G
.
Following [3, 7.1], we introduce the functional
(34)
K
∞
t
,
a
;
G
,
G
=
inf
⁡
a
=
a
0
+
a
1
max
⁡
a
0
,
t
a
1
G
with a
0
∈
G and a
1
∈
G. The inequality
(35)
K
∞
t
,
a
;
G
,
G
≤
K
t
,
a
;
G
,
G
yields
(36)
K
∞
t
,
a
;
G
,
G
t
ϑ
≤
c
0
a
B
τ
α
ϑ
for all a
∈
B
τ
α
(
G
).
By [3, Lemma 7.1
.
2] for every t
>
0 and a
∈
B
τ
α
(
G
) there exists s
>
0 such that
(37)
K
∞
t
,
a
;
G
,
G
=
s
,
lim
⁡
ɛ
↘
s
E
ɛ
,
a
≔
E
s
+
0
,
a
≤
s
t
.
For any fixed a the function E
(
s
,
a
) is decreasing in the variable s (see [3, Lemma 7.1
.
1]). Therefore,
(38)
E
s
,
a
≤
E
s
1
+
0
,
a
≤
s
1
t
,
a
∈
B
τ
α
G
for every s
1
>
0 such that
(39)
s
1
≤
K
∞
t
,
a
;
G
,
G
=
s
for all a
∈
B
τ
α
(
G
). It follows that
(40)
E
s
,
a
ϑ
≤
s
1
ϑ
t
ϑ
≤
s
ϑ
-
1
s
t
ϑ
,
a
∈
B
τ
α
G
.
As a result, we obtain
(41)
s
1
-
ϑ
E
s
,
a
ϑ
≤
K
∞
t
,
a
;
G
,
G
t
ϑ
,
a
∈
B
τ
α
G
.
Using ((36)), we have
(42)
s
1
-
ϑ
E
s
,
a
ϑ
≤
c
0
a
B
τ
α
ϑ
.
Substituting α
=
(
1
-
ϑ
)
/
ϑ, we get
(43)
s
α
E
s
,
a
≤
c
0
1
/
ϑ
a
B
τ
α
,
a
∈
B
τ
α
G
.
On the other hand, if(44)
a
0
=
r
a
0
+
a
0
G
<
s
,
then r
(
a
0
)
<
s
-
a
0
G, where is denoted
(45)
r
a
0
=
inf
⁡
t
>
0
:
a
0
∈
G
t
.
So, a
0
∈
G
t for all numbers t
>
0 such that
(46)
r
a
0
<
t
<
s
-
a
0
G
.
The embedding G
t
⊂
G
s yields a
0
∈
G
s. Hence, the following inequality
(47)
δ
s
,
a
≤
E
s
,
a
,
a
∈
G
,
s
>
0
holds. Now taking c
2
=
c
0
1
/
ϑ in ((43)) and using ((47)), we finally obtain the required inequality ((29)).


## 4. Hardy Spaces of Infinitely Many Variables

We will investigate the Hardy spaceH
χ
p
(
0
<
p
≤
∞
) of complex functions on the infinite-dimensional group
(48)
U
∞
=
⋃
U
m
:
m
∈
N
integrable with respect to a projective limit χ of probability Haar measures χ
m, determined on the corresponding m-dimensional unitary matrix groups U
(
m
).The measureχ is determined on, the so-called, the space of virtual unitary matrices U, being a projective limit of U
(
m
), 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
(49)
U
2
∞
≔
U
∞
×
U
∞
.The Hardy spaceH
χ
p in the case p
=
2 was investigated in [1]. Let us describe the space H
χ
p for all p
∈
(
0
,
∞
] in more detail.LetC
m
(
m
∈
N
) be the m-dimensional complex Hilbert space with the scalar product ·
∣
·
C
m and the canonical orthonormal basis
(50)
E
C
m
=
e
m
1
,
…
,
e
m
m
,
where e
m
l
=
(
0
,
…
,
0,1
︷
l
,
0
,
…
,
0
)
︸
m. Consider the Hilbertian sum with the scalar product x
∣
y
E
=
∑
m
x
m
∣
y
m
C
m,
(51)
E
≔
⨁
m
∈
N
E
m
=
x
=
x
m
:
x
m
∈
C
m
,
x
E
=
∑
m
∈
N
x
m
C
m
2
1
/
2
<
∞
,
where every coordinate x
m
∈
C
m is identified with its image 0
,
…
,
0
,
x
m
,
0
,
…
∈
E
m under the natural embedding C
m
↬
E. Then the system
(52)
E
E
m
=
e
m
1
,
…
,
e
m
m
with e
m
l
~
(
0
,
…
,
0
,
e
m
l
︷
m
+
l
,
0
,
…
)
∈
E
m, (
l
=
1
,
…
,
m
) forms the canonical orthonormal basis in E
m and the canonical orthonormal basis of E has the form
(53)
E
E
=
⋃
E
E
m
:
m
∈
N
.LetU
(
m
) be the group of unitary (
m
×
m
)-matrices with the unit 1
m. We equip every group U
(
m
) with the probability Haar measure χ
m. The right action of the cartesian product
(54)
U
2
m
≔
U
m
×
U
m
over the group U
(
m
), we define as follows
(55)
u
·
g
=
w
-
1
u
v
,
g
=
v
,
w
∈
U
2
m
for all u
,
v
,
w
∈
U
(
m
).We write every matrixu
m
∈
U
(
m
) with n
>
m in the block matrix form
(56)
u
n
=
u
n
-
m
a
b
t
,
where u
n
-
m
∈
U
(
n
-
m
), corresponding to the matrix partition
(57)
n

⨉

n
=
n
-
m
+
m
⨉
n
-
m
+
m
.
In [6, Proposition 0.1], [2, Lemma 3.1] it was proven that the Livšic-type mapping π
m
n
:
U
(
n
)
→
U
(
n
-
m
)
(58)
π
m
n
:
u
n
=
u
n
-
m
a
b
t
⟼
u
m
=
u
n
-
m
-
a
1
+
t
-
1
b
,
t
≠
-
1
m
u
n
-
m
,
t
=
-
1
m
(which is not a group homomorphism) is Borel and surjective onto U
(
n
-
m
) which commutes with the right action of U
2
(
n
-
m
).As is known [6, Theorem 1.6], the pullback of the probability Haar measure χ
m on U
(
m
) under the projections π
m
n is the probability Haar measure χ
n on U
(
n
), that is,
(59)
χ
m
∘
π
m
n
=
χ
n
,
n
>
m
.
Consider the projective limits
(60)
U
=
lim
⁡
⟵
⁡
U
m
,
taken with respect to the surjective Borel projections π
m
n. The canonical projection
(61)
π
m
:
U
⟶
U
m
such that π
m
=
π
m
n
∘
π
n are surjective by surjectivity of π
m
n. The virtual unitary group U acts isometrically over the Hilbert space E by coordinate-wise way,
(62)
u
x
=
π
m
u
x
m
m
∈
N
∈
E
with π
m
(
u
)
(
x
m
)
∈
E
m, for all elements u
∈
U and x
=
x
m
∈
E.Following [2, 6] with the help of the Kolmogorov consistent theorem, we uniquely define a probability measure χ on U, as the projective limit under mapping ((59)),
(63)
χ
=
lim
⁡
⟵
⁡
χ
m
,
which satisfies the equality
(64)
χ
=
χ
m
∘
π
m
,
m
∈
N
.The right action of the infinite-dimensional unitary matrix groupsU
2
(
∞
) over the spaces of virtual unitary matrices U is defined (see [2, Definition 4.5]) as
(65)
π
m
u
·
g
=
w
-
1
π
m
u
v
,
u
∈
U
,
where m is so large that g
=
(
v
,
w
)
∈
U
2
(
m
). The measure χ is invariant under the right actions ((65)). Moreover, χ is a Radon measure (see [1, Lemma 1]).A complex function onU is called cylindrical [2, Definition 4.5] if it has the form
(66)
f
u
=
f
m
∘
π
m
u
,
u
∈
U
for a certain m
∈
N and complex function f
m on U
(
m
). Any such function f is χ-essentially bounded.LetL
χ
∞ denote the closure of the algebraic hull of all cylindrical χ-essentially bounded functions ((72)) with respect to the norm
(67)
f
L
χ
∞
=
ess
sup
⁡
u
∈
U
f
u
.LetL
χ
p
(
0
<
p
<
∞
) be the space of χ-integrable complex functions f on the space of virtual matrices U with the finite quasinorm f
L
χ
p
=
∫
U
‍
|
f
|
p
d
χ
1
/
p which is a quasinormed complete additive group (see [3, Lemma 3.10.3]). Since χ is a probability measure, the contractive embedding L
χ
∞
↬
L
χ
p holds and
(68)
f
L
χ
p
≤
f
L
χ
∞
,
f
∈
L
χ
∞
.As is well known [3], the constant γ in the triangle inequality
(69)
f
+
g
L
χ
p
≤
γ
f
L
χ
p
+
g
L
χ
p
may be chosen equal to 1 for the case 1
≤
p
≤
∞ and 2
(
1
-
p
)
/
p for the case 0
<
p
<
1.Let us denote(70)
m
≔
m
1
,
…
,
m
m
,
k
m
≔
k
m
1
,
…
,
k
m
m
∈
Z
+
m
,
k
m
≔
k
m
1
+
⋯
+
k
m
m
,
k
m
!
≔
k
m
1
!
·
…
·
k
m
m
!
,
m
≔
11
,
21,22
,
…
,
m
1
,
…
,
m
m
.
Consider the system of all finite sequences
(71)
k
≔
k
1
,
…
,
k
r
,
…
∈
⨉
r
=
1
∞
Z
+
r
and let m and supp
⁡
{
k
} denote, respectively, the largest index r and the set of all indexes r in the sequence {
k
} so that the integer vector k
(
r
)
=
k
r
1
,
…
,
k
r
r is nonzero. Take the corresponding system of χ
m-integrable cylindrical functions of the variable u
∈
U,
(72)
e
m
*
k
≔
e
1
*
k
1
·
…
·
e
m
*
k
m
,
e
r
*
k
r
u
=
∏
i
=
1
r
π
r
∘
u
e
r
1
∣
e
r
i
E
k
r
i
.
For any p
∈
(
0
,
∞
) equality ((64)) yields
(73)
e
m
*
k
L
χ
m
p
p
=
∫
U
m
e
m
*
k
∘
π
m
-
1
p
d
χ
m
=
∫
U
e
m
*
k
p
d
χ
=
e
m
*
k
L
χ
p
p
.
Similarly, we have in the case p
=
∞. Let us form the systems of normalized cylindrical functions
(74)
E
m
*
=
e
m
*
k
L
χ
p
-
1
e
m
*
k
:
k
∈
⨉
r
=
1
m
Z
+
r
,
E
*
=
E
m
*
:
m
∈
N
with fixed and nonfixed m. Clearly (see [1]), they belong to the space L
χ
∞.Definition 6.
The Hardy spaceH
χ
p
(
0
<
p
≤
∞
) on the space of virtual unitary matrices U is defined as L
χ
p-closure of the complex linear span of E
* (for p
=
2 see [1, Definition 5]).It is essential to note that the systemE
* forms an orthonormal basis in the Hilbert space H
χ
p in the case p
=
2 [1, Theorem 1].


## 5. Approximations in Hardy Spaces

Now, we will consider a quasinormed groupG, as an additive subgroup in the Hardy space H
χ
p
(
0
<
p
≤
∞
) endowed with the quasinorm ·
L
χ
p. 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 spaceH
χ
p of all cylindrical functions E
m
*
:
m
≤
t of not greater than t first variables, that is,
(75)
G
t
=
span
⁡
E
m
*
:
m
≤
t
.
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 inH
χ
p of all cylindrical functions E
* such that the number
(76)
k
≔
∑
r
∈
supp
⁡
k
k
r
is not greater than t; that is, we choose
(77)
G
t
=
span
⁡
e
m
*
k
:
k
≤
t
,
m
∈
N
.
It corresponds to the approximation of functions by polynomials of a fixed finite degree.
(III)
For nonlinear approximation, we use all not greater thant-terms linear combinations in H
χ
p of cylindrical functions from E
*; that is, we choose
(78)
G
t
=
⋃
k
span
⁡
e
m
*
k
:
#
supp
⁡
k
≤
t
,
where # means cardinality of a set. Notice that, in contrast to linear approximation, the set G
t is not linear. A sum of two elements in G
t will in general need 2
t terms in its representation by E
*.In all three considered cases for any constantγ
≥
1 the embedding G
t
±
G
s
⊂
G
γ
(
t
+
s
) holds for all t
,
s
≥
0. Therefore, we choose the constant γ so that the Hardy space H
χ
p had to be a complete quasinormed additive subgroup. Namely, in what follows we put
(79)
γ
=
1
,
1
≤
p
≤
∞
,
2
1
-
p
/
p
,
0
<
p
<
1
.Moreover, since{
k
} runs over all finite subsequences in ⨉
r
=
1
∞
Z
+
r, the additive subgroups ⋃
‍
G
t
:
t
>
0 are total in H
χ
p.We endowed the additive subgroups(80)
G
=
⋃
G
t
:
t
>
0
⊂
H
χ
p
,
corresponding to each of the cases (I)–(III) of linear and nonlinear approximations, with the quasinorms
(81)
f
=
f
L
χ
p
+
inf
⁡
t
>
0
:
f
∈
G
t
of form ((2)) with a suitable (to choice of p) constant γ
≥
1. In all considered cases for every pair index {
0
<
α
<
∞, 0
<
τ
≤
∞
} or {
0
≤
α
<
∞
,
τ
=
∞
} and an index p
∈
(
0
,
∞
] the additive subgroups
(82)
B
τ
α
G
⊂
H
χ
p
will be denoted by B
τ
,
p
α
(
G
) and endowed with the quasinorm ·
B
τ
,
p
α of form ((9)).Theorem 7.
(i) For everyα, τ the additive subgroup B
τ
,
p
α
(
G
) is complete under quasinorm ((81)) and the union ⋃
α
,
τ
‍
B
τ
,
p
α
(
G
) is total in H
χ
p.
(ii) For everyα, τ there are constants c
1, c
2 such that
(83)
inf
⁡
f
-
f
0
L
χ
p
:
f
0
∈
G
t
≤
c
2
f
B
τ
,
p
α
t
α
for all f
∈
B
τ
,
p
α
(
G
) and
(84)
f
B
τ
,
p
α
≤
c
1
f
α
f
L
χ
p
,
∀
f
∈
G
.Proof.
Reasoning is based on the previous auxiliary statements. The groupsG are total in the space H
χ
p for any index p
∈
(
0
,
∞
] by their definitions for all three cases. Therefore, claim (i) follows from Lemma 2.
To prove assertion (ii), we can apply Theorem5. Namely, inequality ((28)) instantly implies estimation ((83)) for the distance from an element x
∈
B
τ
,
p
α
(
G
) to the additive subgroup G
t
⊂
H
χ
p, 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⊗
h
n
E stand for the complete nth tensor power of a Hilbert space E, endowed with the scalar product
(85)
x
1
⊗
⋯
⊗
x
n
∣
ψ
n
⊗
h
n
E
=
∑
j
x
1
∣
y
1
j
E
⋯
x
n
∣
y
n
j
E
and the norm
(86)
ψ
n
⊗
h
n
E
=
ψ
n
∣
ψ
n
⊗
h
n
E
1
/
2
,
where x
1
⊗
⋯
⊗
x
n, y
1
j
⊗
⋯
⊗
y
n
j
∈
⊗
h
n
E with x
t
j
,
y
t
j
∈
E for all t
=
1
,
…
,
n and finite sums ψ
n
=
∑
j
‍
y
1
j
⊗
⋯
⊗
y
n
j. Put ⊗
h
0
E
=
C. Use the denotation
(87)
x
⊗
n
=
x
⊗
⋯
⊗
x
,
x
∈
E
.ReplacingE by the subspace E
m, we similarly define the tensor product ⊗
h
n
E
m. There is the isometric embedding
(88)
⊗
h
n
E
m
↬
⊗
h
n
E
.
If m
=
1 then ⊗
h
n
C
=
C.Ifσ
:
{
1
,
…
,
n
}
↦
{
σ
(
1
)
,
…
,
σ
(
n
)
} runs over all n-elements permutations S
(
n
) then the symmetric nth tensor power ⊙
h
n
E
m is defined to be a codomain of the symmetrization mapping
(89)
⊗
h
n
E
m
∋
x
1
⊗
⋯
⊗
x
n
⟼
x
1
⊙
⋯
⊙
x
n
,
x
1
⊙
⋯
⊙
x
n
=
1
n
!
∑
σ
∈
S
n
x
σ
1
⊗
⋯
⊗
x
σ
n
,
which is an orthogonal projector. Similarly, the symmetric nth tensor power ⊙
h
n
E can be defined. Clearly, ⊙
h
n
E
m is a closed subspace in ⊙
h
n
E. Moreover, the following isometric embedding holds:
(90)
⊙
h
n
E
m
↬
⊙
h
n
E
.Consider the symmetric Fock spaceF and its closed subspaces F
m of the forms
(91)
F
≔
C
⊕
E
⊕
⊙
h
2
E
⊕
⊙
h
3
E
⊕
⋯
,
F
m
≔
C
⊕
E
m
⊕
⊙
h
2
E
m
⊕
⊙
h
3
E
m
⊕
⋯
.As is well known (see, e.g., [7]), the system of normalized symmetric tensor elements
(92)
E
⊙
h
n
E
m
=
e
m
⊗
k
m
F
m
-
1
e
m
⊗
k
m
:
k
m
=
n
e
m
⊗
k
m
≔
e
m
1
⊗
k
m
1
⊙
⋯
⊙
e
m
m
⊗
k
m
m
,
e
m
⊗
k
m
F
m
2
=
k
m
!
k
m
!
,
indexed by the set k
(
m
)
∈
Z
+
m, forms an orthonormal basis in the closed subspace ⊙
h
n
E
m
⊂
F
m.Similarly, the system of symmetric tensor elements(93)
E
⊙
h
n
E
=
e
m
⊗
k
F
-
1
e
m
⊗
k
:
k
=
n
e
m
⊗
k
≔
e
1
⊗
k
1
⊙
⋯
⊙
e
m
⊗
k
m
,
e
m
⊗
k
F
2
=
∏
r
=
1
m
k
r
!
k
r
!
,
indexed by finite sequences {
k
}
∈
⨉
r
=
1
∞
Z
+
r, forms an orthonormal basis in the closed subspace ⊙
h
n
E
⊂
F.In [1, Lemma 3 and Theorem 2] it is proved that in the antilinear isometric isomorphisms
(94)
F
m
≃
H
χ
m
2
,
F
≃
H
χ
2
hold, where H
χ
m
2 means the classic unitary Hardy space of analytic complex functions on the open unit ball B
m
=
x
m
∈
C
m
:
x
m
C
m
<
1 (see [8, Theorem 5.6
.
8]). It also follows that the system of cylindrical functions E
m
* forms an orthogonal basis in H
χ
m
2.On the other hand, the system of normalized symmetric tensor elements(95)
E
F
m
=
⋃
E
⊙
h
n
E
m
:
n
∈
Z
+
uniquely defines the corresponding system E
m
* of χ-integrable normalized cylindrical functions ((72)), because the equalities
(96)
e
r
*
k
r
u
=
∏
i
=
1
r
π
r
∘
u
e
r
1
∣
e
r
i
E
k
r
i
=
π
m
⊗
k
r
u
e
m
1
⊗
k
r
∣
e
m
⊗
k
r
⊗
h
n
C
m
for any r
=
1
,
…
,
m hold. Moreover, their norms are completely determined by indices {
k
}, namely,
(97)
e
m
*
k
L
χ
2
2
=
∏
r
=
1
m
r
-
1
!
k
r
!
r
-
1
+
k
r
!
.
Thus (see [1, Theorem 2]), the above-mentioned antilinear isometries ((94)) and the isometric embedding F
m
↬
H
χ
2 are uniquely defined by the one-to-one correspondence of normalized basic elements
(98)
e
m
⊗
k
e
m
⊗
k
F
⇄
e
m
*
k
e
m
*
k
L
χ
2
.Now, let us use correspondence ((98)) to construct approximating scales G
t in the Fock space F, which are isometrically equivalent to similar scales in the Hardy space H
χ
2.(I)
Let firstG
t be the linear span in the symmetric Fock space F of all symmetric tensor elements E
F
m
:
m
≤
t, generated by all finite-dimensional spaces E
m of not greater than t dimensions; that is, we choose
(99)
G
t
=
span
⁡
E
F
m
:
m
≤
t
.
(II)
For the second case of linear approximation, we will choose(100)
G
t
=
span
⁡
e
m
⊗
k
:
k
≤
t
.
(III)
For nonlinear approximation, we will choose(101)
G
t
=
⋃
k
span
⁡
e
m
⊗
k
:
#
supp
⁡
k
≤
t
.In all cases (I)–(III) we endowed the corresponding additive subgroups in the symmetric Fock space(102)
G
=
⋃
G
t
:
t
>
0
⊂
F
with the quasinorms
(103)
ψ
=
ψ
F
+
inf
⁡
t
>
0
:
ψ
∈
G
t
.Let us denote byB
τ
,
2
α
(
G
) and endow with the quasinorms ·
B
τ
,
2
α of the form ((9)) all three appropriate to the cases (I)–(III) additive subgroups
(104)
B
τ
α
G
⊂
F
for every pair index {
0
<
α
<
∞, 0
<
τ
≤
∞
} or {
0
≤
α
<
∞
,
τ
=
∞
} and the fixed index p
=
2.Using the isometric equalities ((94)) between the Hardy space H
χ
2 and the symmetric Fock space F, as well as the one-to-one correspondence ((98)) between their normalized basic elements, we conclude that the corresponding quasinormed subgroups B
τ
,
2
α
(
G
) in the spaces H
χ
2 and F are isometric. So, Theorem 7 can be rewritten in the following form.Theorem 8.
(i) For everyα, τ the additive subgroup B
τ
,
2
α
(
G
) is complete under the quasinorm ·
B
τ
,
2
α and the union ⋃
α
,
τ
‍
B
τ
,
2
α
(
G
) is total in F.
(ii) For everyα, τ there are constants c
1, c
2 such that
(105)
inf
⁡
ψ
-
ψ
0
F
:
ψ
0
∈
G
t
≤
c
2
f
B
τ
,
2
α
t
α
for all ψ
∈
B
τ
,
2
α
(
G
) and
(106)
ψ
B
τ
,
2
α
≤
c
1
ψ
α
ψ
F
,
∀
ψ
∈
G
.



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*Source: 631503-2014-08-06.xml*