A Note on Wavelet Subspaces 1 Introduction - Semantic Scholar

A Note on Wavelet Subspaces Ondrej HUTN´ IK1

Abstract. The wavelet subspaces of the space of square integrable functions on the affine group with respect to the left invariant Haar measure dν are studied using the techniques from [9] with respect to wavelets whose Fourier transforms are related to Laguerre polynomials. The orthogonal projections onto each of these wavelet subspaces are described and explicit forms of reproducing kernels are established. Isomorphisms between wavelet subspaces are given.

1

Introduction

It is well known that the one-dimensional wavelet analysis is an intermediate between the function theory on the upper half-plane of one complex variable and the harmonic analysis on the real line, cf. [8]. On the other hand it is also a perfect illustration of a deep analogy between quantum mechanics and signal processing, cf. [1]. For instance, the relation between weighted Bergman spaces on the upper half-plane and the space of wavelet transforms of Hardy space functions with respect to a specific Bergman wavelet is well known, cf. [4]. The key tool in this connection is the (Bergman) transform giving an isometrical isomorphism of the space L2 (G, dν) (the space of all square-integrable functions on the affine group G with respect to the left invariant Haar measure dν) under which the space of wavelet transforms is mapped onto the weighted Bergman space. N. L. Vasilevski developed in [9], and recently summarized in his book [10], an interesting tool of constructing of isometrical isomorphisms between function spaces to study their structure (e.g. Bergman-type spaces, Fock-type spaces, etc.) and Toeplitz operators acting on them. These techniques can be adapted in order to study certain wavelet subspaces of L2 (G, dν). An outline of the method for wavelets has been announced in [5], here we apply it to the fundamental case where the wavelet subspaces arise from the functions whose Fourier transforms are related to Laguerre plynomials. In particular, we study the wavelet subspaces A(k) , resp. A¯(k) , k ∈ N ∪ {0} (with respect to wavelets ψ (k) , resp. ψ¯(k) which will be specified later) from the operator point of view mentioned above and we find a unitary operator U : L2 (G, dν) → L2 (R, dx) ⊗ L2 (R+ , dy) such that for all k ∈ N ∪ {0} U : A(k) → L2 (R+ ) ⊗ Lk , U : A¯(k) → L2 (R− ) ⊗ Lk ,

where Lk is the one-dimensional subspace of L2 (R+ , dy) generated by function ℓk (y) = e−y/2 Lk (y), and Lk (y) is the Laguerre polynomial of degree k. Clearly, 1 Mathematics Subject Classification (2000): Primary 42C40 Key words and phrases: admissible wavelet, wavelet transform, Calder´ on reproducing formula, wavelet subspace, isometry Acknowledgement. This paper was supported by Grant VEGA 2/0097/08.

1

the transform U maps the wavelet subspaces to L2 -type spaces which are much more easier than results obtained in general case, cf. [5]. Moreover, these results reveal that poly-Bergman spaces, cf. [9], and wavelet subspaces share intriguing patterns that may prove usable. Next we investigate the orthogonal projections of L2 (G, dν) onto wavelet subspaces A(k) , resp. A¯(k) , and we give the explicit forms of reproducing kernels in A(k) , resp. A¯(k) , using the isometrical isomorphism U and its construction. Similar results of that kind were a starting point in [6] to study Toeplitz-Hankel type operators. In the last part of this paper we describe isomorphisms from the Hardy space to the considered wavelet subspaces.

2

Wavelets and wavelet subspaces

Here we use the obvious notations: R+ (R− ) is the positive (negative halfline), χ+ (χ− ) is the characteristic function of R+ (R− ). We also denote N0 = N ∪ {0}. It is well known that the one-dimensional wavelet analysis can be explained in terms of square-integrable representation of the affine group, cf. [4]. Let G = {ζ = (u, v); u ∈ R, v > 0} be the “ax + b”-group with the group law (u, v)(u′ , v ′ ) = (vu′ + u, vv ′ ) and the left invariant Haar measure dν(ζ) = v −2 du dv. Let L2 (G, dν) be the space of all functions on G for which Z kf k2G = hf, f iG = |f (ζ)|2 dν(ζ) < ∞. G

Here, h·, ·i always means the inner product on L2 (R), whereas h·, ·iG denotes the inner product on L2 (G, dν) (and its corresponding norm k · kG ). Consider the representation ρ of G on L2 (R) defined by   1 x−u (ρζ f )(x) = fζ (x) = √ f , f ∈ L2 (R). v v Thus ρ is reducible on L2 (R), but irreducible on the Hardy space H2 (R). In what follows we call the function ψ to be an admissible wavelet if it satisfies the admissibility condition Z 2 dξ ˆ |ψ(xξ)| = 1, ξ R+ for almost every x ∈ R, where ψˆ stands for the Fourier transform F : L2 (R) → L2 (R) given by Z F {g}(ξ) = gˆ(ξ) = g(x)e−2πixξ dx. R

The Laguerre polynomials Ln (x) of degree n, n = 0, 1, . . ., and type 0 are given by n  
X ey dn n (−y)k −y n Ln (y) = L(0) (y) = e y = , y ∈ R+ . n n n! dy k k! k=0

Recall that the system of functions ℓn (y) = e−y/2 Ln (y),

y ∈ R+ , n = 0, 1, . . . , 2

forms an orthonormal basis in the space L2 (R+ , dy), i.e. Z ℓm (y)ℓn (y) dy = δmn , m, n = 0, 1, . . . . R+

For k ∈ N0 consider the functions ψ (k) , ψ¯(k) on R which Fourier transforms are given by p ψˆ(k) (ξ) = χ+ (ξ) 2ξe−ξ Lk (2ξ),

ˆ¯(k) (ξ) = ψˆ(k) (−ξ). The inverse Fourier transform computation yields and ψ ψ (k) (x) =

r   k 3 2 √ X Γ( 23 + j) π 2 F1 ( 2 , −k; 1; 1−2πix ) k = , 2 (−2)j 3 2 j (1 − 2πix) 32 +j (1 − 2πix) 2 j=0

where Γ(z) is the Euler gamma function and 2 F1 (a, b; c; z) is the Gauss hypergeometric function, cf. [2]. Also, put ψ¯(k) (x) = ψ (k) (x). According to the Calder´ on reproducing formula, cf. [3], Z Z dv dv (k) (k) f (u) = (ψv ∗ ψv ∗ f )(u) 2 , and h(u) = (ψ¯v(k) ∗ ψ¯v(k) ∗ h)(u) 2 , v v R+ R+ 2 2 2 2 for all f ∈ H+ (R) and h ∈ H− (R), where H+ (R), resp. H− (R) are the Hardy spaces, i.e. 2 H+ (R) = {f ∈ L2 (R); supp fˆ ⊂ [0, ∞)}; H 2 (R) = {f ∈ L2 (R); supp fˆ ⊂ (−∞, 0]}, −

respectively. Here ∗ denotes the usual convolution on L2 (R) and 1 u ψv (u) = √ ψ , v v

(u, v) ∈ G,

2 2 is a dilation of ψ. It is well known that H+ (R) and H− (R) are the only proper invariant subspaces under ρ and the direct sum of both Hardy spaces coincides with the whole space 2 2 L2 (R) = H+ (R) ⊕ H− (R).

Define the subspaces A(k) and A¯(k) of L2 (G, dν) by

2 (R)}; A(k) = {(f ∗ ψv(k) )(u); f ∈ H+ (k) (k) 2 ¯ ¯ A = {(f ∗ ψv )(u); f ∈ H− (R)}.

Note that the spaces A(k) , resp. A¯(k) are, in fact, the spaces of Calder´ on (or 2 2 wavelet) transforms of functions f ∈ H+ (R), resp. f ∈ H− (R), with respect to wavelets ψ (k) , resp. ψ¯(k) . Indeed, Z Z (k) (k) (k) (k) e f (x)ψζ (x) dx = (f ∗ ψv )(u) = f (x)ψζ (x) dx = hf, ψζ i, R

R

e where ψ(x) = ψ(−x) and the fact ψe(k) = ψ (k) is used. Analogously for the (k) ¯ space A . 3

In order to construct the isometrical isomorphism U from Introduction we consider the unitary operator U1 = (F ⊗ I) : L2 (G, dν(ζ)) = L2 (R, du) ⊗ L2 (R+ , v −2 dv) → L2 (R, du) ⊗ L2 (R+ , v −2 dv) with ζ = (u, v) ∈ G, and the unitary operator U2 : L2 (R, du) ⊗ L2 (R+ , v −2 dv) → L2 (R, dx) ⊗ L2 (R+ , dy) given by the rule p   2|x| y F x, . U2 : F (u, v) 7→ y 2|x| Now we may state the following theorem describing the structure of the wavelet subspaces A(k) and A¯(k) inside L2 (G, dν). Theorem 2.1 The unitary operator U = U2 U1 gives an isometrical isomorphism of the space L2 (G, dν) = L2 (R, du) ⊗ L2 (R+ , v −2 dv) onto L2 (R, dx) ⊗ L2 (R+ , dy) under which (i) the space A(k) is mapped onto L2 (R+ )⊗Lk , where Lk is the one-dimensional subspace of L2 (R+ , dy) generated by function ℓk (y) = e−y/2 Lk (y); (ii) the space A¯(k) is mapped onto L2 (R− ) ⊗ Lk with

U : A¯(k) 7→ L2 (R− ) ⊗ Lk .

(k)

Proof. Denote by A1 the image of the space A(k) in the mapping U1 , (k) analogously for the space A¯(k) . Obviously, the space A1 consists of all functions √ √ F (u, v) = v fˆ(u)ψˆ(k) (uv) = χ+ (u) 2uv fˆ(u)e−uv Lk (2uv), and, moreover, kF (u, v)kA(k) = kf (u)kL2(R,du) . 1

Obviously, the inverse operator U2−1 = U2∗ : L2 (R, dx) ⊗ L2 (R+ , dy) → L2 (R, du) ⊗ L2 (R+ , v −2 dv) has the form p U2−1 : F (x, y) 7→ 2|u|vF (u, 2|u|v). 2 Then for each f ∈ H+ (R) one has √ U2 : χ+ (u) 2u v fˆ(u)e−uv Lk (2uv) 7→ χ+ (x)fˆ(x)e−y/2 Lk (y).   (k) (k) Thus, the image A2 = U2 A1 is the set of all functions of the form

F (x, y) = χ+ (x)fˆ(x)e−y/2 Lk (y),

2 f ∈ H+ (R).

Analogously for the space A¯(k) . Since the Fourier transform F gives an iso2 metrical isomorphism of the space L2 (R) under which the Hardy space H+ (R), 2 resp. H− (R), is mapped onto L2 (R+ ), resp. L2 (R− ), the proof of theorem is complete. 2 Remark 2.2 The result of Theorem 2.1 may be viewed as a “wavelet version” of the result of Vasilevski [9] obtained for the Bergman and poly-Bergman spaces. It also mentions the intriguing patterns which wavelet subspaces and poly-Bergman spaces share. Remark 2.3 In the case of a system of admissible wavelets forming all together an orthonormal basis of L2 (R) one gets a (complete) direct sum decomposition of the space L2 (G, dν) as it is obtained in [6], Theorem 1. 4

3

Projections onto wavelet subspaces

Now we are interested in reproducing kernels of A(k) and A¯(k) and orthogonal projections onto them. Recall that the reproducing kernels play a central role in the theory of wavelets. For instance, the reproducing kernel may be used for interpolation, for discretization of the reconstruction formula, or for calibration of wavelets, cf. [1]. (k) Let ζ = (u, v) ∈ G, η = (s, t) ∈ G and denote by Kζ (η) = K (k) (η, ζ) the reproducing kernel of A(k) . Then (k)

(k)

Kη(k) (ζ) = (ψv(k) ∗ ψt )(u − s) = hψζ , ψη(k) i. (k)

(k)

Clearly, Kζ (η) = Kη (ζ). If F ∈ A(k) , then F (ζ) = (f ∗ =

Z

G

ψv(k) )(u)

=

ψv(k)

∗

Z

(k)

(ψt

R+

(k)

(k)

∗ ψt

(k)

(ψv(k) ∗ ψt )(u − s)(f ∗ ψt )(s) (k)

∗ f )(u)

dsdt = t2

(k)

Z

dt t2

G

F (η)Kη(k) (ζ) dν(η),

(k)

i.e. F (ζ) = hF, Kζ iG . Obviously, Kζ (η) = hKζ , Kηk iG . Similarly, we have ¯ (k) (η) of A¯(k) as the reproducing kernel K ζ ¯ (k) (η) = K (k) (η) = K (k) (ζ). K η ζ ζ In [6] similar results for projections and reproducing kernels are computed explicitly (without further details of computation). Here we apply the reverse procedure leading to Theorem 2.1 which allows us to give exact formulas for the kernels and projections on each wavelet subspaces without direct computations and using the construction of isometrical isomorphism U . Here, B(x, y) = Γ(x)Γ(y) Γ(x+y) is the usual Euler Beta function. Theorem 3.1 Let ζ = (u, v) ∈ G and η = (s, t) ∈ G. Then the orthogonal projection P (k) of L2 (G, dν) onto the space A(k) is given by Z (P (k) F )(ζ) = F (η)Kη(k) (ζ) dν(η), G

where k X k X



2(ζ − ζ) (2π − 1)(η − ζ) + (2π + 1)(η − ζ) j=0 l=0  j+1 2(η − η) × , (2π − 1)(η − ζ) + (2π + 1)(η − ζ)

Kη(k) (ζ) =

(k)

κj,l

and (k) κj,l

=

(−1)j+k

k j


k
l

2 B(j + 1, l + 1)

5

.

l+1

(k)

Proof. The orthogonal projection U P (k) U −1 = B2 Lk is obviously given by (k)

B2

: L2 (G, dν) → L2 (R+ ) ⊗

= χ+ (x)I ⊗ Q(k) ,

where (Q(k) H)(y) = hH, ℓk iℓk = ℓk (y)

Z

H(θ)ℓk (θ) dθ

R+

is the orthogonal projection of L2 (R+ ) onto the one-dimensional space Lk generated by the function ℓk (y). Indeed, Z (k) (B2 F )(x, y) = χ+ (x)ℓk (y) F (x, θ)ℓk (θ) dθ. R+

(k)

(k)

= U2−1 B2 U2 , we get Z dθ (k) −uv (B1 F )(u, v) = χ+ (u) 2uv e Lk (2uv) F (u, θ)e−uθ Lk (2uθ) . θ R+

Calculating the projection B1

Now, (k)

(F −1 ⊗ I)B1 F Z Z −vξ = χ+ (ξ) 2vξe Lk (2vξ) R

=

Z

tF (ξ, t)e

−tξ

R+

dt Lk (2tξ) 2 t

F (ξ, t)χ+ (ξ) 2tvξLk (2tξ)Lk (2vξ)e−ξ(t+v−2πiuξ)

G

(k)

Since B2

!

e2πiuξ dξ

dξdt . t2

= U2 (F ⊗ I)P (k) (F −1 ⊗ I)U2−1 , then (k)

B1

(k)

= U2−1 B2 U2 = (F ⊗ I)P (k) (F −1 ⊗ I),

and therefore (k)

(k)

(k)

(F −1 ⊗ I)B1 F = P (k) (F −1 ⊗ I)F = h(F −1 ⊗ I)F, Kζ iG = hF, (F ⊗ I)Kζ iG . Thus, (k)

(F ⊗ I)Kζ (η) = χ+ (ξ) 2tvξLk (2tξ)Lk (2vξ)e−ξ(t+v+2πiuξ) . Since Lk (y) =

k   X k (−y)j j=0

j

where (k)

λj

=

j!

=

k X

(k)

λj y j ,

j=0

  (−1)j k , j! j

then we get (k)

(F ⊗ I)Kζ (η) =

k X k X j=0 l=0

  (k) (k) λj λl 2j+l+1 tl+1 v j+1 χ+ (ξ)ξ j+l+1 e−ξ(v+t+2πiu) . 6

Since χ+ (ξ)ξ m e−ξ(v+t+2πiu) is the Fourier transform (with respect to s = ℜη) of the function Γ(1 + m) , (t + v − 2πi(s − u))1+m then (k)

Kζ (η) =

k X k X

(k) (k)

λj λl 2j+l+1 tl+1 v j+1

j=0 l=0

Γ(2 + j + l) . (t + v − 2πi(s − u))2+j+l

Introducing (k)

    1 (k) (k) (−1)j k (−1)l k Γ(2 + j + l) λj λl Γ(2 + j + l) = 2 Γ(j + 1) j Γ(l + 1) l 2

j+k k k (−1) j l , = 2 B(j + 1, l + 1)

κj,l =

we finally get the formula for the kernel (k) Kζ (η)

=

k X k X

(k) κj,l

j=0 l=0



2t t + v − 2πi(s − u)

l+1 

2v t + v − 2πi(s − u)

j+1

.

(k)

(k)

Using the fact that Kη (ζ) = Kζ (η) and 2v 2(ζ − ζ) = , t + v + 2πi(s − u) (2π − 1)(η − ζ) + (2π + 1)(η − ζ) 2(η − η) 2t , = t + v + 2πi(s − u) (2π − 1)(η − ζ) + (2π + 1)(η − ζ) we complete the proof.

2

Similar results hold for the space A¯(k) .

4

Isomorphisms between wavelet subspaces Now we describe the isomorphisms between the wavelet subspaces A(k) and 2 resp. A¯(k) and H− (R). Given k ∈ N0 , introduce the operator

2 H+ (R),

(k)

(k)

2 R1 = R0 U1 : H+ (R) → L2 (R) ⊗ L2 (R+ ), (k)

where R0

is given by the rule (k)

(R0 f )(x, y) = χ+ (x)f (x)ℓk (y). (k)

(k)

Obviously, the image of R1 coincides with the space A2 . The adjoint operator is (k) ∗ (k) ∗ 2 R1 = U1∗ R0 : L2 (R) ⊗ L2 (R+ ) → H+ (R), 7

where

and clearly



(k) ∗

R0

Z  F (x) = χ+ (x)

(k) ∗

R1

(k)

F (x, τ )ℓk (τ ) dτ, R+

(k)

2 2 R1 = I : H+ (R) → H+ (R),

(k) ∗

R1 R1

(k)

= B2 : L2 (G, dν) → A2 .

Summarizing the above construction we have (k) ∗

Theorem 4.1 The operator R(k) = R1 2 H+ (R), and the restriction

U maps the space L2 (G, dν) onto

2 R(k) |A(k) : A(k) → H+ (R)

is an isometrical isomorphism. The adjoint operator ∗

(k)

R(k) = U ∗ R1

2 : H+ (R) → A(k) ⊂ L2 (G, dν)

2 is an isometrical isomorphism of the space H+ (R) onto A(k) . ∗

Remark 4.2 Operators R(k) and R(k) provide the following decomposi2 tions of the projection P (k) and of the identity operator on H+ (R) ∗

R(k) R(k) = I

2 2 : H+ (R) → H+ (R),

∗

R(k) R(k) = P (k) : L2 (G, dν) → A(k) . Theorem 4.3 The isometrical isomorphism ∗

(k)

R(k) = U ∗ R1

2 : H+ (R) → A(k) ⊂ L2 (G, dν)

is given by   √ Z (k) ∗ R f (x, y) = 2y Proof.

R+

p ξ fˆ(ξ)Lk (2ξy)eiξ(2πx+iy) dξ.

The direct calculation yields   ∗ (k) (k) R(k) f (x, y) = (U ∗ R1 f )(x, y) = (U1∗ U2∗ R0 U1 f )(x, y)   p = (F −1 ⊗ I) χ+ (ξ) 2ξ y fˆ(ξ)ℓk (2ξy)   p = (F −1 ⊗ I) χ+ (ξ) 2ξ y fˆ(ξ)e−ξy Lk (2ξy) √ Z p = 2y ξ fˆ(ξ)Lk (2ξy)eiξ(2πx+iy) dξ. R+

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2

Corollary 4.4 The inverse isomorphism (k) ∗

R(k) = R1

2 U : A(k) → H+ (R)

is given by p Z Z (RF )(ξ) = χ+ (ξ) 2ξ R

dudv Fˆ (u, v)Lk (2vξ)e−iξ(2πu−iv) . v R+

¯ (k) , R ¯ (k)∗ , R ¯ (k) Similar results hold for the space A¯(k) , where operators R 1 (k) ¯ and R are introduced analogously. 0 Remark 4.5 The above results may be viewed as a special case of constructing of operators which is based on a general scheme presented in [10] and applied in Vasilevski’s work in many different settings (e.g., Bergman-type spaces, Fock-type spaces, etc.). (k)

Let us return to spaces A2 = L2 (R+ ) ⊗ Lk , the whole construction goes (k) analogously also for the spaces A¯2 = L2 (R− ) ⊗ Lk . It is convenient to change ∞ e the previously used basis {ℓk (y)}∞ k=0 of L2 (R+ , dy) to the new basis {ℓk (y)}k=0 , where ℓek (y) = (−1)k ℓk (y), k = 0, 1, 2, . . . .

We note that the previously defined one-dimensional spaces Lk are generated by the new basis elements ℓek (y) as well, and that the statements of Theorem 2.1 remain valid without any change. In L2 (R+ , dy) introduce the operators, cf. [10], Z y (S + f )(y) = −f (y) + e−y/2 et/2 f (t) dt 0 Z ∞ − y/2 (S f )(y) = −f (y) + e e−t/2 f (t) dt, y

which are bounded on L2 (R+ , dy) and mutually adjoint. In the space L2 (R, dx)⊗ L2 (R+ , dy) put T2+ = I ⊗ S + ,

T2− = I ⊗ S − .

It is known, cf. [10], that operators S + and S − act on functions ℓek (y) (for each admissible k) as follows (S + ℓek )(y) = ℓek+1 (y), (S − ℓek )(y) = ℓek−1 (y), (S − ℓe0 )(y) = 0.

Thus the operator S + is an isometric operator on L2 (R+ , dy) and is nothing but the unilateral forward shift with respect to the basis {ℓek (y)}∞ k=0 . Its adjoint operator S − is the unilateral backward shift with respect to the same basis, e = and its kernel coincides with the one-dimensional space L0 generated by ℓ(y) e−y/2 . Thus the operator (k)

(k+1)

T2+ |A(k) : A2 → A2 2

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is an isometrical isomorphism and the operator (k+1)

T2− |A(k+1) : A2 2

(k)

→ A2

is its inverse. Iterating these isomorphisms we get Lemma 4.6 For given natural numbers k < l the operator (k)

(l)

(T2+ )l−k |A(k) : A2 → A2 2

is an isometrical isomorphism, and the operator (l)

(k)

(T2− )l−k |A(l) : A2 → A2 2

is its inverse. (k)

By Theorem 2.1 we have that A(k) = U −1 (A2 ) for each k ∈ N0 . Introducing operators T + = U −1 T2+ U, T − = U −1 T2− U acting on L2 (G, dν) and applying Lemma 4.6 we immediately get the following theorem. Theorem 4.7 For given natural numbers k < l the isometrical isomorphism between the wavelet subspaces A(k) and A(l) is given by the operator (T + )l−k |A(k) : A(k) → A(l) and the operator (T − )l−k |A(l) : A(l) → A(k) gives the inverse isomorphism. Remark 4.8 The obtained results may serve as a starting point for investigating the Toeplitz- and small and big Hankel-type operators defined as in [6] by Ta(k,l) = P (k) Ma P (l) , ha(k,l) = P¯ (k) Ma P (l) ,   k X (k,l) (i) Ha = I − P  Ma P (l) , j=0

with anti-analytic symbol a(ζ) on G, where Ma is the operator of pointwise multiplication by a. For a more general approach, see [7].

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References [1] Ali S T, Antoine J-P, Gazeau J-P (2000) Coherent States, Wavelets and Their Generalizations, Graduate Texts in Contemporary Physics, New York: Springer [2] Abramowitz M, Stegun I A (1970) Handbook of Mathematical Functions, New York: Dover [3] Calder´ on A (1964) Intermediate spaces and interpolation, the complex method. Studia Math 24: 113–190 [4] Grossmann A, Morlet J, Paul T (1986) Transforms associated to square integrable group representations II: Examples. Ann Inst Henri Poincar´e 45: 293–309 [5] Hutn´ık O (2008) On the structure of the space of wavelet transforms. C R Acad Sci Paris S´er I Math 346: 649–652 [6] Jiang Q, Peng L (1992) Wavelet transform and Toeplitz-Hankel type operators. Math Scand 70: 247–264 [7] Jiang Q, Peng L (1992) Toeplitz and Hankel type operators on the upper half-plane. Integral Equations Operator Theory 15: 744–767 [8] Paul T (1984) Functions analytic on the half-plane as quantum mechanical states. J Math Phys 25: 3252-3262 [9] Vasilevski N L (1999) On the structure of Bergman and poly-Bergman spaces. Integral Equations Operator Theory 33: 471–488 [10] Vasilevski N L (2008) Commutative Algebras of Toeplitz Operators on the Bergman Space, Series: Operator Theory: Advances and Applications, Vol. 185, Basel: Birkh¨ auser ˇ arik UniOndrej Hutn´ık, Institute of Mathematics, Faculty of Science, Pavol Jozef Saf´ versity in Koˇsice, Current address: Jesenn´ a 5, 041 54 Koˇsice, Slovakia, E-mail address: [email protected]

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