A Note on Wavelet Subspaces

Ondrej HUTNIK1

Abstract. The wavelet subspaces of the space of square integrablefunctions on the affine group with respect to the left invariant Haarmeasure dν are studied using the techniques from [9] with respect towavelets whose Fourier transforms are related to Laguerre polynomi-als. The orthogonal projections onto each of these wavelet subspacesare 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 intermediatebetween the function theory on the upper half-plane of one complex variable andthe harmonic analysis on the real line, cf. [8]. On the other hand it is also aperfect illustration of a deep analogy between quantum mechanics and signalprocessing, cf. [1]. For instance, the relation between weighted Bergman spaceson the upper half-plane and the space of wavelet transforms of Hardy spacefunctions with respect to a specific Bergman wavelet is well known, cf. [4]. Thekey tool in this connection is the (Bergman) transform giving an isometricalisomorphism of the space L2(G, dν) (the space of all square-integrable functionson the affine group G with respect to the left invariant Haar measure dν) underwhich the space of wavelet transforms is mapped onto the weighted Bergmanspace.

N. L. Vasilevski developed in [9], and recently summarized in his book [10],an interesting tool of constructing of isometrical isomorphisms between functionspaces to study their structure (e.g. Bergman-type spaces, Fock-type spaces,etc.) and Toeplitz operators acting on them. These techniques can be adapted inorder to study certain wavelet subspaces of L2(G, dν). An outline of the methodfor wavelets has been announced in [5], here we apply it to the fundamental casewhere the wavelet subspaces arise from the functions whose Fourier transformsare 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) fromthe operator point of view mentioned above and we find a unitary operatorU : 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 functionℓk(y) = e−y/2Lk(y), and Lk(y) is the Laguerre polynomial of degree k. Clearly,

1Mathematics Subject Classification (2000): Primary 42C40Key words and phrases: admissible wavelet, wavelet transform, Calderon reproducing formula,wavelet subspace, isometryAcknowledgement. This paper was supported by Grant VEGA 2/0097/08.

1

the transform U maps the wavelet subspaces to L2-type spaces which are muchmore easier than results obtained in general case, cf. [5]. Moreover, these resultsreveal that poly-Bergman spaces, cf. [9], and wavelet subspaces share intriguingpatterns that may prove usable.

Next we investigate the orthogonal projections of L2(G, dν) onto waveletsubspaces A(k), resp. A(k), and we give the explicit forms of reproducing kernelsin 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-Hankeltype operators. In the last part of this paper we describe isomorphisms fromthe Hardy space to the considered wavelet subspaces.

2 Wavelets and wavelet subspaces

Here we use the obvious notations: R+ (R−) is the positive (negative half-line), 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 explainedin terms of square-integrable representation of the affine group, cf. [4]. LetG = {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

‖f‖2G = 〈f, f〉G =

∫

G

|f(ζ)|2 dν(ζ) <∞.

Here, 〈·, ·〉 always means the inner product on L2(R), whereas 〈·, ·〉G denotesthe inner product on L2(G, dν) (and its corresponding norm ‖ · ‖G).

Consider the representation ρ of G on L2(R) defined by

(ρζf)(x) = fζ(x) =1√vf

(x− u

v

), f ∈ L2(R).

Thus ρ is reducible on L2(R), but irreducible on the Hardy space H2(R). Inwhat follows we call the function ψ to be an admissible wavelet if it satisfies theadmissibility condition ∫

R+

|ψ(xξ)|2 dξξ

= 1,

for almost every x ∈ R, where ψ stands for the Fourier transform F : L2(R) →L2(R) given by

F{g}(ξ) = g(ξ) =

∫

R

g(x)e−2πixξ dx.

The Laguerre polynomials Ln(x) of degree n, n = 0, 1, . . ., and type 0 aregiven by

Ln(y) = L(0)n (y) =

ey

n!

dn

dyn

(e−yyn

)=

n∑

k=0

(n

k

)(−y)k

k!, y ∈ R+.

Recall that the system of functions

ℓn(y) = e−y/2Ln(y), y ∈ R+, n = 0, 1, . . . ,

2

forms an orthonormal basis in the space L2(R+, dy), i.e.

∫

R+

ℓm(y)ℓn(y) dy = δmn, m, n = 0, 1, . . . .

For k ∈ N0 consider the functions ψ(k), ψ(k) on R which Fourier transformsare given by

ψ(k)(ξ) = χ+(ξ)√

2ξe−ξLk(2ξ),

and ˆψ(k)

(ξ) = ψ(k)(−ξ). The inverse Fourier transform computation yields

ψ(k)(x) =√

2

k∑

j=0

(−2)j

(k

j

)Γ(3

2 + j)

(1 − 2πix)32+j

=

√π

2

2F1(32 ,−k; 1; 2

1−2πix )

(1 − 2πix)32

,

where Γ(z) is the Euler gamma function and 2F1(a, b; c; z) is the Gauss hy-

pergeometric function, cf. [2]. Also, put ψ(k)(x) = ψ(k)(x). According to theCalderon reproducing formula, cf. [3],

f(u) =

∫

R+

(ψ(k)v ∗ ψ(k)

v ∗ f)(u)dv

v2, and h(u) =

∫

R+

(ψ(k)v ∗ ψ(k)

v ∗ h)(u) dvv2,

for all f ∈ H2+(R) and h ∈ H2

−(R), where H2

+(R), resp. H2−

(R) are the Hardyspaces, i.e.

H2+(R) = {f ∈ L2(R); supp f ⊂ [0,∞)};

H2−

(R) = {f ∈ L2(R); supp f ⊂ (−∞, 0]},

respectively. Here ∗ denotes the usual convolution on L2(R) and

ψv(u) =1√vψ(uv

), (u, v) ∈ G,

is a dilation of ψ. It is well known that H2+(R) and H2

−(R) are the only proper

invariant subspaces under ρ and the direct sum of both Hardy spaces coincideswith the whole space

L2(R) = H2+(R) ⊕H2

−(R).

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

A(k) = {(f ∗ ψ(k)v )(u); f ∈ H2

+(R)};A(k) = {(f ∗ ψ(k)

v )(u); f ∈ H2−

(R)}.

Note that the spaces A(k), resp. A(k) are, in fact, the spaces of Calderon (orwavelet) transforms of functions f ∈ H2

+(R), resp. f ∈ H2−

(R), with respect to

wavelets ψ(k), resp. ψ(k). Indeed,

(f ∗ ψ(k)v )(u) =

∫

R

f(x)ψ(k)ζ (x) dx =

∫

R

f(x)ψ(k)ζ (x) dx = 〈f, ψ(k)

ζ 〉,

where ψ(x) = ψ(−x) and the fact ψ(k) = ψ(k) is used. Analogously for thespace A(k).

3

In order to construct the isometrical isomorphism U from Introduction weconsider the unitary operator U1 = (F ⊗ I) : L2(G, dν(ζ)) = L2(R, du) ⊗L2(R+, v

−2dv) → L2(R, du) ⊗ L2(R+, v−2dv) with ζ = (u, v) ∈ G, and the

unitary operator U2 : L2(R, du) ⊗ L2(R+, v−2dv) → L2(R, dx) ⊗ L2(R+, dy)

given by the rule

U2 : F (u, v) 7→√

2|x|y

F

(x,

y

2|x|

).

Now we may state the following theorem describing the structure of the waveletsubspaces A(k) and A(k) inside L2(G, dν).

Theorem 2.1 The unitary operator U = U2U1 gives an isometrical isomor-phism of the space L2(G, dν) = L2(R, du) ⊗ L2(R+, v

−2dv) 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-dimensionalsubspace of L2(R+, dy) generated by function ℓk(y) = e−y/2Lk(y);

(ii) the space A(k) is mapped onto L2(R−) ⊗ Lk with

U : A(k) 7→ L2(R−) ⊗ Lk.

Proof. Denote by A(k)1 the image of the space A(k) in the mapping U1,

analogously for the space A(k). Obviously, the space A(k)1 consists of all functions

F (u, v) =√vf(u)ψ(k)(uv) = χ+(u)

√2uv f(u)e−uvLk(2uv),

and, moreover, ‖F (u, v)‖A

(k)1

= ‖f(u)‖L2(R,du).

Obviously, the inverse operator U−12 = U∗

2 : L2(R, dx) ⊗ L2(R+, dy) →L2(R, du) ⊗ L2(R+, v

−2dv) has the form

U−12 : F (x, y) 7→

√2|u|vF (u, 2|u|v).

Then for each f ∈ H2+(R) one has

U2 : χ+(u)√

2u vf(u)e−uvLk(2uv) 7→ χ+(x)f(x)e−y/2Lk(y).

Thus, the image A(k)2 = U2

(A

(k)1

)is the set of all functions of the form

F (x, y) = χ+(x)f (x)e−y/2Lk(y), f ∈ H2+(R).

Analogously for the space A(k). Since the Fourier transform F gives an iso-metrical isomorphism of the space L2(R) under which the Hardy space H2

+(R),resp. H2

−(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 ver-sion” of the result of Vasilevski [9] obtained for the Bergman and poly-Bergmanspaces. It also mentions the intriguing patterns which wavelet subspaces andpoly-Bergman spaces share.

Remark 2.3 In the case of a system of admissible wavelets forming alltogether an orthonormal basis of L2(R) one gets a (complete) direct sum de-composition 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 ofA(k) and A(k) and orthogonalprojections onto them. Recall that the reproducing kernels play a central rolein the theory of wavelets. For instance, the reproducing kernel may be used forinterpolation, for discretization of the reconstruction formula, or for calibrationof wavelets, cf. [1].

Let ζ = (u, v) ∈ G, η = (s, t) ∈ G and denote by K(k)ζ (η) = K(k)(η, ζ) the

reproducing kernel of A(k). Then

K(k)η (ζ) = (ψ(k)

v ∗ ψ(k)t )(u− s) = 〈ψ(k)

ζ , ψ(k)η 〉.

Clearly, K(k)ζ (η) = K

(k)η (ζ). If F ∈ A(k), then

F (ζ) = (f ∗ ψ(k)v )(u) = ψ(k)

v ∗∫

R+

(ψ(k)t ∗ ψ(k)

t ∗ f)(u)dt

t2

=

∫

G

(ψ(k)v ∗ ψ(k)

t )(u − s)(f ∗ ψ(k)t )(s)

dsdt

t2=

∫

G

F (η)K(k)η (ζ) dν(η),

i.e. F (ζ) = 〈F,K(k)ζ 〉G. Obviously, K

(k)ζ (η) = 〈K(k)

ζ ,Kkη 〉G. Similarly, we have

the reproducing kernel K(k)ζ (η) of A(k) as

K(k)ζ (η) = K

(k)ζ (η) = K(k)

η (ζ).

In [6] similar results for projections and reproducing kernels are computedexplicitly (without further details of computation). Here we apply the reverseprocedure leading to Theorem 2.1 which allows us to give exact formulas for thekernels and projections on each wavelet subspaces without direct computationsand 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 orthogonalprojection P (k) of L2(G, dν) onto the space A(k) is given by

(P (k)F )(ζ) =

∫

G

F (η)K(k)η (ζ) dν(η),

where

K(k)η (ζ) =

k∑

j=0

k∑

l=0

κ(k)j,l

(2(ζ − ζ)

(2π − 1)(η − ζ) + (2π + 1)(η − ζ)

)l+1

×(

2(η − η)

(2π − 1)(η − ζ) + (2π + 1)(η − ζ)

)j+1

,

and

κ(k)j,l =

(−1)j+k(kj

)(kl

)

2B(j + 1, l+ 1).

5

Proof. The orthogonal projection UP (k)U−1 = B(k)2 : L2(G, dν) → L2(R+)⊗

Lk is obviously given by

B(k)2 = χ+(x)I ⊗Q(k),

where

(Q(k)H)(y) = 〈H, ℓk〉ℓk = ℓk(y)

∫

R+

H(θ)ℓk(θ) dθ

is the orthogonal projection of L2(R+) onto the one-dimensional space Lk gen-erated by the function ℓk(y). Indeed,

(B(k)2 F )(x, y) = χ+(x)ℓk(y)

∫

R+

F (x, θ)ℓk(θ) dθ.

Calculating the projection B(k)1 = U−1

2 B(k)2 U2, we get

(B(k)1 F )(u, v) = χ+(u) 2uv e−uvLk(2uv)

∫

R+

F (u, θ)e−uθLk(2uθ)dθ

θ.

Now,

(F−1 ⊗ I)B(k)1 F

=

∫

R

χ+(ξ) 2vξe−vξLk(2vξ)

(∫

R+

tF (ξ, t)e−tξLk(2tξ)dt

t2

)e2πiuξ dξ

=

∫

G

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

t2.

Since B(k)2 = U2(F ⊗ I)P (k)(F−1 ⊗ I)U−1

2 , then

B(k)1 = U−1

2 B(k)2 U2 = (F ⊗ I)P (k)(F−1 ⊗ I),

and therefore

(F−1⊗I)B(k)1 F = P (k)(F−1⊗I)F = 〈(F−1⊗I)F,K(k)

ζ 〉G = 〈F, (F⊗I)K(k)ζ 〉G.

Thus,

(F ⊗ I)K(k)ζ (η) = χ+(ξ) 2tvξLk(2tξ)Lk(2vξ)e−ξ(t+v+2πiuξ).

Since

Lk(y) =

k∑

j=0

(k

j

)(−y)j

j!=

k∑

j=0

λ(k)j yj ,

where

λ(k)j =

(−1)j

j!

(k

j

),

then we get

(F ⊗ I)K(k)ζ (η) =

k∑

j=0

k∑

l=0

λ(k)j λ

(k)l 2j+l+1tl+1vj+1

(χ+(ξ)ξj+l+1e−ξ(v+t+2πiu)

).

6

Sinceχ+(ξ)ξme−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(k)ζ (η) =

k∑

j=0

k∑

l=0

λ(k)j λ

(k)l 2j+l+1tl+1vj+1 Γ(2 + j + l)

(t+ v − 2πi(s− u))2+j+l.

Introducing

κ(k)j,l =

1

2λ

(k)j λ

(k)l Γ(2 + j + l) =

(−1)j

Γ(j + 1)

(k

j

)(−1)l

Γ(l + 1)

(k

l

)Γ(2 + j + l)

2

=(−1)j+k

(kj

)(kl

)

2B(j + 1, l+ 1),

we finally get the formula for the kernel

K(k)ζ (η) =

k∑

j=0

k∑

l=0

κ(k)j,l

(2t

t+ v − 2πi(s− u)

)l+1(2v

t+ v − 2πi(s− u)

)j+1

.

Using the fact that K(k)η (ζ) = K

(k)ζ (η) and

2v

t+ v + 2πi(s− u)=

2(ζ − ζ)

(2π − 1)(η − ζ) + (2π + 1)(η − ζ),

2t

t+ v + 2πi(s− u)=

2(η − η)

(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) andH2

+(R), resp. A(k) and H2−

(R).Given k ∈ N0, introduce the operator

R(k)1 = R

(k)0 U1 : H2

+(R) → L2(R) ⊗ L2(R+),

where R(k)0 is given by the rule

(R(k)0 f)(x, y) = χ+(x)f(x)ℓk(y).

Obviously, the image of R(k)1 coincides with the space A

(k)2 . The adjoint operator

isR

(k)1

∗

= U∗

1R(k)0

∗

: L2(R) ⊗ L2(R+) → H2+(R),

7

where (R

(k)0

∗

F)

(x) = χ+(x)

∫

R+

F (x, τ)ℓk(τ) dτ,

and clearly

R(k)1

∗

R(k)1 = I : H2

+(R) → H2+(R),

R(k)1 R

(k)1

∗

= B2 : L2(G, dν) → A(k)2 .

Summarizing the above construction we have

Theorem 4.1 The operator R(k) = R(k)1

∗

U maps the space L2(G, dν) ontoH2

+(R), and the restriction

R(k)|A(k) : A(k) → H2+(R)

is an isometrical isomorphism. The adjoint operator

R(k)∗ = U∗R(k)1 : H2

+(R) → A(k) ⊂ L2(G, dν)

is an isometrical isomorphism of the space H2+(R) onto A(k).

Remark 4.2 Operators R(k) and R(k)∗ provide the following decomposi-tions of the projection P (k) and of the identity operator on H2

+(R)

R(k)R(k)∗ = I : H2+(R) → H2

+(R),

R(k)∗R(k) = P (k) : L2(G, dν) → A(k).

Theorem 4.3 The isometrical isomorphism

R(k)∗ = U∗R(k)1 : H2

+(R) → A(k) ⊂ L2(G, dν)

is given by

(R(k)∗f

)(x, y) =

√2y

∫

R+

√ξf(ξ)Lk(2ξy)eiξ(2πx+iy) dξ.

Proof. The direct calculation yields

(R(k)∗f

)(x, y) = (U∗R

(k)1 f)(x, y) = (U∗

1U∗

2R(k)0 U1f)(x, y)

= (F−1 ⊗ I)(χ+(ξ)

√2ξ y f(ξ)ℓk(2ξy)

)

= (F−1 ⊗ I)(χ+(ξ)

√2ξ yf(ξ)e−ξyLk(2ξy)

)

=√

2y

∫

R+

√ξf(ξ)Lk(2ξy)eiξ(2πx+iy) dξ.

2

8

Corollary 4.4 The inverse isomorphism

R(k) = R(k)1

∗

U : A(k) → H2+(R)

is given by

(RF )(ξ) = χ+(ξ)√

2ξ

∫

R

∫

R+

F (u, v)Lk(2vξ)e−iξ(2πu−iv) dudv

v.

Similar results hold for the space A(k), where operators R(k), R(k)∗, R(k)1

and R(k)0 are introduced analogously.

Remark 4.5 The above results may be viewed as a special case of con-structing 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-typespaces, Fock-type spaces, etc.).

Let us return to spaces A(k)2 = L2(R+) ⊗ Lk, the whole construction goes

analogously also for the spaces A(k)2 = L2(R−) ⊗Lk. It is convenient to change

the previously used basis {ℓk(y)}∞k=0 of L2(R+, dy) to the new basis {ℓk(y)}∞k=0,where

ℓk(y) = (−1)kℓk(y), k = 0, 1, 2, . . . .

We note that the previously defined one-dimensional spaces Lk are generatedby the new basis elements ℓk(y) as well, and that the statements of Theorem 2.1remain valid without any change.

In L2(R+, dy) introduce the operators, cf. [10],

(S+f)(y) = −f(y) + e−y/2

∫ y

0

et/2f(t) dt

(S−f)(y) = −f(y) + ey/2

∫∞

y

e−t/2f(t) dt,

which are bounded on L2(R+, dy) and mutually adjoint. In the spaceL2(R, dx)⊗L2(R+, dy) put

T+2 = I ⊗ S+, T−

2 = I ⊗ S−.

It is known, cf. [10], that operators S+ and S− act on functions ℓk(y) (for eachadmissible k) as follows

(S+ℓk)(y) = ℓk+1(y),

(S−ℓk)(y) = ℓk−1(y),

(S−ℓ0)(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 {ℓk(y)}∞k=0. Its adjointoperator S− is the unilateral backward shift with respect to the same basis,and its kernel coincides with the one-dimensional space L0 generated by ℓ(y) =e−y/2. Thus the operator

T+2 |

A(k)2

: A(k)2 → A

(k+1)2

9

is an isometrical isomorphism and the operator

T−

2 |A

(k+1)2

: A(k+1)2 → A

(k)2

is its inverse. Iterating these isomorphisms we get

Lemma 4.6 For given natural numbers k < l the operator

(T+2 )l−k|

A(k)2

: A(k)2 → A

(l)2

is an isometrical isomorphism, and the operator

(T−

2 )l−k|A

(l)2

: A(l)2 → A

(k)2

is its inverse.

By Theorem 2.1 we have that A(k) = U−1(A(k)2 ) for each k ∈ N0. Introducing

operatorsT+ = U−1T+

2 U, T− = U−1T−

2 U

acting on L2(G, dν) and applying Lemma 4.6 we immediately get the followingtheorem.

Theorem 4.7 For given natural numbers k < l the isometrical isomorphismbetween 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 inves-tigating the Toeplitz- and small and big Hankel-type operators defined as in [6]by

T (k,l)a = P (k)MaP

(l),

h(k,l)a = P (k)MaP

(l),

H(k,l)a =

I −

k∑

j=0

P (i)

MaP

(l),

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

10

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[2] Abramowitz M, Stegun I A (1970) Handbook of Mathematical Functions,New York: Dover

[3] Calderon A (1964) Intermediate spaces and interpolation, the complexmethod. Studia Math 24: 113–190

[4] Grossmann A, Morlet J, Paul T (1986) Transforms associated to squareintegrable group representations II: Examples. Ann Inst Henri Poincare45: 293–309

[5] Hutnık O (2008) On the structure of the space of wavelet transforms. C RAcad Sci Paris Ser I Math 346: 649–652

[6] Jiang Q, Peng L (1992) Wavelet transform and Toeplitz-Hankel type oper-ators. Math Scand 70: 247–264

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[8] Paul T (1984) Functions analytic on the half-plane as quantum mechanicalstates. J Math Phys 25: 3252-3262

[9] Vasilevski N L (1999) On the structure of Bergman and poly-Bergmanspaces. Integral Equations Operator Theory 33: 471–488

[10] Vasilevski N L (2008) Commutative Algebras of Toeplitz Operators on theBergman Space, Series: Operator Theory: Advances and Applications, Vol.185, Basel: Birkhauser

Ondrej Hutnık, Institute of Mathematics, Faculty of Science, Pavol Jozef Safarik Uni-versity in Kosice, Current address: Jesenna 5, 041 54 Kosice, Slovakia,E-mail address: ondrej.hutnik@upjs.sk

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