New integral operators arising from new bilateral generating functions ...

NEW INTEGRAL OPERATORS ARISING FROM NEW BILATERAL GENERATING FUNCTIONS FOR THE WEIGHTED UNIVARIATE COMPLEX HERMITE POLYNOMIALS A. BENAHMADI, A. EL KACHKOURI, AND A. GHANMI

arXiv:1707.06309v1 [math.CV] 19 Jul 2017

Dedicated to Professor Ahmed Intissar on the occasion of his 65th birthday

A BSTRACT. We obtain some refinements and new results for the class of weighted univariate complex Hermite polynomials. Mainly, we deal with a generalized integral representation. New bilateral generating functions are obtained and many interesting new integral operators connecting many interesting functional Hilbert spaces are derived as applications.

1

I NTRODUCTION

The univariate complex Hermite polynomials go back to Itô (1952) who introduced them in the context of complex Markov process [22]. They also have been used as a basic tool in the nonlinear analysis of travelling wave tube amplifiers and appear in calculating the effects of nonlinearities in broadband radio frequency communication systems [8]. Their expansions have been successfully applied to microwave amplifier saturation characteristics by the Prometheus team, and demonstrated excellent curve fitting over the 20dB range of the Rayleigh decibel density. In the last decade, they were intensively investigated in a considerable number of papers in connection with many branches of mathematics and physics. For example in studying singular values of the Cauchy transform [21], coherent states theory [6, 5], combinatory [20, 19] and signal processing [26, 9]. Many interesting and useful mathematical properties of these polynomials were thoroughly studied and reviewed (see for example [16, 10, 19] and the references therein). In [17], we have explored two widest generalizations of the classical Mehler’s formula for the univariate complex Hermite polynomials. As applications, we have derived an explicit closed expression of the heat kernel associated to a specific magnetic Laplacian on the complex plane as well as an integral reproducing property of these polynomials by a like Fourier transform. A basic tool was the integral representation established in Section 2 of ∂ m+n ν Hm,n (z, z) = (−1)m+n eνzz m n e−νzz (1.1) ∂z ∂z for z ∈ C and arbitrary fixed positive real number ν > 0. The incorporation of the parameter ν is fairly interesting for its physical meaning. In fact, it can be interpreted as the magnitude of a constant magnetic field applied perpendicularly to the Euclidean plane. In the present paper, we continue and complete the investigation in our previous concerning the class of complex orthogonal polynomials defined by their Rodrigues’ formula (1.1). Our purpose is twofold. Firstly, we review and obtain some refinements of some basic results obtained so far in a more general setting. Secondly, we investigate a generalized integral representation (Theorem 2.4) and three new interesting bilateral generating functions. The two first ones involves the products of complex Hermite polym n ν ν ν (z, z)H ν nomials tn Hm,n n,m′ (w, w) (Theorem 3.1) and u t Hm,n (z, z)Hn,m′ (w, w) (Theorem 3.4). The ν (z; z). As applications, we deal with some third one, Theorem 3.14, involves the product ξ n Hnµ (x)Hm,n 2 remarkable integral transforms connecting the L2 (R; e−x dx) to the one-dimensional Bargmann-Fock space This research work was partially supported by the Simons Foundation and the Hassan II Academy of Sciences and Technology. 1

2

A. BENAHMADI, A. EL KACHKOURI, AND A. GHANMI

F 2,ν (C) and more generally to the generalized Bargmann-Fock spaces that are L2 -eigenspaces of a mag2 2 netic Laplacian ∆ν acting on L2 (C; e−ν|z| dxdy). An integral transform mapping L2 (C; e−ν|z| dxdy) to the two-dimensional Bargmann-Fock space F 2,ν (C2 ) is also introduced. The paper is organized as follows. In Section 2, we present an interesting integral representation for ν (z; z). We establish in Section 3 some new bilateral generating functions. An orthogonal hilberthe Hm,n 2 tian decomposition of the Hilbert space L2 (C; e−ν|z| dλ) in terms of the L2 -eigenspaces of the magnetic Schrödinger operator ∆ν is reviewed and presented in Section 4. Section 5 is concerned with some new interesting integral operators between some well-known functional Hilbert spaces.

2

AN

INTEGRAL REPRESENTATION

The integral representation we obtain is, in some how, a generalization the one obtained by Ismail [19, Theorem 5.1], Z

(−i)m+n 2 2 n (2.1) ξ m ξ e|z| −|ξ| +2iℜ(hξ,zi)) dλ(ξ). π C Ismail’s proof is based on the generating function of Hm,n (z; z). In the sequel, we provide a generalized integral representation involving real and complex parameters using elementary results giving an integral representation of the gaussian. To this end, notice that the following parametrization ν = αβ µ , where µ > 0 and α, β ∈ C such that αβ > 0,is used so that Hm,n (z; z) =

ν Hm,n (z; z)

m+n

= (−1)

e

αβ |z|2 µ

∂ m+n ∂z m ∂z n

− αβ |z|2 µ

e

µ,α,β =: Hm,n (z; z).

(2.2)

The fundamental tool we use to reintroduce the class of univariate complex Hermite polynomials is the following elementary result. Key Lemma 2.1. Let µ > 0 be a fixed positive real number and α, β ∈ C two complex numbers. Then Z αβ π −µ|ξ|2 +αξ+βξ dλ(ξ) = (2.3) eµ. e µ C In particular, for every z ∈ C, we have Z π − αβ 2 |z|2 e µ e−µ|ξ| +αξz−βξz dλ(ξ) = . (2.4) µ C Proof. Formula (2.3) is quite easy to check by writing ξ as ξ = x + iy with x, y ∈ R, and next making use of Fubini’s theorem as well as the explicit formula for the gaussian integral Z

−µx2 +bx

e

R

dx =

1

π µ

2

b2

e 4µ ;

µ > 0, b ∈ C.

(2.5)

Remark 2.2. By considering the particular case α = −β = i, the identity (2.4) reads simply |z|2 µ

Z

µ 2 e−µ|ξ| +2iℜhξ,zidλ(ξ). π C This is the well-known fact that the Fourier transform reproduces the gaussian function. e−

=

(2.6)

Remark 2.3. The integral formula (2.4) suggests the consideration of the complex Hermite polynomials µ,α,β (z; z) defined by (2.2). Hm,n The main result of this section is the following.

NEW INTEGRAL OPERATORS ARISING FROM NEW BILATERAL GENERATING FUNCTIONS FOR THE WEIGHTED UNIVARIATE COMPLEX HERMI

Theorem 2.4. For given µ > 0 and α, β ∈ C such that αβ > 0, we have Z µ n αβ |z|2 −µ|ξ|2 +αhξ,zi−βhξ,zi µ,α,β Hm,n (z; z) = (−α)m (β)n ξ m ξ e µ dλ(ξ). π C

(2.7)

Proof. The integral involved in left hand-side of (2.4) converges uniformly in z on every disc D(0, r) of C. Thus, by differentiate repeatedly the both sides of (2.4), with respect to z and z, we obtain the integral µ,α,β (z; z) given through (2.7). representation for the Hm,n Remark 2.5. By taking for example µ = 1 and α = −β = i, so that ν = αβ/µ = 1, the integral representation (2.7) reduces further to (2.1) which is the one obtained by Ismail [19, Theorem 5.1]. As immediate consequence of Theorem 2.4, we see that for given real number ν > 0 and given complex numbers a, b ∈ C such that ab > 0, we have abν ν (z; z). an bm Hm,n (az; bz) = Hm,n

(2.8)

More particularly, if a = b = ν −1/2 , we have m+n

1 ν

2

ν Hm,n

z z √ ;√ ν ν

= Hm,n (z; z),

(2.9)

ν corresponding to ν = 1. This transition formula can be used to transport some where Hm,n is the Hm,n ν (z; z). For example, well established properties and identities related to Hm,n (z; z) to the context of Hm,n making use of [16, Proposition 3.4 (b)] one obtains

Lemma 2.6. We have the generating function +∞ X

uk ν Hk,n(z, z) = ν n (z − u)n eνuz . k! k=0

(2.10)

The integral representation in Theorem 2.4 combined with the reproducing property (2.3) in Lemma 2.1, can also be used to check the following result in a easy way. Theorem 2.7. We have the exponential generating function ∞ X um v n

m,n=0

m!n!

ν Hm,n (z; z) = eν(uz+vz−uv) .

(2.11)

Remark 2.8. The result of Theorem 2.7 is well-known for the special case ν = 1 (see for example [31, 16]). It follows from the well-established fact (2.10). The identity (2.11) for arbitrary ν > 0 can be obtained from the one corresponding to ν = 1 by the transport formula (2.9). A direct simpler proof of (2.10) as well as (2.11) can be handled making use of the integral representation (2.7). From Theorem 2.7, we deduce the following Corollary 2.9. The particular case of u = v = w, yields ∞ X wm w n

m,n=0

3

N EW

m!n!

ν Hm,n (z; z) = e−ν|w|

2 +2νℜ(zw)

.

(2.12)

BILATERAL GENERATING FUNCTIONS

In the present section, we derive some new bilinear generating functions. Notice for instance that it is easy to see that the Rodrigues’ formula (1.1) infers ν Hm,n (z, z) = (−1)m ν n eνzz

∂ m n −νzz . z e ∂z m

(3.1)

4


Subsequently, we can check that ν m −ν|z−ξ| Hm,m ′ (z − ξ, z − ξ) = (−1) e

2

∂ m m′ m′ −ν|z−ξ|2 (z ν e − ξ) . ∂z m

(3.2)

Accordingly, we can prove the following Theorem 3.1. For every t in the unit circle and z, w ∈ C, we have Gνm,m′ (t; z, w) :=

+∞ X

tn ν m′ ν Hm,m′ (z − tw, z − tw)eνtwz . H ν (z, z)Hn,m ′ (w, w) = (−t) n m,n n!ν n=0

(3.3)

Proof. Using successively the variant (3.1) of the Rodrigues’ formula as well as the result of Lemma 2.6, one gets Gνm,m′ (t; z, w) =

m tn n −νzz ν m n νzz ∂ z e ¯ Hn,m (−1) ν e ′ (w, w) m n n! ν ∂z n=0 +∞ X

i ∂ m h m′ m′ νtzw −νzz ν (w e . − tz) e ∂z m Now, if t is assumed to belong to the unit circle, the above identity can be rewritten as

= (−1)m eνzz

i ∂ m h m′ m′ −ν|z−tw|2 ν (z − tw) e . ∂z m In the right hand-side of the previous equality we recognize (3.2). Thus, the expression of Gνm,m′ (t; z, w) reduces further to the desired result (3.3). ′

Gνm,m′ (t; z, w) = tm eνzz e−νtw(z−tw) (−1)m

Remark 3.2. We recover the result of Proposition 3.6 in [16] by taking t = 1 = ν in (3.3). This can be used to prove the first Mehler’s formula in [17]. As immediate consequence of Theorem 3.1, we show Corollary 3.3. For every t in the unit circle and z ∈ C, we have +∞ X

tn 2 2 νt|z|2 ν . |Hm,n (z, z)|2 = m!(νt)m L(0) m (ν|1 − t| |z| )e n n!ν n=0

Particularly, we have

(3.4)

+∞ X

ν (z, z)|2 |Hm,n 2 = m!ν m eν|z| . n n!ν n=0

(3.5)

(0)

ν ¯ = (−ν)m m!Lm (ν|ξ|2 ), we get Proof. By taking m = m′ in (3.3) and using Hm,m (ξ, ξ) +∞ X

tn ν 2 νtwz (w, w) = (−νt)m m!L(0) H ν (z, z)Hn,m . m (ν|z − tw| )e n m,n n!ν n=0

(3.6)

Then (3.4) follows by considering the particular case z = w. Added to (3.3), we prove the following bilinear generating functions of exponential type. Theorem 3.4. For every t in the unit circle and complex numbers u, z, w ∈ C such that ν|u| < 1, we have +∞ X

um tn ′ ν m′ (z − tw − u)m eνtzw+νu(z−tw) H ν (z, z)Hn,m ′ (w, w) = (−νt) n m,n m!n!ν m,n=0

and +∞ X

−ν 2 tu|z − tw|2 1 um tn ν ν H (z, exp z)H (w, w) = m,n n,m m!n!ν n (1 − νtu) 1 − νtu m,n=0

!

eνtwz .

(3.7)

(3.8)


Proof. The identity (3.7) follows by twice application of (2.10). It can also be obtained easily by combining (3.3) and (2.10). To prove (3.8), we start from (3.6) to get +∞ X

+∞ X um tn ν 2 νtwz ν z)H (w, w) = (νtu)m L(0) H (z, . n,m m (ν|z − tw| )e m,n n m!n!ν m,n=0 m=0

(3.9)

In the right hand-side of (3.9), we recognize the well-known generating function for the Laguerre polynomials, to wit ([27, p. 135]): ∞ X

z n L(α) n (x) =

n=0

Thus, we get +∞ X

−xz 1 exp ; |z| < 1. 1+α (1 − z) 1−z

1 um tn −ν 2 tu|z − tw|2 ν ν z)H (w, w) = H (z, exp n,m m,n ν n m!n! (1 − νtu) 1 − νtu m,n=0 −ν 2 tu|z − tw|2 1 exp = (1 − νtu) 1 − νtu

! !

eνtwz eνtwz .

The following identity is clearly a particular case of (3.8) by setting z = w. Corollary 3.5. For every t in the unit circle and complex numbers u, z ∈ C such that ν|u| < 1, we have +∞ X

um tn −ν 2 tu|z|2 |1 − t|2 1 ν 2 exp |H (z, z)| = m,n m!n!ν n (1 − νtu) 1 − νtu m,n=0

!

2

eνt|z| .

(3.10)

In particular +∞ X

2

eν|z| um 2 ν z)| = |H (z, . m,n m!n!ν n (1 − νu) m,n=0

(3.11)

By equating the left hand-sides of (3.5) and (3.11), we deduce following curious identity. Corollary 3.6. For every complex numbers u, z ∈ C such that ν|u| < 1, we have

ν 2 +∞ X |Hm ′ ,n (z, z)| um ′ m′ ν (1 − νu) = m !ν |Hm,n (z, z)|2 . n n n!ν m!n!ν m,n=0 n=0 +∞ X

(3.12)

We conclude this section by proving the following bilateral generating function involving the weighted univariate complex Hermite polynomials and the real Hermite polynomials µ 2

Hnµ (x) := (−1)n e 2 x For µ = 1 we denote Hnµ (x) simply Hn (x).

dn −µx2 e ; dxn

Theorem 3.7. We have

µ > 0.

(3.13)

!

+∞ X

ν (z; z) ξ n Hnµ (x)Hm,n ν 2 2 √ √ √ µξz + √ z − µx . = ( µξ)m e−µ(ξ z −2xξz) Hm n n!ν 2 µξ n=0

(3.14)

Proof. Making use of Hm,n (z, z) = e−∆C (z m z n ) as well as the well-known generating function for the real Hermite polynomials ([27, Eq. (1) p.187]), +∞ X

ξ n Hnµ (x) 2 = e−µξ +2µxξ , n! n=0

6


we obtain +∞ X tn Hnµ (x)Hm,n (z, z) n!ν n n=0

+∞ X

(tz/ν)n Hnµ (x) z n! n=0 m

−∆C

=e

Now, by utilizing the fact

!

2

tz

= eµx e−∆C z m e−µ[ ν −x]

2

.

∂ j −(az−b)2 2 = (−1)j aj e−(az−b) Hj (az − b) e j ∂z as well as the well-known identity m X m

j=0

j

!

Hj (x)(2ξ)m−j = Hm (x + ξ),

we can rewrite the above sum as +∞ X tn H µ (x)Hm,n (z, z) n n!ν n n=0

µx2

=e

m X m

j=0

j

!

z m−j (−1)j

∂ j −µ[ tz −x]2 ν e ∂z j

!m−j ! √ m µt m −µ[ t2 z22 −2x tz ] X µt √ m ν ν ν e = z − µx Hj √ z ν µt ν j j=0 ! √ √ m tz t2 z 2 µt µt ν √ z + √ z − µx . = e−µ[ ν 2 −2x ν ] Hm ν ν 2 µt √

Finally, the desired result follows thanks to the transport formula (2.9). As immediate consequence we claim the following Corollary 3.8. We have +∞ X

ν (z; z) ζ m ξ n Hnµ (x)Hm,n 2 2 2 = e−µz (ξ +ζ )+2µzξ(x+ζξ)+νζz−2µζξx. n m!n!ν m,n=0

(3.15)

ν Remark 3.9. Further summation formulas for the Hm,n of Mehler type are derived in [17].

4

AN

2

ORTHOGONAL HILBERTIAN DECOMPOSITION OF

L2 (C; e−ν|z| dλ)

We begin by discussing the action of the first order differential operators ∂ ∂ Aν = − νz and Bν = − νz ∂z ∂z ν (z, z). The operators A and B are considered with on the univariate complex Hermite polynomials Hm,n ν ν 2 2,ν 2 −ν|z| ∞ dλ), of all square integrable C0 (C) as its regular domain in the standard Hilbert space L := L (C; e functions on C with respect to the usual Lebesgue measure dλ(z) = dxdy; z = x + yi. Namely, we assert ν (z, z) in the sense that we have Lemma 4.1. The operators Aν and Bν are creation operators for the Hm,n ν ν Aν Hm,n (z, z) = −Hm+1,n (z, z).

ν ν (z, z). Bν Hm,n (z, z) = −Hm,n+1

(4.1) (4.2)

∂ ν ν (z, z)−H ν H (z, z) = νzHm,n m+1,n (z, z) which infers ∂z m,n ν (z, z) = H ν (z, z). (4.1). The identity (4.2) holds by conjugation since Hm,n n,m

Proof. Starting from (1.1) one can easily check

ν (z, z) can be realized in terms of A and B as follows. Accordingly, the polynomials Hm,n ν ν


Corollary 4.2. We have ν n n Hm,n (z, z) = (−1)m Am ν (ν z )

= =

(4.3)

n

(−1) Bνn (ν m z m ) n (−1)m+n Am ν Bν · (1).

(4.4) (4.5)

More generally, we have ν ν (z, z). Hm+j,n+k (z, z) = (−1)j+k Ajν Bνk Hm,n

(4.6)

ν (z, z) = (−1)j H ν Proof. By induction, the identity (4.1) yields Ajν Hm,n m+j,n (z, z) for any nonnegative k ν ν k integer j. Similarly, we have Bν Hm,n (z, z) = (−1) Hm,n+k (z, z). Hence (4.6) follows by combining these two identities. The equalities (4.3), (4.4) and (4.5) follow as immediate consequence of (4.6), since ν (z, z) = z m , H ν (z, z) = z n and H ν (z, z) = 1. Hm,0 0,n 0,0

Remark 4.3. The equalities (4.3), (4.4) and (4.5) can be taken for alternative definitions for the complex Hermite polynomials. ∂ ∂ ν (z, z) in the sense that we and are annihilation operators for the Hm,n Lemma 4.4. The operators ∂z ∂z have ∂ ν ν (z, z). (4.7) H (z, z) = mνHm−1,n ∂z m,n ∂ ν ν (z, z). (4.8) H (z, z) = nνHm,n−1 ∂z m,n ∂ and Bνn = ∂z we get (4.7). The identity (4.8) follows from (4.7) by complex conjugation.

Proof. By derivation of both sides of (4.4) and next use the fact that

∂ − νz ∂z

n

commute,

Corollary 4.5. We have the three terms recurrence formulas ν ν ν (z, z) − nνHm,n−1 (z, z) Hm+1,n (z, z) = νzHm,n

ν Hm,n+1 (z, z)

=

ν νzHm,n (z, z) −

(4.9)

ν mνHm−1,n (z, z).

(4.10)

Now, using the fact that the considered first order differential operator Aν (resp. Bν ) has as adjoint the ∂ ∂ 2 resp. Bν∗ = − in the Hilbert space L2,ν := L2 (C; e−ν|z| dλ), one can show that operator A∗ν = − ∂z ∂z ν (z, z) belong to L2 (C; e−ν|z|2 dλ). More precisely, we assert the following. the polynomials Hm,n 2

ν Lemma 4.6. The square norm of Hm;n in L2 (C; e−ν|z| dλ) is given by

π

ν 2 m!n!ν m+n .

Hm;n 2,ν = L

(4.11)

ν

2

Proof. By the definition of the square norm in L2,ν := L2 (C; e−ν|z| dλ) and the alternative definition (4.3), we see that

E D

ν 2 n ν

Hm;n 2,ν = (−1)m ν n Am ν (z ), Hm;n L

L2,ν

D

ν = (−1)m ν n z n , (A∗ν )m Hm;n

E

L2,ν

.

ν Hence the result (4.11) follows by means of (A∗ν )m Hm;n = (−1)m m!ν m+n z n , which follows by induction from (4.7), as well as the well-established fact kz n k2L2,ν = πn!/ν n+1 .

Now, by considering the second order differential operators ∆ν = −∆ + νE,

and

f = −∆ + νE, ∆ ν

(4.12)

8


∂2 is the usual Laplace-Beltrami operator on C, E is the Euler first order differential operator ∂z∂z ∂ ∂ given by E = z and E = z is its complex conjugate, we show easily the following ∂z ∂z Proposition 4.7. We have where ∆ =

ν ν ∆ν Hm,n (z, z) = nνHm,n (z, z) and

f H ν (z, z) = mνH ν (z, z). ∆ ν m,n m,n

(4.13)

Proof. It is easy to see from (4.1) combined with (4.7) and (4.2) combined with (4.8) that the polynomials ν (z, z) are eigenfunctions of ∆ and ∆ f in (4.12). Hm,n ν ν As immediate consequence, the orthogonality property Z

C

2

ν ν (z; z)e−ν|z| dλ(z) = 0 Hm;n (z; z)Hj;k

(4.14)

for every pairs of nonnegative integers (m, n), (j, k) such that (m, n) 6= (j, k), can be handled easily since ν (z, z) are commune eigenfunctions of the Laplacians ∆ and ∆ f and since eigenfuncthe polynomials Hm,n ν ν tions for distinct eigenvalues of a symmetric operator are orthogonal. The operator ∆ν is connected to the so-called special Hermite operator given explicitly by ([29]) Lν = −

(

ν ∂ ∂2 ∂ ν2 + − z¯ z − |z|2 ∂z∂ z¯ 2 ∂z ∂ z¯ 4

)

(4.15)

2

by means of the unitary map Mν : L2 (C; dλ) −→ L2 (C; e−ν|z| dλ) (called ground-state transformation), ν 2 Mν ψ(z) = e 2 |z| ψ(z), being indeed Lν since ∆ν Mν = Mν (Lν − ν2 ). The operator Lν is an unbounded symmetric operator on C0∞ (R2 ) and is an essentially self-adjoint operator in the free Hilbert space L2 (C; dλ). It describes in physics the quantum behavior of a charged particle confined in the plane, under the action of an external constant magnetic field. Notice that Lν can be realized as a Schrödinger operator, √ (4.16) Hθ f := −∆f − 2 −1(df |θ) + id∗ θ + |θ|2 f,

on the Euclidean plane R2 = C, where θ is the canonical vector potential θ = iν(¯ z dz − zd¯ z ); ν ∈ R. The spectral theory of Lν on L2 (C; dλ) is well-known (for a systematic study one can refer for example to [23, 4, 12, 18]). The spectrum of Lν consists then of eigenvalues of infinite multiplicity (Landau levels) of the form Ek := (k + 1/2)ν; k = 0, 1, 2, · · · . The associated eigenfunctions are known to be expressible ν (z, z in terms of the complex Hermite polynomials Hj,k ¯) which form a complete orthogonal system of the 2

Hilbert space L2 (C; e−ν|z| dλ) (see for example [22, 21, 3] for ν = 1). The above discussion leads to the following Theorem 4.8. The L2 -eigenspace 2

Fn2,ν (C) = {f ∈ L2 (C; e−ν|z| dλ);

∆ν f = nf }

(4.17)

of ∆ν associated to the eigenvalue n is a reproducing kernel Hilbert space, whose reproducing kernel is given by ν (z − w, z − w)eνhw,zi. Kkν (z, w) = Hk,k

(4.18)

Moreover, we have the orthogonal hilbertian decomposition 2

L2 (C; e−ν|z| dλ) =

∞ M

n=0 −ν|z|2

Fn2,ν (C).

dλ) coincides with the well-known Bargmann-Fock Remark 4.9. The null-subspace of ∆ν in L2 (C; e 2 space, F02,ν (C) = F 2,ν (C) := Hol(C) ∩ L2 (C; e−ν|z| dλ). Notice also that for every fixed n, the polynomiν (z, z ¯), for varying m, constitute an orthogonal basis of the L2 -eigenspace Fn2,ν (C), so that (4.18) als Hm,n follows easily using the bilinear generating function (3.3) with m = m′ = k and t = 1.


5

N EW

INTEGRAL OPERATORS

In this section, we can readily employ the obtained generating and bilinear generating functions to introduce some new integral transforms. The proofs of the theorems below are subject to a general principle issue from the framework of coherent states transform [14, 11, 15]. In the sequel, we give a brief review of this principle. Let (HX ; ωX ) be an infinite complex functional Hilbert space on X with an orthogonal basis {en }n with respect to the inner scaler product Z

hφ, ψiHX :=

X

φ(x)ψ(x)ωX (x)dx

for given weight measure ωX . In a similar way we consider (HY ; ωY ) with an orthogonal basis {fn }n and assume that HY is in addition a reproducing kernel Hilbert space with reproducing kernel K(y, y ′ ). Associated to the data (HX ; ωX ; {en }n ) and (HY ; ωY ; {fn }n ), we perform the following kernel function T : X × Y −→ C defined by ∞ X en (x)fn (y) T (x, y) := . ken kHX kfn kHY n=0 It is straightforward to check that hT (·, y), T (·, y)iHX reduces further to K(y, y ′ ), the reproducing kernel function of HY . Moreover, the map y ∈ Y 7−→ T (·, y) ∈ HX defines a quantization of Y into HX . Thus, we can consider the integral transform T (φ)(y) :=

Z

X

T (x, y)φ(x)ωX (x)dx

for every φ ∈ HX . This transform maps HX into HY and satisfies T and subsequently T (φ) = Moreover, we have

∞ P

n=0

kφk2HX

ek kek kHX

!

=

βn fn ∈ HY for every φ =

=

∞ X

n=0

2

|αn |

ken k2HX

=

∞ X

n=0

fk kfk kHY

∞ P

n=0

αn en ∈ HX , where βn := αn ken kHX /kfn kHY .

|βn |2 kfn k2HY = kT (φ)k2HY .

Thereby, T defines a isometric linear transform from HX onto HY and the function x 7−→ hφ, T (x, ·)iHY belongs to HX for every φ ∈ HY , since hφ, T (x, ·)iHY =

*

∞ X

en (x)fn φ, ken kHX kfn kHY n=0

+

= HY

∞ X

hφ, fn iHY en (x). ken kHX kfn kHY n=0

In addition, we have the following integral representation (a resolution of the identity of HY ) φ(y) =

Z

X

for every φ ∈ HY .

T (x, y) hφ, T (x, ·)iHY ωX (x)dx = T hφ, T (x, ·)iHY (y)

The first integral operator to be introduced is related to the exponential generating function (2.11). In fact the kernel function is given by T ν (z|u, v) =

3/2 X ∞ ν ν m+n um v n

π

m,n=0

m!n!

ν (z; z) = Hm,n

3/2

ν π

eν(uz+vz−uv) ,

ν (z; z) are respectively orthogonal bases of the two-dimensional Bargmannwhere em,n (u, v) = um v n and Hm,n 2 Fock space F 2,ν (C2 ) and the Hilbert space L2,ν := L2 (C; e−ν|ξ| dλ) with square norms kem,n k2L2,ν = 2 π 2 m!n! π m+n . Therefore, we assert ν ν m+n and kHm,n kL2,ν (C2 ) = ν m!n!ν

10


Theorem 5.1. The integral operator T ν (ψ)(z, w) :=

3/2

ν π

e−νzw

Z

e−ν(|ξ|

2 −zξ−wξ)

ψ(ξ)dλ(ξ)

(5.1)

C

2

defines an isometric isomorphism from L2 (C; e−ν|ξ| dλ) onto the two-dimensional Bargmann-Fock space F 2,ν (C2 ). Moreover, we have ν

T

ν (Hm,n )(z, w)

=

1/2

ν π

ν m+n z m wn . 2

Remark 5.2. This new transform made the quantum mechanical configuration space L2 (C; e−ν|ξ| dλ) unitary isomorphic to the phase space F 2,ν (C2 ) as do the standard two-dimensional Segal-Bargmann transform ([7, 13, ?, 25]) B2α [ψ](z, w)

=

3 Z

2α π

2

R2

2

2

e−α(x1 +x2 +z

√ 2(z.x1 +wx2 ))

2 +w 2 −2

ψ(x1 , x2 )dx1 dx2 ;

(z, w) ∈ C2 .

The next integral transform relies isometrically the two generalized Bargmann-Fock spaces Fn2,ν (C) and It is connected to the generating function (3.3) whose the kernel function is given by

Fn2,ν ′ (C).

+∞ ν (z, z)H ν X Hm,n ¯ ν m,n′ (w, w) √ ′ m!ν m π n!n′ !ν n+n m=0 ν νwz ν √ . = Hn,n ′ (z − w, z − w)e ′ ′ n+n π n!n !ν

ν Tn,n ′ (z, w) :=

Namely, we assert Theorem 5.3. The integral operator ν Tn,n ′ (ψ)(w)

:=

(−1)n ν √ ′ π n!n′ !ν n+n

Z

e−ν(|z|

2 −zw)

C

is unitary from Fn2,ν (C) onto Fn2,ν ′ (C). Moreover, we have ν ν Tn,n ′ (Hm,n )(w) =

n!ν n n′ !ν n′

1/2

ν Hn,n ′ (z − w, z − w)ψ(z)dλ(z)

(5.2)

ν Hm,n ′ (w, w).

(5.3)

Remark 5.4. The result of the previous Theorem leads to following interesting integral reproducing property ν . More precisely, for n = n′ we have T ν (H ν )(w) = H ν (w, w). More for the polynomials Hm,n n,n m,n m,n explicitly

(−1)n ν πn!ν n

Z

e−ν(|z|

2 −zw)

C

ν ν ν (z, z)dλ(z) = Hm,n (w, w). Hn,n (z − w, z − w)Hm,n

(5.4)

Remark 5.5. By setting n = 0 we recover the following integral operator mapping the Bargmann-Fock space to a generalized Bargmann-Fock space Fn2,ν (C). Indeed, we have ν T0,n (ψ)(w) :=

ν √ π n!ν n

Z

C

(z − w)n e−ν(|z|

2 −zw)

ψ(z)dλ(z).

(5.5)

The last transform to be introduced is based on the bilateral generating function (3.14). In fact the kernel 2 function associated to L2 (R; e−µx dx) and generalized Bargmann-Fock space Fn2,ν (C) is given by ν Tm (x; z)

3/4

1/2 +∞ X

ν (z, z) Hnν (x)Hn,m √ 2n ν n n! n=0 ! √ 1/2 3/4 √ 1 ν ν(z + z) √ − ν2 z 2 + 2νxz √ e − νx . = Hm π 2m m! 2

:=

ν π

1 m!ν m

(5.6)

Notice here that the univariate real Hermite polynomials Hnµ (x) defined through (3.13) form an orthogonal 2 basis of Hilbert space L2 (R; e−νx dx) with norm given explicitly by kHnν k2L2 (R;e−νx2 dx) = (π/ν)1/2 2n ν n n!.


Theorem 5.6. The integral operator T B νm (φ)(z) :=

3/4

ν π

1 m 2 m!

1/2 Z

e−νx

2− ν z2+ 2

√ 2νxz

R

Hm

r

√ ν (z + z) − νx φ(ξ)dx. 2

(5.7)

2

defines an isometric isomorphism from L2 (R; e−νx dx) onto the generalized Bargmann-Fock space Fn2,ν (C) defined through (4.17). Moreover, we have T

Bνm (Hnν )(z)

=

1/4

ν π

2n m!ν m

1/2

ν Hn,m (z, z).

(5.8)

Remark 5.7. By setting m = 0 and ν = 1, we recover the known Segal-Bargmann transform from the 2 L2 (R; e−νx dx) onto the Bargmann-Fock space F 2,ν (C) [7, 28]. Indeed, we have 3/4 Z

2 √ 1 2 z e−x − 2 + 2xz φ(ξ)dx =: B(φ)(z). (5.9) π R This transform is closely connected to the Heisenberg group and has found wide application in quantum optics, in signal processing and in harmonic analysis on phase space [13].

Remark 5.8. For ν = 1 and arbitrary m, the formula (5.7) reads 3/4

1/2 Z

2 √ −x2 − z2 + 2xz

z+z Hm √ − x φ(ξ)dx =: Bm (φ)(z). e 2 R Up to a multiplicative constant, this is exactly the generalized Segal-Bargmann transform given in [24]. This transform coincides with the isometric operator considered by Vasilevski and linking the space of square integrable functions on the real line with the so-called true-poly-Fock spaces [30, Theorem 2.5] (see also [1, 2]). The proof presented here is direct and more simpler. The crucial tool is the bilateral generating function (3.14). ν π

1 m 2 m!

R EFERENCES [1] Abreu L.D., Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions. Appl. Comput. Harmon. Anal. 29 (2010), no. 3, 287–302. [2] Abreu L.D., Balazs P., de Gosson M., Mouayn Z., Discrete coherent states for higher Landau levels. Ann. Physics 363 (2015) 337–353. [3] Agorram F., Benkhadra A., El Hamyani A., Ghanmi A., Complex Hermite functions as Fourier-Wigner transform. Integral Transforms Spec. Funct. 27 (2016), no. 2, 94-100. [4] Askour N., Intissar A., Mouayn Z., Explicit formulas for reproducing kernels of generalized Bargmann spaces of C n . J. Math. Phys. (5) 41 (2000) 3057–3067. [5] Ali S.T., Bagarello F., Gazeau J-P., Quantizations from reproducing kernel spaces. Ann. Physics 332 (2013), 127–142 [6] Ali S.T., Bagarello F., Honnouvo G., Modular structures on trace class operators and applications to Landau levels, J. Phys. A, 43, no. 10 (2010) 105202, 17 pp. [7] Bargmann V., On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14 (1961) 187-214. [8] Barrett M.J., Nonlinear analysis of travelling wave tube amplifiers using complex Hermite polynomials. Preprint 1990. [9] Dallinger R., Ruotsalainen H., Wichman R., Rupp M., Adaptive pre-distortion techniques based on orthogonal polynomials. In Conference Record of the 44th Asilomar Conference on Signals, Systems and Computers, IEEE (2010) pp 1945-1950. [10] Dunkl C., Xu Y. Orthogonal polynomials of several variables. Encyclopedia of Mathematics and its Applications, 155. 2nd ed. Cambridge: Cambridge University Press; 2014. [11] Driver B., On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall isomorphisms. J. Funct. Anal. 133 (1995) 69–128. [12] Ferapontov E.V., Veselov A.P., Integrable Schrödinger operators with magnetic fields: factorization method on curved surfaces. J. Math. Phys. (2) 42 (2001) 590–607. [13] Folland G.B., Harmonic analysis in phase space. Annals of Mathematics Studies, 122. Princeton University Press, Princeton, NJ; 1989. [14] Hall B.C., The Segal-Bargmann "coherent-state" transform for Lie groups. J. Funct. Anal. 122 (1994) 103–151. [15] Gazeau J.-P., Coherent states in quantum physics. WILEY-VCH Verlag GmbH & Co. KGaA Weinheim 2009. [16] Ghanmi A., Operational formulae for the complex Hermite polynomials Hp,q (z, z¯). Integral Transforms Spec. Funct., Volume 24, Issue 11 (2013) pp 884-895.

12


[17] Ghanmi A., Mehler’s formulas for the univariate complex Hermite polynomials and applications. To appear in Math. Methods Appl. Sci. [18] Ghanmi A., Intissar A., Asymptotic of complex hyperbolic geometry and L2 -spectral analysis of Landau-like Hamiltonians. J. Math. Phys. (3) 46 (2005) 032107. [19] Ismail M.E.H., Analytic properties of complex Hermite polynomials. Trans. Amer. Math. Soc. 368 (2016), no. 2, 1189-1210. [20] Ismail M.E.H., Simeonov P., Complex Hermite polynomials: their combinatorics and integral operators. Proc. Amer. Math. Soc. 143 (2015), no. 4, 1397–1410. 2 [21] Intissar A., Intissar A., Spectral properties of the Cauchy transform on L2 (C; e−|z| dλ), J. Math. Anal. Appl. 313, no 2 (2006) 400-418. [22] Itô K., Complex multiple Wiener integral. Jap. J. Math., 22 (1952) 63-86. [23] Matsumoto H., Ueki N., Spectral analysis of Schrödinger operators with magnetic fields. J. Funct. Anal. (1) 140 (1996) 218– 255. [24] Mouayn Z., Coherent state transforms attached to generalized Bargmann spaces on the complex plane. Math. Nachr. 284 (2011) no. 14-15, 1948–1954. [25] Neretin, Lectures on gaussian integral operators and classical groups. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2011. [26] Raich R., Zhou G., Orthogonal polynomials for complex Gaussian processes. IEEE Trans. Signal Process., vol. 52 (2004) no. 10, pp. 2788-2797. [27] Rainville E.D., Special functions, Chelsea Publishing Co., Bronx, N.Y., (1960). [28] Segal I., Mathematical characterization of the physical vacuum for a linear Bose-Einstein field. Illinois J. Math. 6 (1962) 500–523. [29] Thangavelu S., Lectures on Hermite and Laguerre Expansions. Princeton University Press, 1993. [30] Vasilevski N.L., Poly-Fock spaces. Oper. Theory, Adv. App. 117 (2000) 371–386. [31] Wünsche A., Laguerre 2D-functions and their application in quantum optics, J. Phys. A 31 (1998), no. 40, 8267-8287

C ENTER OF M ATHEMATICAL R ESEARCH AND A PPLICATIONS OF R ABAT (C E R E MAR), A NALYSIS AND S PECTRAL G EOMETRY (A.G.S.), L ABORATORY OF M ATHEMATICAL A NALYSIS AND A PPLICATIONS (L.A.M.A.), D EPARTMENT OF M ATHEMATICS , P.O. B OX 1014, FACULTY OF S CIENCES , M OHAMMED V U NIVERSITY IN R ABAT, M OROCCO E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]