aa r X i v : . [ m a t h . F A ] M a y HOLOMORPHIC HERMITE FUNCTIONS AND ELLIPSES
HIROYUKI CHIHARAA
BSTRACT . In 1990 van Eijndhoven and Meyers introduced systems of holomorphic Hermite functionsand reproducing kernel Hilbert spaces associated with the systems on the complex plane. Moreoverthey studied the relationship between the family of all their Hilbert spaces and a class of Gelfand-Shilovfunctions. After that, their systems of holomorphic Hermite functions have been applied to studyingquantization on the complex plane, combinatorics, and etc. On the other hand, the author recentlyintroduced systems of holomorphic Hermite functions associated with ellipses on the complex plane.The present paper shows that their systems of holomorphic Hermite functions are determined by somecases of ellipses, and that their reproducing kernel Hilbert spaces are some cases of the Segal-Bargmannspaces determined by the Bargmann-type transforms introduced by Sj¨ostrand.
1. I
NTRODUCTION
Let < s < . We denote by X s ( C ) the set of all holomorphic functions on C satisfying anintegrability condition k ϕ k s = Z C | ϕ ( z ) | exp (cid:18) − − s s | z | + 1 + s s ( z + ¯ z ) (cid:19) L ( dz ) < ∞ , where L ( dz ) is the Lebesgue measure on C ≃ R . The function space X s ( C ) is a Hilbert spaceequipped with an inner product ( ϕ, ψ ) s = Z C ϕ ( z ) ψ ( z ) exp (cid:18) − − s s | z | + 1 + s s ( z + ¯ z ) (cid:19) L ( dz ) , ϕ, ψ ∈ X s ( C ) . The quantity k·k s coincides with the norm induced by the inner product. In [1] van Eijndhoven andMeyers first introduced the function space X s ( C ) and holomorphic Hermite functions ψ sn defined by ψ sn ( z ) = b nn ( s ) − / e − z / H n ( z ) , n = 0 , , , . . . ,b mn ( s ) = π √ s − s n (cid:18) s − s (cid:19) n n ! δ mn , m, n = 0 , , , . . . .H n ( z ) = ( − n e z (cid:18) ddz (cid:19) n e − z , n = 0 , , , . . . . They proved the following properties.
Theorem 1 (van Eijndhoven and Meyers [1]) . • X s ( C ) is a reproducing kernel Hilbert space with a reproducing formula ϕ ( z ) = Z C K s ( z, ζ ) ϕ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ ) for z ∈ C and φ ∈ X s ( C ) , where the integral kernel is given by K s ( z, ζ ) = 1 − s πs exp (cid:18) − s s z ¯ ζ − s s ( z + ¯ ζ ) (cid:19) . Mathematics Subject Classification.
Primary 33C45; Secondary 46E20, 46E22, 35S30.
Key words and phrases.
Bargmann transform, Segal-Bargmann spaces, holomorphic Hermite functions.Supported by the JSPS Grant-in-Aid for Scientific Research • { ψ sn } ∞ n =0 is a complete orthonormal system of X s ( C ) . • [ and β ∈ R are fixed constants satisfying ( α, β ) = (1 , . For ρ > , weconsider an elliptic disk of the form E ρ ( α, β ) = { x + iξ ∈ C | ( x, ξ ) ∈ R , | αx + i ( βx + ξ ) | ρ } . Note that ∂E ρ ( α, β ) is an ellipse whose major and minor axes join at the origin of C . For z = x + iξ ,set ζ = αx + i ( βx + ξ ) = α + 1 + iβ z + α − iβ z, OLOMORPHIC HERMITE FUNCTIONS AND ELLIPSES 3 Ψ α,β ( z ) = exp (cid:16) µ z (cid:17) , µ = µ ( α, β ) = 1 − α − β + 2 iβ α + β , Ψ α,βn ( z ) = (cid:26) e λz / (cid:18) ddz (cid:19) n e − λz / (cid:27) Ψ α,β ( z ) , n = 1 , , , . . . ,λ = λ ( α, β ) = 2 α (1 + α + β )(1 − α − β − iβ ) . More precisely, { Ψ α,β } ∞ n =1 is generated by the creation operators, which is the adjoint of the annihi-lation operator for Ψ α,β . Then we have | Ψ α,β ( z ) | e −| z | / = exp (cid:18) − | ζ | α + β (cid:19) , which shows that the function Ψ α,β and the system { Ψ α,βn } ∞ n =1 are concerned with the elliptic disk E ρ ( α, β ) in some sense. One of the results in [7] is the following. Theorem 2 ( [7, Theorem 4.2]) . The family { Ψ α,βn } ∞ n =0 is a complete orthogonal system of H B ( C ) . One of the purposes of the present paper is to understand Theorem 1 and its subsequent results inthe framework of the standard Segal-Bargmann space H B ( C ) .The standard Bargmann transform B , the standard Segal-Bargmann space H B ( C ) and related ob-jects are generalised. In fact, Sj¨ostrand constructed more general framework and applied it to studyingmicrolocal analysis. See, e.g., [9, 10] and references therein. In what follows we recall Sj¨ostrand’stheory restricted on R quickly. The generalisation of the Bargmann transform B is given as a globalFourier integral operator of the form T f ( z ) = C φ Z R e iφ ( z,x ) f ( x ) dx, z ∈ C , where φ ( z, x ) is a complex-valued quadratic phase function of the form φ ( z, x ) = a z + bzx + c x for an appropriate function f with assumptions b = 0 and Im c > , and C φ = 2 − / π − / | b | (Im c ) − / .We call T a Bargmann-type transform. Note that T can be also extended for all the tempered distribu-tions on R since | e iφ ( z,x ) | = O ( e − (Im c ) x / ) . The case of a = i/ , b = − i and c = i corresponds tothe standard Bargmann transform B . Set Φ( z ) = | bz | c − b z + ¯ b ¯ z c − az − ¯ a ¯ z i , Ψ( z, ζ ) = | b | zζ c − b z + ¯ b ζ c − az − ¯ aζ i . We denote the set of all square integrable holomorphic functions on C with respect to a weightedmeasure e − z ) L ( dz ) by H Φ ( C ) , which is a Hilbert space equipped with an inner product ( ϕ, ψ ) H Φ = Z C ϕ ( z ) ψ ( z ) e − z ) L ( dz ) , ϕ, ψ ∈ H Φ ( C ) . Set k ϕ k H Φ = p ( ϕ, ϕ ) H Φ for short. Similarly we define L ( C ) which is a Hilbert space consistingof all square integrable functions on C with respect to the weighted measure e − z ) L ( dz ) . Wedenote its inner product and norm by ( · , · ) L and k·k L respectively. The operator T gives a Hilbert H. CHIHARA space isomorphism of L ( R ) onto H Φ ( C ) , that is, T is bijective and ( T f, T g ) H Φ = ( f, g ) L for f, g ∈ L ( R ) . The inverse of T is the formal adjoint T ∗ which is concretely given by T ∗ ϕ ( x ) = C φ Z C e − iφ ( z,x ) ϕ ( z ) e − z ) L ( dz ) , x ∈ R . Moreover, H Φ ( C ) is a closed subspace of L ( C ) , and the projection operator is given by T ◦ T ∗ F ( z ) = C Φ Z C e z, ¯ ζ ) F ( ζ ) e − ζ ) L ( dζ ) , F ∈ L ( C ) , where C Φ = | b | / π Im c . In particular, ϕ = T ◦ T ∗ ϕ for ϕ ∈ H Φ ( C ) .The purpose of the present paper is to understand the function space X s ( C ) and the family ofholomorphic Hermite functions { ψ sn } ∞ n =0 , and the subsequent results in the framework of Sj¨ostrand’smicrolocal analysis. More precisely, in Section 2 we first study the properties of X s ( C ) and { ψ sn } ∞ n =0 in the framework of the standard Bargmann transform. In particular, we shall understand { ψ sn } ∞ n =0 from a view point of ellipses originated in [7]. Finally, in Section 3 we study X s ( C ) and { ψ sn } ∞ n =0 ,and some of subsequent results in the framework of Sj¨ostrand.2. X s ( C ) AND ELLIPSES
In this section we shall understand Theorem 1 from a view point of [7, Section 4]. Our results inthe present section are the following.
Theorem 3. X s ( C ) and { ψ sn } ∞ n =0 are essentially determined by the ellipse ∂E ρ ( √ s, in the frame-work H B ( C ) . More precisely, we have a Hilbert space isomorphism X s ( C ) ∋ ϕ ( z ) r s − s ϕ (cid:18) s − s z (cid:19) exp (cid:18)
14 1 + s − s z (cid:19) ∈ H B ( C ) (1) whose inverse is given by H B ( C ) ∋ ψ ( z ) r − s s ψ (cid:18) − s s z (cid:19) exp (cid:18) − s s z (cid:19) ∈ X s ( C ) , (2) and for n = 0 , , , . . . , ψ sn ( z ) × exp (cid:18) − − s s | z | + 1 + s s z (cid:19) = − r − s s ! n b nn ( s ) − / Ψ √ s, n r − s s z ! × exp − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r − s s z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3) Moreover, the reproducing kernel K s ( z, ζ ) of X s ( C ) can be also obtained by the Hilbert space iso-morphism and the reproducing formula for H B ( C ) . Recall the definition of ∂E ρ ( √ s, , that is, ∂E ρ ( √ s,
0) = { x + iξ ∈ C | ( x, ξ ) ∈ R , sx + ξ = ρ } . We here remark that { ∂E ρ ( √ s, | < s < , ρ > } is the set of all ellipses whose major and minoraxes are contained in the real and imaginary axes respectively. Proof of Theorem . First we shall show that (1) is a Hilbert space isomorphism of X s ( C ) onto H B ( C ) and its inverse is given by (2). Let ϕ ( z ) be a Lebesgue measurable function on C . Set P ( z ) = ϕ ( z ) e z / , which corresponds to the holomorphic Hermite polynomials H n ( z ) . By using anidentity of the form −
12 = − s s + (1 − s ) s , OLOMORPHIC HERMITE FUNCTIONS AND ELLIPSES 5 we deduce that k ϕ k s = Z C | ϕ ( z ) | exp (cid:18) − − s s | z | + 1 + s s ( z + ¯ z ) (cid:19) L ( dz )= Z C | P ( z ) e − z / | exp (cid:18) − − s s | z | + 1 + s s ( z + ¯ z ) (cid:19) L ( dz )= Z C (cid:12)(cid:12)(cid:12)(cid:12) P ( z ) exp (cid:18) − s s z + (1 − s ) s z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − − s s | z | + 1 + s s ( z + ¯ z ) (cid:19) L ( dz )= Z C (cid:12)(cid:12)(cid:12)(cid:12) P ( z ) exp (cid:18) (1 − s ) s z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − − s s | z | (cid:19) L ( dz ) . If we change the variable by z = r s − s ζ, then we have k ϕ k s = s − s Z C (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18)r s − s ζ (cid:19) exp (cid:18)
14 1 − s s ζ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) e −| ζ | / L ( dζ ) (4) = s − s Z C (cid:12)(cid:12)(cid:12)(cid:12) ϕ (cid:18)r s − s ζ (cid:19) exp (cid:18) s − s ζ + 14 1 − s s ζ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) e −| ζ | / L ( dζ )= s − s Z C (cid:12)(cid:12)(cid:12)(cid:12) ϕ (cid:18)r s − s ζ (cid:19) exp (cid:18)
14 1 + s − s ζ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) e −| ζ | / L ( dζ ) . (5)If ϕ ∈ X s ( C ) , then ϕ (cid:18)r s − s z (cid:19) exp (cid:18)
14 1 + s − s z (cid:19) is holomorphic in C and belongs to H B ( C ) . Hence (5) shows that (1) is an injective and isometricmapping of X s ( C ) to H B ( C ) . In the same way one can show that (2) is an inverse of (1). Thus (1) isa Hilbert space isomorphism of X s ( C ) onto H B ( C ) , and its inverse is given by (2).Next we show that X s ( C ) and { ψ sn } ∞ n =0 are essentially determined by the ellipse ∂E ρ ( √ s, inthe framework H B ( C ) . This follows from (4) and the correspondence (3). For this reason, it sufficesto show the correspondence (3). If we choose ( α, β ) = ( √ s, , then we have for n = 0 , , , . . . , µ ( √ s,
0) = 1 − s s , λ ( √ s,
0) = 2 s − s , Ψ √ s, n ( z ) = (cid:26) exp (cid:18) s − s z (cid:19) (cid:18) ddz (cid:19) n exp (cid:18) − s − s z (cid:19)(cid:27) exp (cid:18)
14 1 − s s z (cid:19) = (cid:18) − r s − s (cid:19) n H n (cid:18)r s − s z (cid:19) exp (1 − s ) s (cid:18)r s − s z (cid:19) ! . By using this, we deduce that ψ sn ( z ) exp (cid:18) − − s s | z | + 1 + s s z (cid:19) = b nn ( s ) − / H n ( z ) e − z / exp (cid:18) − − s s | z | + 1 + s s z (cid:19) = b nn ( s ) − / H n ( z ) exp (cid:18) − − s s | z | + (1 − s ) s z (cid:19) H. CHIHARA = − r − s s ! n b nn ( s ) − / Ψ √ s, n r − s s z ! exp (cid:18) − (1 − s ) s z (cid:19) × exp (cid:18) − − s s | z | + (1 − s ) s z (cid:19) = − r − s s ! n b nn ( s ) − / Ψ √ s, n r − s s z ! exp (cid:18) − − s s | z | (cid:19) = − r − s s ! n b nn ( s ) − / Ψ √ s, n r − s s z ! exp − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r − s s z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which is a desired equation (3).Finally, we show that the reproducing kernel K s ( z, ζ ) can be obtained by the reproducing formula ϕ = B ◦ B ∗ ϕ for ϕ ∈ H B ( C ) . Let ϕ ∈ X s ( C ) . Then ϕ (cid:18) s − s z (cid:19) exp (cid:18)
14 1 + s − s z (cid:19) ∈ H B ( C ) . Substitute this into the reproducing formula for H B ( C ) . Then we have ϕ (cid:18) s − s z (cid:19) exp (cid:18)
14 1 + s − s z (cid:19) = 12 π Z C e z ¯ ζ ϕ (cid:18) s − s ζ (cid:19) exp (cid:18)
14 1 + s − s ζ (cid:19) e −| ζ | / L ( dζ ) . Hence, ϕ (cid:18) s − s z (cid:19) = 12 π Z C exp (cid:18) z ¯ ζ −
14 1 + s − s z + 14 1 + s − s ζ (cid:19) ϕ (cid:18) s − s ζ (cid:19) e −| ζ | / L ( dζ ) . By using the change of variable z r s − s z for z and ζ , we deduce that ϕ ( z ) = 1 − s πs Z C exp (cid:18) − s s z ¯ ζ − s s ( z − ζ ) (cid:19) ϕ ( ζ ) exp (cid:18) − − s s | ζ | (cid:19) L ( dζ )= 1 − s πs Z C exp (cid:18) − s s z ¯ ζ − s s ( z + ¯ ζ ) (cid:19) × ϕ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ )= Z C K s ( z, ζ ) ϕ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ ) . This completes the proof. (cid:3) X s ( C ) AND THE B ARGMANN - TYPE TRANSFORMS
In this section we study the function space X s ( C ) and some related topics from a view point ofSj¨ostrand’s theory of microlocal analysis based on the Bargmann-type transforms. We can choose thephase function φ ( z, x ) so that Φ( z ) = 1 − s s | z | − s s ( z + ¯ z ) . (6) OLOMORPHIC HERMITE FUNCTIONS AND ELLIPSES 7
Indeed, if the constants a , b and c satisfy − s s = | b | c , s s = b c + a i , (7)then (6) holds. There are uncountably many choices of the triple ( a, b, c ) satisfying (7). For example,the choice of a = is , b = ± i p − s , c = t + is ( t ∈ R ) satisfies the condition (6). Moreover, if the condition (7) is satisfied, then Ψ( z, ζ ) = 1 − s s zζ − s s ( z + ζ ) , and C Φ = | b | π Im c = 1 − s πs . Thus we have just proved the following theorem.
Theorem 4.
If we choose ( a, b, c ) satisfying (7) , then X s ( C ) = H Φ ( C ) and T ◦ T ∗ F ( z ) = Z C K s ( z, ζ ) F ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ ) , F ∈ L ( C ) . The reproducing kernel Hilbert space X s ( C ) and the system of holomorphic Hermite functions { ψ sn } ∞ n =0 have been applied to studying quantization on the complex plane, counting and combina-torics, and etc. In the study of quantization on C , Twareque Ali, G ´orska, Horzela and Szafraniecconstructed a Hilbert space isomorphism of X s ( C ) onto H B ( C ) and its inverse concretely. Moreprecisely, they constructed an integral transform of X s ( C ) onto H B ( C ) . They obtained its integralkernel by using the generating function of Hermite functions. Their idea works well since a system ofmonomials { z n / √ π n +1 n ! } ∞ n =0 is a complete orthonormal system of H B ( C ) . See their paper [5, Sec-tion III] for the detail.Recall that B : L ( R ) → H B ( C ) and T : L ( R ) → H Φ ( C ) are Hilbert space isomorphisms.By using this fact, one can construct uncountably many Hilbert space isomorphisms of X s ( C ) onto H B ( C ) . Then we have the following theorem. Theorem 5.
If we choose ( a, b, c ) satisfying (7) , then B ◦ T ∗ is a Hilbert space isomorphism of X s ( C ) onto H B ( C ) . Finally we will see some examples of Theorem 5.
Theorem 6. • If a = i/s , b = − i √ − s and c = is , then X s ( C ) = H Φ ( C ) , and for any ψ ∈ X s ( C ) B ◦ T ∗ ψ ( z ) = Z C G ( z, ζ ) ψ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ ) ,G ( z, ζ ) = √ − s / πs / exp r − s s z ¯ ζ + 1 − s s ) z − − s + s s ¯ ζ ! . • If a = is , b = √ − s and c = is , then X s ( C ) = H Φ ( C ) , and for any ψ ∈ X s ( C ) B ◦ T ∗ ψ ( z ) = Z C G ( z, ζ ) ψ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ ) ,G ( z, ζ ) = √ − s / πs / exp − i r − s s z ¯ ζ + 1 − s s ) z −
12 ¯ ζ ! . H. CHIHARA
Proof.
We will check only the first example. The second one can be proved in the same way. We hereomit the detail. Suppose that a = i/s , b = i √ − s and c = is . Then T ∗ ψ ( x ) = √ − s / π / s / Z C exp (cid:18) − s ¯ ζ + p − s ¯ ζx − s x (cid:19) × ψ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ ) , ψ ∈ X s ( C ) . We will reduce the explicit formula of B ◦ T ∗ ψ . Applying B on T ∗ ψ , we have B ◦ T ∗ ψ ( z ) = √ − s π / s / Z C (cid:18)Z R exp (cid:18) zx − x + p − s ¯ ζx − s x (cid:19) dx (cid:19) × exp (cid:18) − z − s ¯ ζ (cid:19) ψ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ ) . Since zx − x + p − s ¯ ζx − s x = − s (cid:26) x −
11 + s ( z + p − s ¯ ζ ) (cid:27) + 12(1 + s ) ( z + p − s ¯ ζ ) , we deduce that B ◦ T ∗ ψ ( z ) = √ − s π / s / Z C Z R exp − s (cid:26) x −
11 + s ( z + p − s ¯ ζ ) (cid:27) ! dx ! × exp (cid:18) − z − s ¯ ζ − s ) ( z + p − s ¯ ζ ) (cid:19) × ψ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ )= √ − s π / s / Z C (cid:18)Z R exp (cid:18) − s x (cid:19) dx (cid:19) × exp r − s s z ¯ ζ + 1 − s s ) z − − s + s s ¯ ζ ! × ψ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ )= √ − s / πs / Z C exp r − s s z ¯ ζ + 1 − s s ) z − − s + s s ¯ ζ ! × ψ ( ζ ) exp (cid:18) − − s s | ζ | + 1 + s s ( ζ + ¯ ζ ) (cid:19) L ( dζ ) . This completes the proof. (cid:3) A CKNOWLEDGEMENTS
The author is very grateful to Professor Franciszek Hugon Szafraniec for being interested in themanuscript [7] and for teaching the author recent topics on holomorphic Hermite functions kindly.This was a chance to write the present paper since the author knew nothing about them.
OLOMORPHIC HERMITE FUNCTIONS AND ELLIPSES 9 R EFERENCES [1] Van Eijndhoven SJL, Meyers JL. New orthogonality relations for the Hermite polynomials and related Hilbert spaces.J Math Anal Appl. 1990;146:89–98.[2] Gelfand IM, Shilov GE. Generalized Functions Vol.2. New York (NY): Academic Press; 1968.[3] Szafraniec FH. Analytic models of the quantum harmonic oscillator. Contemp Math. 1998;212:269–276.[4] Gazeau JP, Szafraniec FH. Holomorphic Hermite polynomials and a non-commutative plane. J Phys A: Math Theor.2011;44:495201 13pp.[5] Twareque Ali S, G´orska K, Horzela A, Szafraniec FH. Squeezed states and Hermite polynomials in a complex variable.J Math Phys. 2014;55:012107 11pp.[6] Ismail MEH, Simeonov P. Complex Hermite polynomials: their combinatorics and integral operators. Proc Amer MathSoc. 2014;143:1397–1410.[7] Chihara H. Bargmann-type transforms and modified harmonic oscillators. arXiv:1702.06646.[8] Folland GB. Harmonic Analysis in Phase Space. Princeton (NJ): Princeton University Press; 1989.[9] Sj¨ostrand J. Function spaces associated to global I-Lagrangian manifolds. In: Morimoto M, Kawai T, editors. Structureof Solutions to Differential Equations, Katata/Kyoto 1995. River Edge (NJ): World Scientific Publishing,; 1996. p.369–423.[10] Chihara H. Bounded Berezin-Toeplitz operators on the Segal-Bargmann space. Integral Equations Operator Theory.2009;63:321–335.D
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