The Bargmann Transform and Windowed Fourier Localization

Transkript

1 Integr. equ. oper. theory 57 (007, c 006 Brhäuser Verlag Basel/Swtzerland X/ , publshed onlne December 6, 006 DOI 0.007/s Integral Equatons and Operator Theory The Bargmann Transform and Wndowed Fourer Localzaton Mn-Ln Lo Abstract. We consder the relatonshp between Gabor-Daubeches wndowed Fourer localzaton operators L w ϕ and Berezn-Toepltz operators T ϕ,usngthe Bargmann sometry β. For wndow w a fnte lnear combnaton of Hermte functons, and a very general class of functons ϕ, we prove an equvalence of the form βl w ϕ β C M ϕc T (I+Dϕ by obtanng the exact formulas for the operator C and the lnear dfferental operator D. Mathematcs Subect Classfcaton ( B35, 4C40, 8R30. Keywords. Berezn-Toepltz operator, Bargmann sometry, wndowed Fourer localzaton.. Introducton We begn wth two Hlbert spaces. The frst s the usual space L (R n,dv of Lebesgue square-ntegrable complex functons on real Eucldean space. The second s the space H (C n,dµ of entre functons on complex n-space whch are squarentegrable wth respect to the measure dµ(z (π n exp{ z /dv(z (dv s Lebesgue volume measure. Consder the natural orthonormal bass for L (R n,dv consstng of Hermte functons ([8], p.5; [], pp. 05, 06 w π n 4 (! H (xe x, where ranges over all n-tuples of non-negatve ntegers,!!! n! + + n,az a z + a z + + a n z n,z zz,a z a z wth z z z Ths analyss s based on some papers by L. A. Coburn [8, 9, 0].

2 398 Lo IEOT for z the complex conugate of z and H (x ( e x D e x, for D ( x ( x n n. There s a correspondng natural orthonormal bass for H (C n,dµ consstng of monomals e (! / z, where z z z zn n. In what follows, we wll also wrte n ( z ( z n n as well as n n ( z ( z n n. For f n H (C n,dµ, [3] began a systematc study of densely defned operators of the form: (T ϕ f(z e z a/ ϕ(af(adµ(a, C n e ( z/ ϕ( s assumed to be n L (C n,dµ for all z n C n. The (possbly unbounded operator T ϕ s called the Berezn-Toepltz operator wth symbol ϕ. Note that the Berezn-Toepltz operator T ϕ maps ts doman nto H (C n,dµ. The orthogonal proecton operator from L (C n,dµ onto the closed subspace H (C n,dµsgven by (Pf(z f,e a z/ e z a/ f(adµ(a. C n Restrcton to f n H (C n,dµ shows that H (C n,dµ s a reproducng ernel Hlbert space wth reproducng ernel e z a/.thus,form ϕ the operator of multplcaton by ϕ onl (C n,dµ, we see that T ϕ PM ϕ P.Moreover,T z and T z M z, at least on fnte lnear combnatons of the {e. There s an nterestng connecton between Berezn-Toepltz operators on H (C n,dµandgabor-daubeches wndowed Fourer localzaton operators on L (R n,dv. To be precse, the Gabor-Daubeches localzaton operator L w ϕ wth symbol (or weght functon ϕ and wndow w, s densely defned on L (R n,dvby L w ϕf,g (π n C n ϕ(aβf,w a βw W a βw,βg dv(a, where β s the Bargmann sometry ([]; [8], pp. 40, 47, whch s surectve, wth βw e and β : L (R n,dv H (C n,dµ, (W a f(z a (zf(z a e z a/ a /4 f(z a. n Typcally, the wndow w s chosen to be a unt vector n L (R n,dv. For w 0 π n 4 e x, βw 0 and t s not dffcult to verfy that βl w0 ϕ β T ϕ. Thus, the Bargmann sometry allows the amalgamaton of the substantal wor already done n analyzng T ϕ [,6,3,4,5,7,8,,0,]andL w0 ϕ [, 3, 4, 5, 6, 7].

3 Vol. 57 (007 Wndowed Fourer Localzaton 399 In a paper by L. A. Coburn [0], he checed for ϕ ether an arbtrary polynomal n z and z, or a functon n B a (C n (the algebra of Fourer-Steltes transforms of compactly supported, complex, bounded regular Borel measures on C n, that βl w (,0,,0 ϕ β T (I+ ϕ, βl w (,0,,0 ϕ β T (I+4 +( ϕ. Hs most general result was: Let p(z be an arbtrary polynomal n H (C n,dµandletϕ be a functon such that e ( a/ ϕ( snl (C n,dµ for all a n C n. Then, for wndow w a w wth rangng over a fnte set of non-negatve mult-ndces and a, βl w ϕ β p(a C M ϕ Cp(a, for a precsely determned operator C C(w. It follows that there s a lnear dfferental operator D D(w, wth coeffcents whch are polynomals n z and z and wth no dervatve- free term, such that βl w ϕ β p(z C M ϕ Cp(z T (I+Dϕ p(z, for ϕ an arbtrary polynomal n z and z. Coburn conectured that D(w was actually a constant coeffcent lnear dfferental operator and that the stated result held for all ϕ n B a (C n as well. In the next secton of ths artcle, we begn wth proofs of the result outlned above for w Hermte functon w,andϕ n the Schwartz space S. Thethrd secton deals wth more general wndows w whch are fnte lnear combnatons of Hermte functons for the same type of symbol ϕ as n secton two. In the fnal secton, we extend the above result to ϕ n a more general class of functons E(C n whch ncludes the space B a (C n, as well as all polynomals ϕ.. Computatons for Hermte Wndows For p(z an arbtrary polynomal n H (C n,dµ, and β the Bargmann sometry, our frst result s Theorem.. For ϕ such that e ( a/ ϕ( s n L (C n,dµ for all a n C n, we have βl w ϕ β p(z! C M ϕc p(z wth C 0 ( (I P M z PM z P, where 0 means sum over all wth 0, 0,, 0 n n and ( ( ( ( n n. Formally (up to doman consderatons,! C M ϕc { ( ( ( + T! z T z ϕ z. T z,0

4 400 Lo IEOT Proof. Note that βw e! z.weobservethat p(z,e z a/ (z a [ ( ] e ( a z/ z ā p(zdµ(z C n 0 { ( ā PM z p (a 0 { ( (I P M z PM z Pp (a 0 ( C p (a. Now, for p(z and q(z arbtrary polynomals n z, βl w ϕ β p, q ϕ(ap(z,ez a/ (z a e z a/ (z a,q(z dµ(a C! n C! ϕ(a( C p (a ( C q (adµ(a n! M ϕc p, C q and ϕ(a ( C p (a s easly seen to be n L (C n,dµsnce ( C p (a sapolynomal n a, ā. Thus, We chec drectly that βl w ϕ β p(z! C M ϕ C p(z. C M ϕc p(z { ( PM z PM z (I P 0 { 0 { 0 { 0 M ϕ ( (I P M z PM z P p(z ( T z PM z ( M z T z 0 0 ( T z T z P M ϕ ( ( T z T z p(z

5 Vol. 57 (007 Wndowed Fourer Localzaton 40 { 0 { 0 { 0 {,0 ( T z PM z + T z P M ϕ ( M z T z + T z p(z ( T z PM z ( ( M ϕ { 0 ( + T z T z ϕ z T z p(z. ( M z T z p(z The Schwartz space S conssts of those C functons whch, together wth all ther dervatves, vansh at nfnty faster than any power of z. More precsely, for any nonnegatve nteger N and any mult-ndex α, we defne and then f (N,α sup z C n ( + z N α f(z ; S { f C : f (N,α < for all N,α. Lemma.. For ϕ n S, ϕ(z χ a (zˇϕ(adv(a, C n where χ a (z exp{im(z a. Proof. For ϕ n S, ˇϕ s n S snce the nverse Fourer transform s an somorphsm of S onto tself. By the Fourer Inverson Theorem, ϕ(z χ C n a (zˇϕ(adv(a. Usng Theorem., Lemma., and the dentty W a e a /4 T χa( from [3], we show Theorem.3. For ϕ n S, andp(z an arbtrary polynomal n H (C n,dµ, we have βl w ϕ β p(z T { m0 m ( m ( m! ( m ϕ p(z. Proof. By drect calculaton,! C M χa( C p(z! e a { 4,0! e a { 4,0 ( ( ( ( ( + T z T z W a T z T z p(z ( + z [K a (z(z a p(z a]

6 40 Lo IEOT (to smplfy the notaton, we use c to denote! e a /4 { c,0 { c l0 0 ( ( ( ( + z ( + z 0 l0 ( l [K a (z(z a ] +l p(z a l ( ( l [K a (z(z a ] +l p(z a l (Rendexng by m + l { ( c ( + z 0 ( m ( ( r ( r r m r m0 r0 {{ { c 0 0 ( m m r0 ( r ( m r, f m 0; 0, otherwse. ( + z p(z a [K a (z(z a ] { c ( + z p(z a ck a (zp(z a m0 (ā m m m0! ( + z (z a m ( m! {{ m ( m m! ( m! e a 4 K a (zp(z a [z (z a] m {{ ( a m ( m { m0 ( m ( m! m0 ( ā ( m (ā m p(z a m [K a (z(z a ] m K a (z m (z a m {{{{ ( ā m! K a(z ( m! (z a m m ( m! ( m+ m m a m e a /4 W a p(z,

7 Vol. 57 (007 Wndowed Fourer Localzaton 403 so that { m0 m ( m!! C M χ a( C p(z ( ā m ( a m m T χa( p(z. Integratng both sdes of the operator equaton above wth respect to dσ(a, for dσ of the form dσ ˇϕdv for ϕ n S (see Lemma.gves! C M ϕc p(z T { m0 m ( m ( m! ( m ϕ p(z for ϕ(z χ a (zdσ(a. C n Operator ntegrals are understood n the wea sense and Fubn s Theorem s requred. Note that m m ( m! ( m ϕ(z m0 C n { m0 m ( m! (ā m m a m χ a (zdσ(a by dfferentaton under the ntegral. By Theorem., we now have βl w ϕ β p(z! C M ϕc p(z T { m0 m ( m ( m! ( m ϕ p(z. 3. Computatons for more general wndows For w a fnte lnear combnaton of Hermte functons w, we can replcate the analyss of Theorem. and Theorem.3. Theorem 3.. Let p(z be an arbtrary polynomal n H (C n,dµ and let ϕ be a functon such that e ( a/ ϕ( s n L (C n,dµ for all a n C n. Then, for wndow w A w wth rangng over a fnte set of non-negatve mult-ndces, we have ( βl w ϕβ A ( Ā p(z h! C M ϕ h h! C h p(z wth (C p(a p(z,e z a/ (z a, h

8 404 Lo IEOT and h rangngoverthesamefntesetofnon-negatvemult-ndces. Formally, h ( h C M ϕc h p(z ( + T z T z ϕ z T z h p(z. Proof. Note that βw 0 0 A! z.weobservethat p(z,e z a/ (βw(z a p(z,e z a/ A! (z a Ā! p(z,ez a/ (z a ( Hence, for p(z, q(z arbtrary polynomals n z, βl w ϕβ p, q Ā! C p (a. ϕ(ap(z,e z a/ (βw(z a e z a/ (βw(z a,q(z dµ(a C n ( Ā ( h ϕ(a C n h h! C Ā hp (a! C q (adµ(a M ϕ h h Ā h h h! C hp, Ā! C q and ϕ(a(c h p(a s easly seen to be n L (C n,dµ, snce (C h p(asapolynomal n a,ā. Thus, ( βl w A ( Ā h ϕ β p(z! C M ϕ h h h! C h p(z. We chec drectly that (smlar to the proof of Theorem. CM ϕ C h p(z [ ( ] ( T z PM z 0 h 0 0 ( ( h M ϕ h ( h 0 ( + T z T z ϕ z T z h p(z. ( M z T z h p(z Theorem 3.. For ϕ n S, p(z an arbtrary polynomal n H (C n,dµ, andw A w wth rangng over a fnte set of non-negatve mult-ndces, we have βl w ϕβ p(z T A { Ā h h, (! h+ mn(h, h 0 ( ( h! p(z, h ϕ h!

9 Vol. 57 (007 Wndowed Fourer Localzaton 405 wth h rangng over the same fnte set of non-negatve mult-ndces as. Proof. From Theorem 3., and usng the dentty W a e a /4 T χa(, we fnd CM χa( C h p(z h ( ( h 0 0 h e a / ( + T z T z T χa( T z T z h p(z h ( ( h ( + z [ K a (z(z a h p(z a ] (to smplfy notaton, we use c to denote h e a /4 n ths proof h ( h c ( +h z ( h l0 0 ( l [ K a (z(z a ] h ( l p(z a l (usng step ( n Theorem.3 c ( +h z 0 ( h t ( ( h h r ( r h r t r t0 r0 {{ c 0 ( h t t r0 ( r ( r t ( +h z c ( +h z 0 ( h (ā 0 ( h ck a (zp(z a h Ka (z, f t 0; 0, otherwse. h ( h 0! (! h t [ K a (z(z a ] t p(z a h K a (z (z a p(z a (z a p(z a (ā h ( ( h+ 0 ( ( (! (usng the dentty (!!! (! z (z a

10 406 Lo IEOT ck a (zp(z a 0 mn(h, ( h+ Thus, we have 0 mn(h, ( h+ 0 ( h ( ā ( ( h ( +h ( h! ( ā C M χ a( C h p(z ( h! ( ā!(z (z a h a e a /4 W a p(z h a T χa( p(z. Integratng both sdes of the operator equaton above wth respect to dσ(a, for dσ of the form dσ ˇϕdv for ϕ S (see Lemma. gves CM ϕ C h p(z T { ( h+ mn(h, 0 ( ( h! p(z h ϕ for ϕ(z χ a (zdσ(a. C n Operator ntegrals are understood n the wea sense and Fubn s Theorem s requred. Note that mn(h, ( h ( h+! h ϕ 0 ( h (ā h a! χ a (zdσ(a mn(h, ( h+ C n 0 by dfferentaton under the ntegral. By Theorem 3., t follows that βl w ϕβ p(z A Ā h h,! h h! C M ϕ C h p(z T A { Ā h h, (! h+ mn(h, h 0 ( ( h! p(z. h ϕ h! By applyng the same proof above to ϕ(z C n χ a (zdσ(a, where σ s a compactly supported, regular, bounded complex-valued Borel measure, we can prove that the result of Theorem 3. holds for ϕ n B a (C n as well.

11 Vol. 57 (007 Wndowed Fourer Localzaton Computatons for more general symbols In ths secton, we use cut-off functons to prove that Coburn s conectured result holds for ϕ n E(C n, where E(C n { ϕ C (C n : for any mult-ndex, there exst constants M M( andα α( such that (D ϕ(z Me α z. Frst, we construct approprate cut-off functons. Tae a C - functon η on C n whch has the followng propertes: ( 0 η on C n ; ( η(z when z ; ( η(z 0 when z. Clearly, η s n Cc (Cn. Now, for each m N, letη m (z η( z m, then ( 0 η m on C n ; ( η m (z when z m; ( η m (z 0 when z m. Note that η m s stll n Cc (Cn. If ϕ s n E(C n, then η m ϕ s n Cc (Cn S. By Theorem 3., we now have Lemma 4.. For ϕ n E(C n, p(z an arbtrary polynomal n H (C n,dµ, and w A w wth rangng over a fnte set of non-negatve mult-ndces, we have βl w η mϕβ p(z T A { Ā h h, (! h+ mn(h, h 0 ( ( h! p(z, h (η mϕ h! wth h rangng over the same fnte set of non-negatve mult-ndces as. In addton to Lemma 4., we also need the followng lemmas to prove Coburn s conectured result for ϕ n E(C n. Lemma 4.. For p, q polynomals n H (C n,dµ, there exst postve constants K, ɛ such that W a p, q Ke ɛ a. Proof. W a p, q C n e z a/ a C n e z a/ a /4 p(z aq(zdµ(z /4 p(z aq(z(π n e z / dv(z. By completng the square, we can rewrte z / z a/+ a /4 wth ts real part equal to ( z + z a /4. For p, q polynomals n H (C n,dµ, we can fnd postve constants M and M such that p(z M e z /8,

12 408 Lo IEOT and q(z M e z /8. Thus, we have W a p, q (π n p(z a q(z e z /4 z a /4 dv(z C n (π n M M e ( z + z a /8 dv(z C n (π n M M e ( z /+ a /4/8 dv(z C n (π n M M e a /3 /6 dv(z. C n e z Lemma 4.3. For ϕ n E(C n,andw, f, g fnte lnear combnatons of Hermte functons, L w ηmϕf,g L w ϕf,g as m. Hence, for p, q polynomals n H (C n,dµ, βl w ηmϕ β p, q βl w ϕ β p, q as m. Proof. Let L w ηmϕ f,g (π n C n η m (aϕ(aβf,w a βw W a βw,βg dv(a. h m (a η m (aϕ(aβf,w a βw W a βw,βg, observe that And h m (a h(a ϕ(aβf,w a βw W a βw,βg pontwse, as m. h m (a ϕ(a ce ɛ a for some c, ɛ > 0 by Lemma 4. Me α a ɛ a for some M,α > 0snceϕ s n E(C n. Snce Me α a ɛ a s n L (C n,dv, we can apply the domnated convergence theorem to the sequence {h m to obtan lm L w f,g m ηmϕ lm m (π n h m (adv(a C n (π n C n h(adv(a L w ϕf,g.

13 Vol. 57 (007 Wndowed Fourer Localzaton 409 For p(z, q(z polynomals n H (C n,dµ, lm m βl w ηmϕβ p, q lm m L w ηmϕβ p, β q L w ϕβ p, β q βl w ϕ β p, q. Lemma 4.4. For ϕ n E(C n, p(z an arbtrary polynomal n H (C n,dµ, we have T (I+Dηmϕp(z T (I+Dϕ p(z n L (C n,dµ as m, where the dfferental operator (I + D { h, mn(h, A Ā h! ( h h h! ( h+! h, wth h, rangng over a fnte set of non-negatve mult-ndces. Hence, for p(z, q(z polynomals n H (C n,dµ, T(I+Dηmϕp, q T (I+Dϕ p, q as m. 0 Proof. It suffces to show that T α β η mϕp(z T α β ϕp(z nl (C n,dµasm, forp(z an arbtrary polynomal n H (C n,dµ. β (η m (zϕ(z β ( β 0 β η m (z ϕ(z η m (z β ϕ(z+ 0 <β η m (z β ϕ(z+ 0 <β ( β ( z β η m ϕ(z ( ( β β ( z ( β η m m ϕ(z. α β (η m (zϕ(z α ( η m (z β ϕ(z + η m (z α β ϕ(z+ 0 <β ( ( β m 0 <β 0 <α ( α ( ( β β [ ( z ] α ( β η m m ϕ(z ( m β α [ ( β η α ( ( z α η β ϕ(z+ m ( z ] m ϕ(z

14 40 Lo IEOT η m (z α β ϕ(z+ 0 <β 0 <α ( ( β β m ( ( α α ( m ( z α η β ϕ(z+ m 0 α ( ( α α ( α β η ( z ϕ(z. m m For each fxed par of mult-ndces α and β, ( α β η (z s bounded for all z C n.thus,for n 0, ( n ( α m β η ( z m γ δ ϕ(zp(z 0nL (C n,dµ for any polynomal p(z nh (C n,dµ, as m.wethenhave α β (η m (zϕ(z p(z η m (z ( α β ϕ (zp(z 0asm. Snce η m (z pontwse and ( α β ϕ (zp(z snl (C n,dµforp(z a polynomal n H (C n,dµandϕ(z ne(c n, η m (z ( α β ϕ (zp(z ( α β ϕ (zp(z 0asm, by the domnated convergence theorem. Thus, lm m α β (η m (zϕ(z p(z ( α β ϕ(z p(z lm m α β (η m (zϕ(z p(z η m (z ( α β ϕ(z p(z So, +η m (z ( α β ϕ(z p(z ( α β ϕ(z p(z lm α β (η m (zϕ(z p(z η m (z ( α β ϕ(z p(z m + lm η m(z ( α β ϕ(z p(z ( α β ϕ(z p(z m 0. α β (η m (zϕ(z p(z ( α β ϕ(z p(z nl (C n,dµasm. Snce the proecton operator s contnuous, lm T m α β (η m(zϕ(zp(z lm m PM α β (η m(zϕ(zp(z lm P ( α β (η m (zϕ(z p(z m P ( ( α β ϕ(z p(z PM α ϕ(zp(z β T α ϕ(zp(z, β where the convergence s n L (C n,dµ. By Lemma 4., Lemma 4.3, and Lemma 4.4, we have now extended the result for ϕ n E(C n.

15 Vol. 57 (007 Wndowed Fourer Localzaton 4 Theorem 4.5. For ϕ n E(C n, p(z an arbtrary polynomal n H (C n,dµ, and w A w wth rangng over a fnte set of non-negatve mult-ndces, we have βl w ϕβ p(z T A { Ā h h, (! h+ mn(h, h 0 ( ( h! p(z, h ϕ h! wth h rangng over the same fnte set of non-negatve mult-ndces as. Corollary 4.6. For ϕ an arbtrary polynomal n z and z, p(z an arbtrary polynomal n H (C n,dµ, and w A w wth rangng over a fnte set of non-negatve mult-ndces, we have βl w ϕβ p(z T A { Ā h h, (! h+ mn(h, h 0 ( ( h! p(z, h ϕ h! wth h rangng over the same fnte set of non-negatve mult-ndces as. Proof. Ths follows drectly from Theorem 4.5 snce ϕ s n E(C n. Acnowledgment The author would le to than Dr. Lews A. Coburn and Dr. Jngbo Xa for ther valuable advce. References [] V. Bargmann, On a Hlbert space of analytc functons and an assocated ntegral transform, Communcatons on Pure and Appled Mathematcs, 4 (96, [] F. A. Berezn, Covarant and contravarant symbols of operators, Math. USSR Izv., 6 (97, 7-5. [3] C. A. Berger and L. A. Coburn, Toepltz operators and quantum mechancs, Journal of Functonal Analyss, 68 (986, [4] C. A. Berger and L. A. Coburn, Toepltz operators on the Segal-Bargmann space, Transactons AMS, 30 (987, [5] C. A. Berger and L. A. Coburn, Heat Flow and Berezn-Toepltz Estmates, Amercan Journal of Mathematcs,6 (994, [6] D. Borthwc, Mcrolocal technques for semclasscal problems n geometrc quantzaton. Perspectves on Quantzaton (edtors: L. A. Coburn and M. A. Reffel Contemporary Math. 4, AMS, Provdence, R.I. (998, [7] L. A. Coburn, The measure algebra of the Hesenberg group, Journal of Functonal Analyss, 6 (999, [8] L. A. Coburn, On the Berezn-Toepltz calculus, Proceedngs of the AMS, 9 (00, [9] L. A. Coburn, The Bargmann sometry and Gabor-Daubeches wavelet localzaton operators, Systems, approxmaton, sngular ntegrals and related topcs (edtors: A. Borchev and N. Nols Operator Theory: Advances and Applcatons 9, Brhauser, Basel (00, [0] L. A. Coburn, Symbol calculus for Gabor-Daubeches Wndowed Fourer localzaton operators, preprnt.

16 4 Lo IEOT [] L. A. Coburn and J. Xa, Toepltz algebras and Reffel deformatons, Communcatons n Mathematcal Physcs, 68 (995, [] I. Daubeches, Tme frequency localzaton operators: A geometrc phase space approach, IEEE Transactons on Informaton Theory, 34 (988, [3] I. Daubeches, The wavelet transform, tme-frequency localzaton and sgnal analyss, IEEE Transactons on Informaton Theory, 36 (990, [4] I. Daubeches, Ten Lectures on Wavelets, CBMS-NSF Regonal Conference Seres 6, SIAM, Phladelpha, 99. [5] F. De Mar, H. G. Fechtnger and K. Nowa, Unform egenvalue estmates for tmefrequency localzaton operators, Journal of the London Mathematcal Socety, 65 (00, [6] J. Du and M. W. Wong, Gaussan functons and Daubeches operators, Integral Equatons and Operator Theory, 38 (000, -8. [7] H. G. Fechtnger and K. Nowa, A Szego type theorem for Gabor-Toepltz localzaton operators, The Mchgan Mathematcal Journal, 49 (00, 3-. [8] G. B. Folland, Harmonc analyss n phase space, Annals of Mathematcs Studes, Prnceton Unv. Press, Prnceton, 989. [9] G. B. Folland, Real analyss, Wley, New Yor, 999. [0] V. Gullemn, Toepltz operators n n-dmensons, Integral Equatons and Operator Theory, 7 (984, [] G. Szegö, Orthogonal polynomals, AMS Colloquum Publcatons, 3, AMS, Provdence, RI., 975. [] J. Xa and D. Zheng, Standard devaton and Schatten class Hanel operators on the Segal-Bargmann space, Indana Unversty Mathematcs Journal, 53 (004, Mn-Ln Lo Department of Mathematcs Calforna State Unversty, San Bernardno 5500 Unversty Parway San Bernardno, CA 9407 USA e-mal: mlo@csusb.edu Submtted: October 3, 005 Revsed: October 9, 006