Uncertainty Principles for the Segal-Bargmann Transform

Transcription

1 Journal of Mathematical Research with Applications Sept, 017, Vol 37, No 5, pp DOI:103770/jissn: Uncertainty Principles for the Segal-Bargmann Transform Fethi SOLTANI Department of Mathematics, Faculty of Science, Jazan University, P O Box 77, Jazan 4514, Saudi Arabia Abstract We recall some properties of the Segal-Bargmann transform; and we establish for this transform qualitative uncertainty principles: local uncertainty principle, Heisenberg uncertainty principle, Donoho-Stark s uncertainty principle and Matolcsi-Szücs uncertainty principle Keywords Segal-Bargmann space; Segal-Bargmann transform; local uncertainty principle; Heisenberg uncertainty principle; Donoho-Stark s uncertainty principle MR(010) Subject Classification 30H0; 3A15 1 Introduction Many uncertainty principles have already been proved for the Fourier transform: Heisenberg- Pauli-Weyl inequality [1 3], Cowling-Price s inequality [], local uncertainty inequality [4 6], Donoho-Stark s inequality [7], Benedicks inequality [8] and Matolcsi-Szücs inequality [9] Laeng- Morpurgo [10] and Morpurgo [11] obtained Heisenberg inequality involving a combination of L 1 - norms and L -norms Folland-Sitaram [1], next Nemri-Soltani [13 15] proved a general forms of the Heisenberg-Pauli-Weyl inequality and the Donoho-Stark s inequality And one expects that the results in this work will be useful when discussing the uncertainty principles for the Segal-Bargmann transform Let H(C) denote the space of entire functions on C, and L (Ω), the space of measurable functions f on C satisfying f L (Ω) := C f(z) dω(z) <, where z = x + iy and dω(z) := e z π dxdy The Fock space F (C) (called also Segal-Bargmann space [16]) is the space of functions in H(C) L (Ω), equipped with the norm f F (C) := f L (Ω) This space was introduced by Bargmann in [17] and it was the tool of many works [16,18,19] Some uncertainty principles have already been proved on the Segal-Bargmann space F (C) by Kehe Zhu in his paper [0] Let L (µ) be the space of measurable functions on R, for which f L (µ) := f(z) dµ(z) <, Received September 8, 016; Accepted February 7, 017 address: fethisoltani10@yahoocom R

2 564 Fethi SOLTANI where dµ(z) := 1 π dz The Segal-Bargmann transform B is defined on L (µ) by B(f)(w) := 1/4 R exp ( w + z + wz ) f(z)dµ(z), w C This transform (see [17]) is an isometric isomorphism of L (µ) onto F (C) and has a pointwise inversion formula Next, it is the background of some applications in this work; especially, we give Heisenberg-Pauli-Weyl uncertainty principle, Donoho-Stark uncertainty principle and Matolcsi-Szücs uncertainty principle for the Segal-Bargmann transform B Building on the ideas of Faris [4] and Price [5,6] for the Fourier transform, we show a local uncertainty principle for the Segal-Bargmann transform B More precisely, we will show the following result Let W be a measurable subset of C such that 0 < m(w ) <, and a > 0 If f L (µ), then K 1 (a)(m(w )) a x a f L (µ), a (0, 1 ), χ W B(f) L (Ω) K (a)(m(w )) 1/ f 1 1 L (µ) x a f 1 L (µ), a > 1/, K 1 ( 1 4 )(m(w ))1/4 f 1/ L (µ) x 1/ f 1/ L (µ), a = 1/, where χ W is the characteristic function of the set W, dm(z) := 1 π dxdy and K 1(a), K (a) are positive constants given explicitly by Theorem 31 We shall use the local uncertainty principle for the Segal-Bargmann transform B to show for it the following Heisenberg-Pauli-Weyl uncertainty principle f L (µ) K(a, b) x a f b +b L (µ) w b B(f) +b L (Ω), a, b > 0, where K(a, b) is a positive constant given explicitly by Theorem 33 Next, by using Clarksontype inequality and Nash-type inequality for the Segal-Bargmann transform B we establish for it Heisenberg uncertainty principle involving L 1 (µ) and L (µ)-norms Finally, based on the techniques of Donoho-Stark [7], we will show uncertainty principles of concentration-type on L (µ) and on L 1 L (µ) for the Segal-Bargmann transform B And based on the ideas of Ghobber-Jaming [1] we deduce uncertainty principle of Matolcsi-Szücs-type involving L 1 (µ) and L (µ)-norms for the Segal-Bargmann transform B The contents of the paper are as follows In Section, we recall some properties of the Fock space F (C) and the Segal-Bargmann transform B In Section 3, we establish local uncertainty principle and Heisenberg-Pauli-Weyl uncertainty principle for the Segal-Bargmann transform B In Section 4, we show Heisenberg uncertainty principle involving L 1 (µ) and L (µ)-norms for the Segal-Bargmann transform B The last section is devoted to present Donoho-Stark uncertainty principle and Matolcsi-Szücs uncertainty principle for the Segal-Bargmann transform B Segal-Bargmann transform We denote by H(C) the space of entire functions on C

3 Uncertainty principles for the Segal-Bargmann transform 565 dω(z), the measure defined on C, by dω(z) := e z dxdy, z = x + iy π L q (Ω), 1 q <, the space of measurable functions f on C satisfying [ 1/q f Lq (Ω) := f(z) dω(z)] q < C We define the pre Hilbert space F (C), to be the space of functions in H(C) L (Ω), equipped with the inner product f, g F (C) := f(z)g(z)dω(z), and the squared norm f F (C) := C C f(z) dω(z) The following properties are proved in [17] (a) If f, g F (C) with f(z) = n=0 a nz n and g(z) = n=0 b nz n, then f, g F (C) = a n b n n! (b) The function K given for w, z C, by n=0 K(w, z) = e wz, is the reproducing kernel for the Fock space F (C), that is, (i) for every w C, the function z K(w, z) belongs to F (C); (ii) for all w C and f F (C), we have f, K(w, ) F (C) = f(w) (c) If f F (C), then f(w) e w / f F (C), w C (d) The space F (C) equipped with the inner product, F (C) is a Hilbert space; and the set { wn n! } n N forms a Hilbertian basis for the space F (C) We denote by L p (µ), 1 p <, the space of measurable functions on R, for which [ 1/p f Lp (µ) := f(z) dµ(z)] p < Here, µ is the measure defined on R by R dµ(z) := 1 π dz U the kernel given for w C and z R, by U(w, z) := 1/4 exp( w + z + wz) The kernel U satisfies the following properties [17]

4 566 Fethi SOLTANI (a) For w C and z R, we have (b) For w C and z R, we have U(w, z) 1/4 e w / (1) U(w, z) = where h n are the Hermite functions given by n=0 h n (z) n! w n, n/ h n (z) := 1/4 ( n n!) 1/ e z / ( 1) k k! (n k)! (z)n k, with n/ being the integer part of n/ (c) For all v, w C, we have e vw = R k=0 U(v, z)u(w, z)dµ(z) (d) For all w C, the function z U(w, z) L (µ), and U(w, ) L (µ) = e w () The kernel U gives rise to an integral transform B, which is called the Segal-Bargmann transform on C, and defined for f in L (µ), by B(f)(w) := U(w, z)f(z)dµ(z), w C From relations (1) and () the Segal-Bargmann transform satisfies the following properties R Theorem 1 (i) For f L 1 (µ) and w C we have (ii) For f L (µ) and w C we have The following theorem is proved in [17] Theorem F (C) In particular, we have B(f)(w) 1/4 e w / f L1 (µ) B(f)(w) e w / f L (µ) The Segal-Bargmann transform B is an isometric isomorphism of L (µ) onto B(f) F (C) = f L (µ), f L (µ) Let β denote the family of entire functions g such that { w sup g(w) exp( w C 4 + λ w )} <, for every λ > 0 Since { wn n! } n N β, the family β is dense in F (C) The family β simplifies the presentation of the inverse operator B 1 (see [17])

5 Uncertainty principles for the Segal-Bargmann transform 567 Theorem 3 (i) If g β, we have B 1 (g)(z) = C g(w)u(w, z)e w dm(w), z R (ii) If g F (C) and z R, we have B 1 (g)(z) = lim g n (w)u(w, z)e w dm(w), n where the sequence {g n } β and converges to g in F (C) 3 Heisenberg uncertainty principle C in L (µ)-sense, In this section we begin by establishing local uncertainty principle for the Segal-Bargmann transform B, more precisely we will show the following theorem Theorem 31 Let W be a measurable subset of C such that 0 < m(w ) <, and a > 0 If f L (µ), then K 1 (a)(m(w )) a x a f L (µ), a (0, 1 ), χ W B(f) L (Ω) K (a)(m(w )) 1/ f 1 1 L (µ) x a f 1 L (µ), a > 1/, K 1 ( 1 4 )(m(w ))1/4 f 1/ L (µ) x 1/ f 1/ L (µ), a = 1/, where 1 4a (1 )a 1 K 1 (a) = a π a/ a, K (a) = (4π)1/4 ( 1) sin( π ) Proof (i) Let f L (µ) The first inequality holds if x a f L (µ) = Assume that x a f L (µ) < By Minkowski s inequality, for all r > 0, But χ W B(f) L (Ω) χ W B(fχ x <r ) L (Ω) + χ W B(fχ x >r ) L (Ω) χ W B(fχ x <r ) L (Ω) + B(fχ x >r ) F (C) ( χ W B(fχ x <r ) L (Ω)) = By Theorem 1 (i), for w C, we have W B(fχ x <r )(w) dω(w)) 1/ B(fχ x <r )(w) 1/4 e w / fχ x <r L1 (µ) Thus, χ W B(fχ x <r ) L (Ω) 1/4 (m(w )) 1/ fχ x <r L 1 (µ) Hence, it follows from Theorem that χ W B(f) L (Ω) 1/4 (m(w )) 1/ fχ x <r L1 (µ) + fχ x >r L (µ) On the other hand, by Hölder s inequality and hypothesis a < 1/, fχ x <r L1 (µ) x a χ x <r L (µ) x a f L (µ)

6 568 Fethi SOLTANI Moreover, 1/4 r a+(1/) π 1/4 (1 ) 1/ x a f L (µ) fχ x >r L (µ) r a x a f L (µ) Thus, we deduce that χ W B(f) L (Ω) [ r a + 1/ (m(w )) 1/ π 1/4 (1 ) 1/ r a+(1/)] x a f L (µ) We choose r = π 1 (m(w )) 1, we obtain the first inequality (ii) The second inequality holds if f L (µ) = or x a f L (µ) = Assume that f L (µ) + x a f L (µ) < From [6] and hypothesis a > 1/, we have f L1 (µ) (π)1/4 ( 1) 1 4a sin( π ) f 1 1 L (µ) x a f 1 Then, according to this inequality we deduce that χ W B(f) L (Ω) 1/4 (m(w )) 1/ f L 1 (µ) L (µ) K (a)(m(w )) 1/ f 1 1 L (µ) x a f 1 L (µ), which gives the second inequality (iii) Let r > 0 From the inequality ( x r )1/4 1 + ( x r )1/, it follows that Optimizing in r, we get x 1/4 f L (µ) r 1/4 f L (µ) + r 1/4 x 1/ f L (µ) Thus, we deduce that x 1/4 f L (µ) f 1/ L (µ) x 1/ f 1/ L (µ) χ W B(f) L (Ω) K 1 ( 1 4 )(m(w ))1/4 x 1/4 f L (µ) which gives the result for a = 1/ Remark 3 Let a > 0 If f L (µ), then K 1 ( 1 4 )(m(w ))1/4 f 1/ L (µ) x 1/ f 1/ L (µ), B(f) L 1, (Ω) K 1(a) x a f L (µ), 0 < a < 1/, B(f) L, (Ω) K (a) f 1 1 L (µ) x a f 1 L (µ), a > 1/, B(f) L 4, (Ω) K 1 ( 1 4 ) f 1/ L (µ) x 1/ f 1/ L (µ), a = 1/, where L s,q (Ω) is the Lorentz-space defined by the norm f L s,q (Ω) := sup W C 0<m(W )< ( (m(w )) 1 s 1 q χw f L q (Ω))

7 Uncertainty principles for the Segal-Bargmann transform 569 We shall use the local uncertainty principle (Theorem 31) to obtain the following Heisenberg- Pauli-Weyl uncertainty principle for the Segal-Bargmann transform B, more precisely we will show the following theorem Theorem 33 Let a, b > 0 If f L (µ), then where K(a, b) = f L (µ) K(a, b) x a f b +b L (µ) w b B(f) [ ( b ) +b + ( b ) ] b 1/(K1 +b (a)) b [ b 1 b+1 + b b b+1 +b L (Ω), +b, a (0, 1 ), b > 0, ] ab+a +b (K (a)) b +b, a > 1/, b > 0, [ 1 ] (b) b+1 + (b) b b+1 b+1 b+ (K 1 ( 1 b 4 )) b+1, a = 1/, b > 0 Here K 1 (a) and K (a) are the constants given by Theorem 31 Proof (i) Let 0 < a < 1/, b > 0 and r > 0 Then B(f) F (C) = B(f)(w) dω(w) + Firstly, From Theorem 31, we get w >r w >r B(f)(w) dω(w) B(f)(w) dω(w) r b w b B(f) L (Ω) B(f)(w) dω(w) (K 1 (a)) r 4a x a f L (µ) Combining the precedent relations, by Theorem we obtain f L (µ) (K 1(a)) r 4a x a f L (µ) + r b w b B(f) L (Ω) Choosing r = ( b ) 1 w 4a+b ( b B(f) L (Ω) K 1 (a) x a f ) 1 +b L, we get the first inequality (µ) (ii) Let a > 1/, b > 0 and r > 0 From Theorem 31, we get B(f)(w) dω(w) (K (a)) r f (1/a) L (µ) x a f 1/a L (µ) Thus, Choosing r = ( B(f) F (C) (K (a)) r f (1/a) L (µ) x a f 1/a L (µ) + r b w b B(f) L (Ω) b w b B(f) L (Ω) (K (a)) f (1/a) L x a f 1/a (µ) L (µ) ) 1 b+, we get the second inequality (iii) Let a = 1/, b > 0 and r > 0 From Theorem 31, we get B(f)(w) dω(w) (K 1 ( 1 4 )) r f L (µ) x 1/ f L (µ) Therefore, Choosing r = ( B(f) F (C) (K 1( 1 4 )) r f L (µ) x 1/ f L (µ) + r b w b B(f) L (Ω) b w b B(f) L (Ω) ) 1 (K 1 ( 1 4 )) f L (µ) x 1/ f L (µ) b+1, we get the third inequality

8 570 Fethi SOLTANI Remark 34 Let f L (µ) In particular case (a = b = 1), we obtain the following Heisenberg s inequality for the Segal-Bargmann transform B, Let Λ be all f L (µ) such that f 3 L (µ) 8 π x f L (µ) w B(f) L (Ω) (31) 1 (f) = x f L (µ) f L (µ), (f) = w B(f) L (Ω) f L (µ) We obtain a characterization of the region of Heisenberg s inequality (see Figure 1), {( 1 (f), (f)), f Λ} {(x, y), x, y > 0, xy 1 8 π } Figure 1 Region of the concentrated Heisenberg s inequality (31) Theorem 35 (Nash-type inequality) Let b > 0 If f L (µ), then f L (µ) D(b) w b B(f) L (Ω), where D(b) = [ ] b 1 b+1 + b b b+1 b+1 Proof Let f L (µ), b > 0 and r > 0 Then B(f) F (C) = B(f)(w) dω(w) + Firstly, w >r From Theorem 1 (ii), we get w >r B(f)(w) dω(w) B(f)(w) dω(w) r b w b B(f) L (Ω) B(f)(w) dω(w) r f L (µ) Combining the precedent relations, by Theorem we obtain f L (µ) r f L (µ) + r b w b B(f) L (Ω) Choosing r = b 1 b+ ( w b B(f) L (Ω) f L (µ) ) 1 b+1, we get the desired inequality

9 Uncertainty principles for the Segal-Bargmann transform Heisenberg principle involving L 1 and L -norms In this section we will establish uncertainty principles for the Segal-Bargmann transform B involving L 1 and L -norms More precisely we will show the following theorems Theorem 41 (Clarkson-type inequality) Let a > 0 If f L 1 L (µ), then where f L1 (µ) D 1 (a) f +1 L (µ) x a f 1 D 1 (a) = ( π ) a (+1) Proof Let f L 1 L (µ) and r > 0 Then +1 L 1 (µ), [ () () +1 ] f L 1 (µ) = fχ x <r L 1 (µ) + (1 χ x <r )f L 1 (µ) Firstly, (1 χ x <r )f L1 (µ) r a x a f L1 (µ) By Hölder s inequality, we get fχ x <r L1 (µ) (µ( x < r)) 1/ f L (µ) ( ) 1/ f L r (µ) π Combining the precedent relations, we obtain f L 1 (µ) ( ) 1/ f L r π (µ) + r a x a f L 1 (µ) Choosing r = ( x a f L 1 (µ) ) ( +1 π )1/4 f L, we get the desired inequality (µ) Theorem 4 (Nash-type inequality) Let b > 0 If f L 1 L (µ), then where f L (µ) D (b) f b b+1 L 1 (µ) w b B(f) 1 D (b) = [ ( b ) 1 b b+1 + ( ) b+1 L (Ω), ] b b+1 1/ Proof Let f L 1 L (µ), b > 0 and r > 0 Then B(f) F (C) = B(f)(w) dω(w) + Firstly, w >r From Theorem 1 (i), we get w >r B(f)(w) dω(w) B(f)(w) dω(w) r b w b B(f) L (Ω) B(f)(w) dω(w) r f L 1 (µ) Combining the precedent relations, by Theorem we obtain f L (µ) r f L 1 (µ) + r b w b B(f) L (Ω) Choosing r = ( b ) 1 b+ ( w b B(f) L (Ω) f L 1 (µ) ) 1 b+1, we get the desired inequality By combining the Clarkson-type inequality (Theorem 41) and the Nash-type inequality (Theorem 4) we obtain the following uncertainty inequality of Heisenberg-type

10 57 Fethi SOLTANI Theorem 43 Let a, b > 0 If f L 1 L (µ), then (i) f 1 b+1 L 1 (µ) f 1 +1 L (µ) C 1 x a f 1 +1 L 1 (µ) w b B(f) 1 b+1 L (Ω), where C 1 = D 1 (a)d (b) (ii) f L (µ) C x a b +b+1 f L 1 (µ) w b B(f) +1 +b+1 L (Ω), where (iii) f L 1 (µ) C 3 x a f b+1 Remark 44 C = (D 1 (a)) b(+1) +b+1 (D (b)) (+1)(b+1) +b+1 +b+1 L 1 (µ) w b B(f) +b+1 L (Ω), where C 3 = (D 1 (a)) (+1)(b+1) +b+1 (D (b)) (b+1) +b+1 The uncertainty principles given by Theorem 4, are the analogs of the results obtained by Laeng-Morpurgo [10] and Morpurgo [11] for the Fourier transform, and the results obtained by Ghobber [] for the Dunkl transform In particular case, if a = b, we obtain the following Heisenberg s inequalities for the Segal-Bargmann transform B (i) For a > 0 and f L 1 L (µ): f L 1 (µ) f L (µ) C x a f L 1 (µ) w B(f) L (Ω), where C = (D 1 (a)d ()) +1 If a = b = 1: f L1 (µ) f L (µ) ( 3) 9 Let Λ be all f L 1 L (µ) such that 1 (f) = x f L 1 (µ) f L1 (µ) 4 π x f L 1 (µ) w B(f) L (Ω) (41), (f) = w B(f) L (Ω) f L (µ) We obtain a characterization of the region of Heisenberg s inequality (see Figure ), {( 1 (f), (f)), f Λ} {(x, y), x, y > 0, xy 4 π ( 3) 9 } Figure Region of the concentrated Heisenberg s inequality (41) (ii) For a = b = 1 and f L 1 L (µ): f 5 L (µ) ( ) 3 ( 3) 1 x f L π 1 (µ) w B(f) 3 L (Ω) (4)

11 Uncertainty principles for the Segal-Bargmann transform 573 Let Λ be all f L 1 L (µ) such that 1 (f) = x f L 1 (µ) f L (µ), (f) = w B(f) L (Ω) f L (µ) We obtain a characterization of the region of Heisenberg s inequality (see Figure 3), {( 1 (f), (f)), f Λ} {(x, y), x, y > 0, x y 3 π ( ) 3 ( } 3) 1 Figure 3 Region of the concentrated Heisenberg s inequality (4) (iii) For a = b = 1 and f L 1 L (µ): Let Λ be all f L 1 L (µ) such that f 5 L (µ) 3 1 π( ) 11 x f 3 L 1 (µ) w B(f) L (Ω) (43) 1 (f) = x f L 1 (µ) f L (µ), (f) = w B(f) L (Ω) f L (µ) We obtain a characterization of the region of Heisenberg s inequality (see Figure 4), π( ) {( 1 (f), (f)), f Λ} {(x, y), x, y > 0, x 3 y } Figure 4 Region of the concentrated Heisenberg s inequality (43)

12 574 Fethi SOLTANI 5 Donoho-Stark uncertainty principle Let E be a measurable subset of R We say that a function f L p (µ), p = 1,, is ε- concentrated on E in L p (µ)-norm, if f χ E f Lp (µ) ε f Lp (µ) (51) Let W be a measurable subset of C and let f L (µ) We say that B(f) is η-concentrated on W in L (Ω)-norm, if B(f) χ W B(f) L (Ω) η B(f) F (C) (5) The Donoho-Stark uncertainty principle for the Segal-Bargmann transform B is given by the following theorem Theorem 51 (Donoho-Stark-type inequality) Let E be a measurable subset of R, W be a measurable subset of C and f L (µ) If f is ε-concentrated on E in L (µ)-norm, B(f) is η-concentrated on W in L (Ω)-norm and ε + η < 1, then µ(e)(m(w ) 1 (1 η ε) (53) Proof Let f L (µ) Assume that µ(e) < and m(w ) < From (51), (5) and Theorem it follows that B(f) χ W B(χ E f) L (Ω) B(f) χ W B(f) L (Ω) + χ W B(f χ E f) L (Ω) η B(f) F (C) + B(f χ E f) F (C) (η + ε) f L (µ) Then the triangle inequality shows that But B(f) F (C) χ W B(χ E f) L (Ω) + B(f) χ W B(χ E f) L (Ω) χ W B(χ E f) L (Ω) + (η + ε) f L (µ) ( χ W B(χ E f) L (Ω) = By Theorem 1 (i) and Hölder s inequality we have W B(χ E f)(w) dω(w)) 1/ B(χ E f)(w) 1/4 e w / χ E f L1 (µ) 1/4 e w / f L (µ)(µ(e)) 1/ Thus, χ W B(χ E f) L (Ω) 1/4 (µ(e)) 1/ (m(w )) 1/ f L (µ), and B(f) F (C) 1/4 (µ(e)) 1/ (m(w )) 1/ f L (µ) + (η + ε) f L (µ) By applying Theorem, we obtain (µ(e)) 1/ (m(w )) 1/ 1/4 (1 η ε), which gives the desired result Remark 5 Let 1 (E) = µ(e), (W ) = m(w ) From Theorem 51, we obtain a characterization of the region of Donoho-Stark s inequality (see Figure 5), { ( 1 (E), (W )), E R, W C } { (x, y), x, y > 0, xy 1 (1 η ε) }

13 Uncertainty principles for the Segal-Bargmann transform 575 Figure 5 Region of the concentrated Donoho-Stark s inequality (53) Theorem 53 (Donoho-Stark-type inequality) Let E be a measurable subset of R, W be a measurable subset of C and f L 1 L (µ) If f is ε-concentrated on E in L 1 (µ)-norm, B(f) is η-concentrated on W in L (Ω)-norm and ε, η < 1, then µ(e)m(w ) 1 (1 ε) (1 η) Proof Let f L 1 L (µ) Assume that µ(e) < and m(w ) < From (51) we have Thus f L1 (µ) ε f L1 (µ) + χ E f L1 (µ) ε f L1 (µ) + (µ(e)) 1/ f L (µ) f L1 (µ) (µ(e))1/ f L (µ) (54) 1 ε On the other hand, from (5) it follows that Thus by Theorem, B(f) L (m) B(f) χ W B(f) L (Ω) + χ W B(f) L (Ω) η B(f) F (C) + 1/4 (m(w )) 1/ f L1 (µ) f L (µ) 1/4 (m(w )) 1/ f L 1 η 1 (µ) (55) Combining (54) and (55) we obtain the result of this theorem Remark 54 The uncertainty principles given by Theorems 51 and 53, are the analogs of the results obtained by Donoho-Stark [7] for the Fourier transform, by Kawazoe-Mejjaoli [3] for the Dunkl transform and by Soltani [4] for the Sturm-Liouville transform In the following we establish uncertainty inequality of Matolcsi-Szücs-type for the Segal- Bargmann transform B Theorem 55 (Matolcsi-Szücs-type inequality) Let f L 1 L (µ) If A f = {x R : f(x) 0} and A B(f) = {w C : B(f)(w) 0}, then µ(a f )m(a B(f) ) 1 Proof Let f L 1 L (µ) We put W = A B(f), then by Theorem 1 (i) and Hölder s inequality we obtain B(f) L (Ω) = χ W B(f) L (Ω) 1/4 (m(w )) 1/ f L1 (µ)

14 576 Fethi SOLTANI Then Theorem gives the desired result 1/4 (m(w )) 1/ (µ(a f )) 1/ f L (µ) Remark 56 The uncertainty principle given by Theorem 55 is the analogs of the result obtained by Matolcsi-Szücs [9] and Benedicks [8] for the Fourier transform Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper References [1] A BONAMI, B DEMANGE, P JAMING Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms Rev Mat Iberoamericana, 003, 19(1): 3 55 [] M COWLING, J F PRICE Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl inequality SIAM J Math Anal, 1984, 15(1): [3] I I HIRSCHMAN A note on entropy Amer J Math, 1957, 79: [4] W G FARIS Inequalities and uncertainty inequalities J Mathematical Phys, 1978, 19(): [5] J F PRICE Inequalities and local uncertainty principles J Math Phys, 1983, 4(7): [6] J F PRICE Sharp local uncertainty principles Studia Math, 1987, 85: [7] D L DONOHO, P B STARK Uncertainty principles and signal recovery SIAM J Appl Math, 1989, 49(3): [8] M BENEDICKS On Fourier transforms of functions supported on sets of finite Lebesgue measure J Math Anal Appl, 1985, 106(1): [9] T MATOLCSI, J SZÜCS Intersection des mesures spectrales conjuguées C R Acad Sci Paris Sér A-B, 1973, 77: A841 A843 [10] E LAENG, C MORPURGO An uncertainty inequality involving L1-norms Proc Amer Math Soc, 1999, 17(1): [11] C MORPURGO Extremals of some uncertainty inequalities Bull London Math Soc, 001, 33(1): 5 58 [1] G B FOLLAND, A SITARAM The uncertainty principle; A Mathematical Survey J Fourier Anal Appl, 1997, 3(3): [13] A NEMRI, F SOLTANI L p uncertainty principles for the Fourier transform with numerical aspect Appl Anal, 016, 95(5): [14] F SOLTANI, A NEMRI Analytical and numerical approximation formulas for the Fourier multiplier operators Complex Anal Oper Theory, 015, 9(1): [15] F SOLTANI, A NEMRI Analytical and numerical applications for the Fourier multiplier operators on R n (0, ) Appl Anal, 015, 94(8): [16] C A BERGER, L A COBURN Toeplitz operators on the Segal-Bargmann space Trans Amer Math Soc, 1987, 301(): [17] V BARGMANN On a Hilbert space of analytic functions and an associated integral transform Comm Pure Appl Math, 1961, 14: [18] F SOLTANI Toeplitz and translation operators on the q-fock spaces Adv Pure Math, 011, 1: [19] F SOLTANI Operators and Tikhonov regularization on the Fock space Int Trans Spec Funct, 014, 5(4): [0] Kehe ZHU Uncertainty principles for the Fock space Preprint 015, arxiv: V1 [1] S GHOBBER, P JAMING Uncertainty principles for integral operators Studia Math, 014, 0(3): [] S GHOBBER Uncertainty principles involving L 1 -norms for the Dunkl transform Int Trans Spec Funct, 013, 4(6): [3] T KAWAZOE, H MEJJAOLI Uncertainty principles for the Dunkl transform Hiroshima Math J, 010, 40(): [4] F SOLTANI Donoho-Stark uncertainty principle associated with a singular second-order differential operator Int J Anal Appl, 014, 4(1): 1 10