Indian J. Pure Appl. Math., 48(1): 147-161, March 2017 c Indian National Science Academy 

DOI: 10.1007/s13226-017-0216-9

QUANTITATIVE UNCERTAINTY PRINCIPLES FOR THE SHORT TIME FOURIER TRANSFORM AND THE RADAR AMBIGUITY FUNCTION H. Lamouchi∗ and S. Omri∗∗ ∗ Institut

pr´ eparatoire aux e´tudes d’ing´ enieurs de Tunis, 2 Rue

Jawaher Lel Nehru - 1089 Montfleury - Tunisie ∗∗ Institut

pr´ eparatoire aux e´tudes d’ing´ enieurs El Manar,

Compus universitaire El Manar, 2092 Tunis, Tunisie e-mails: [email protected]; [email protected] (Received 11 April 2016; after final revision 17 July 2016; accepted 30 September 2016) Logarithmic uncertainty principle and Beckner’s uncertainty principle in terms of entropy are proved for the short time Fourier transform and the radar ambiguity function, also a Heisenberg inequality for generalized dispersion and Price’s local uncertainty principle are obtained. Key words : Short time Fourier transform; Radar ambiguity function; uncertainty principle; time-frequency analysis; localization; logarithmic Beckner’s theorem; entropy; generalized dispersion; Heisenberg inequality; Price’s theorem; local uncertainty principle. 1. I NTRODUCTION The uncertainty principles in harmonic analysis state that a nonzero function f and its Fourier transform f cannot be at the same time simultaneously and sharply localized, that is, it’s impossible for a nonzero function and its Fourier transform to be simultaneously small. There are many formulations of this general fact where the smallness and the localization have been interpreted differently and by several ways. For more details about uncertainty principles, we refer the reader to [4, 8]. For an arbitrary function f ∈ L2 (Rd ) and a nonzero function g ∈ L2 (Rd ) called a window function, the  d by [6] short time Fourier transform (STFT) of f with respect to g is defined on Rd × R  (1.1) Vg (f )(x, ω) = f (z)g(z − x)eiz|ω dμd (z), Rd

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where . | . is the classical inner product on Rd defined by z|ω =

d 

zi ωi and dμd (z) =

i=1

dz d

(2π) 2

is

the normalized Lebesgue measure. The STFT plays an important role in time-frequency analysis namely by providing an interesting way to study the local frequency spectrum of signals. Relation (1.1) shows that unlike the classical Fourier transform, the STFT gives a simultaneous representation of the space and the frequency variables. In signal analysis, the short time Fourier transform is closely related to other common and  d , by [6] known time frequency distributions as the radar ambiguity function defined on Rd × R  A(f )(x, ω) =

 x  −iz|ω x  e f z− dμd (z). f z+ 2 2 Rd

(1.2)

The radar ambiguity function occurs naturally in many radar applications, for more details about its physical aspect, we refer the reader to [6]. In fact, a standard change of variables shows that A(f )(x, ω) = e 2 x|ω Vf (f )(x, ω). i

(1.3)

Roughly speaking, the uncertainty principles for the STFT say that for a given function f ∈ L2 (Rd ), the STFT Vg (f ) cannot be concentrated in the time-frequency plane. In this context, Lieb [14] proved an analogue of Donoho-Strak uncertainty principle for the STFT and the radar ambiguity function. We cite Fernandez, Galbis and Wilczok [5, 13] who studied the annihilating sets for the STFT. Other uncertainty principles have been also showed for the STFT namely by Grochening and Zimmerman [7] who established an analogue of Hardy and Benedick’s theorem for the STFT. Heisenberg inequality, Cowling-price theorem as well as Gelfand-Shilov theorem have been also showed for the STFT and the radar ambiguity function by Bonami, Demange, and Jaming [1]. Recently Lamouchi and Omri [10] proved a quantitative version of Shapiro’s and the umbrella theorems for the STFT. Our purpose in this work is to prove two logarithmic uncertainty principles due to Beckner for both of the STFT and the radar ambiguity function. We also generalize the Heisenberg inequality proved by Bonami, Demange and Jaming [1] and we prove Price’s local uncertainty principle for these two transforms. More precisely, our first main result will be the logarithmic Beckner’s uncertainty principle for the radar ambiguity function. Indeed for every nonzero function f ∈ S(Rd ), we have A(f ) ∈ S(R2d )

UNCERTAINTY PRINCIPLES and

149

 

ln |(x, ω)||A(f )(x, ω)|4 dμ2d (x, ω) d Rd ×R      d 1 ψ + ln 2 |A(f )(x, ω)|4 dμ2d (x, ω),  2 2 d d  R ×R

(1.4)

where ψ denotes the logarithmic derivative of Euler’s function Γ. Next, we will prove an analogue of logarithmic Beckner’s uncertainty principle in terms of entropy for the STFT and the radar ambiguity function, that is for every f, g ∈ L2 (Rd ) such that g is nonzero, we have    E(|Vg (f )|2 )  f 22,Rd g22,Rd d − ln f 22,Rd g22,Rd . As consequence of the uncertainty principle in terms of entropy, we will prove a Heisenberg type inequality for generalized dispersion by showing that for every positive real numbers p, q, there is a nonnegative constant Dp,q such that for every f, g ∈ L2 (Rd ), we have  

 q   p+q |x| |Vg (f )(x, ω)| dμ2d (x, ω) p

d Rd ×R

2

d Rd ×R



2

|ω| |Vg (f )(x, ω)| dμ2d (x, ω) q

p p+q

 Dp,q f 22,Rd g22,Rd . Finally, we will prove Price’s uncertainty principle for the STFT and the radar ambiguity function,  d and for every ξ, p ∈ R, 0 < ξ < d, p > 1, that is for every finite measurable subset Σ of Rd × R there is a nonnegative constant Mξ,p such that for every f, g ∈ L2 (Rd ) such that g is nonzero, we have

  Σ

|Vg (f )(x, ω)|p dμ2d (x, ω) 2pd

1

p−

2pd

p−

2pd

f 2,R(d+ξ)(p+1) g2,R(d+ξ)(p+1) .  Mξ,p (μ2d (Σ)) (p+1) |(x, ω)|ξ Vg (f ) (d+ξ)(p+1) d d d d 2,R ×R

2. T HE S HORT T IME F OURIER T RANSFORM According to relation (1.1), fix a window function g ∈ L2 (Rd ), for every f ∈ L2 (Rd ) the short time  d by [6] Fourier transform of f with respect to g, is defined on the time-frequency plane Rd × R  Vg (f )(x, ω) = f (z)g(z − x)eiz|ω dμd (z). Rd

For every x, ω ∈ Rd , we denote by Mω and Tx the modulation and the translation operators defined respectively on L2 (Rd ) by ∀z ∈ Rd , Mω h(z) = eiz|ω h(z),

(2.1)

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and ∀z ∈ Rd , Tx h(z) = h(z − x).

(2.2)

Then by relations (2.1) and (2.2), we deduce that ∀z ∈ Rd , Mω (Tx h)(z) = eiz|ω h(z − x),

(2.3)

∀z ∈ Rd , Tx (Mω h)(z) = e−ix|ω eiz|ω h(z − x).

(2.4)

and

Again by relation (2.2), the STFT may be expressed as Tx g(ω). Vg (f )(x, ω) = f

(2.5)

and by relations (1.1) and (2.1), we have  Vg (f )(x, ω) =

Rd

f (z)eiz|ω g(z − x)dμd (z)

= Mω f ∗ g(x), where ∗ denotes the usual convolution product on Rd . It’s known [6] that for every f, g ∈ L2 (Rd ) the STFT Vg (f ) is uniformly continuous and bounded  d and satisfies on the time-frequency plane Rd × R Vg (f )∞,Rd ×R d  f 2,Rd g2,Rd .

(2.6)

Moreover according to [6], for all f1 , f2 , g1 , g2 ∈ L2 (Rd ) the functions Vg1 (f1 ) and Vg2 (f2 )  d ) and we have the following orthogonality relation belong to L2 (Rd × R Vg1 (f1 )|Vg2 (f2 )Rd ×R d = f1 |f2 Rd g1 |g2 Rd .

(2.7)

In particular, for every f, g ∈ L2 (Rd ) we get Vg (f )2,Rd ×R d = f 2,Rd g2,Rd .

(2.8)

Given f, g ∈ L2 (Rd ) and ξ, λ, y, z ∈ Rd , then by relations (2.3) and (2.4) we have  VMξ Tz g (Mλ Ty f )(x, ω) = (Mλ Ty f )(u)Tx (Mξ Tz g) (u)e−iω|u dμd (u) d R  ix|ξ f (u − y)g(u − (x + z))e−iu|ω−λ+ξ dμd (u) =e Rd  ix|ξ −iy|ω−λ+ξ =e e f (t)g(t − (x − y + z))e−it|ω−λ+ξ dμd (t) Rd

ix|ξ −iy|ω−λ+ξ

=e

e

Vg (f )(x − y + z, ω − λ + ξ).

(2.9)

UNCERTAINTY PRINCIPLES

151

d Relation (2.9) shows in particular that for every (y, λ), (x, ω) ∈ Rd × R VMλ Ty g (Mλ Ty f )(x, ω) = eix|λ e−iy|ω Vg (f )(x, ω).

(2.10)

We have also the following switching property  g(z)f (z − x)e−iz|ω dμd (z) Vf (g)(x, ω) = d R  −ix|ω =e f (y)g(y + x)e−iy|−ω dμd (z) Rd

−ix|ω

=e

Vg (f )(−x, −ω).

(2.11)

Let λ be a positive real number, for every measurable function f , we denote by fλ the dilate of f defined by fλ (x) = f (λx), then for every f ∈ L2 (Rd ), fλ belongs to L2 (Rd ) and by a standard computation we deduce that Vg (fλ ) is given by Vg (fλ )(x, ω) =

ω 1 Vg 1 (f )(λx, ). d λ λ λ

(2.12)

Finally given two positive real numbers a, b and let f and g be the gaussian functions defined 2

2

respectively by f (x) = (4a) 4 e−a|x| and g(x) = (4b) 4 e−b|x| , then by a standard calculus we have d f  d = g d = 1 and for every (x, ω) ∈ Rd × R d

2,R

2,R

Vg (f )(x, ω) = (16ab)

d 4

 Rd

d 4

2



2

−b|x|2

= (16ab) e

d 4

= (16ab) e

2

Rd

e

e−(a+b)|z| e2bz|x e−iω|z dμd (z)

b2 |x|2 a+b



e−(a+b)(|z|

2 −2z|

b b x+| a+b x|2 a+b

Rd

−ab

2

= (16ab) 4 e a+b |x| d

2

e−a|z| e−b|z−x| e−iω|z dμd (z)

= (16ab) 4 e−b|x| d

d

−ab |x|2 a+b



2

e−(a+b)|z− a+b x| e−iω|z dμd (z) b

Rd

) e−iω|z dμ (z) d

b −iω| a+b x



e

Rd

2

e−(a+b)|y| e−iω|y dμd (z).

Hence, d

Vg (f )(x, ω) =

(4ab) 4

−ab

2

(a + b) 2

b

|ω|2

− 4(a+b)

|x| −iω| a+b x a+b e e d e

.

(2.13)

152

H. LAMOUCHI AND S. OMRI In particular by relation (2.13) we get d

|Vg (f )(x, ω)|2 =

|ω|2 (4ab) 2 −2ab |x|2 − 2(a+b) a+b e e . (a + b)d

(2.14)

3. U NCERTAINTY P RINCIPLES FOR THE STFT

In [2] Beckner used Stein-Weiss and Pitt’s inequalities to obtain a logarithmic estimate of the uncertainty, he showed that for every f ∈ S(Rd ) we have  Rd

2



ln |x||f (x)| dx +

Rd

ln |y||f(y)|2 dy 

    d + ln 2 ψ |f (x)|2 dx, 4 Rd

(3.1)

where f is the classical Fourier transform defined on Rd by ∀y ∈ R , f(y) =



d

Rd

f (x)e−ix|y dμd (x),

and ψ is the logarithmic derivative of Γ function defined by ψ(x) =

Γ (x) . Γ(x)

The previous inequality is known as Beckner’s logarithmic uncertainty principle. In the following we are interested to generalize inequality (3.1) to the radar ambiguity function A(f ) for f ∈ S(Rd ). Theorem 3.1 — Let f ∈ S(Rd ) be a nonzero function, then A(f ) ∈ S(R2d ), and we have   d Rd ×R

ln |(x, ω)||A(f )(x, ω)|4 dμ2d (x, ω)      d 1 ψ + ln 2 |A(f )(x, ω)|4 dμ2d (x, ω),  2 2 d Rd ×R

(3.2)

where ψ denotes the logarithmic derivative of Euler’s function Γ.  d ). P ROOF : Let f, g ∈ S(Rd ), then the function h(x, z) = f (z)g(z − x) belongs to S(Rd × R  d ) is invariant under partial Fourier transform then by relation (2.5) we deduce that Since S(Rd × R  d ). Let φ be the function defined on Rd × R  d by Vg (f ) ∈ S(Rd × R φ(x, ω) = Vg (f )(x, ω)Vf (g)(x, ω),

UNCERTAINTY PRINCIPLES

153

 d ) and according to relations (2.7), (2.10) and (2.11), we have then φ ∈ S(Rd × R    Vf (g)(x, ω)Vg (f )(x, ω)e−ix|y e−iω|λ dμ2d (x, ω) φ(y, λ) = d d    R R Vf (g)(x, ω)Vg (f )(x, ω)ei(x|−y−ω|λ) dμ2d (x, ω) = d d  R R = VM−y Tλ g (M−y Tλ f )(x, ω)Vf (g)(x, ω)dμ2d (x, ω) Rd

d R

= VM−y Tλ g (M−y Tλ f )|Vf (g)Rd ×R d = M−y Tλ f |gRˆ d M−y Tλ g|f Rd    −iy|z g(z)f (z − λ)e dμd (z) = Rd

Rd

 =

Rd

 f (z)g(z −

  g(z)f (z −

λ)e−i−y|z dμd (z)

Rd

λ)e−iy|z dμd (z) −i−y|z

f (z)g(z − λ)e

 dμd (z)

= Vg (f )(λ, −y)Vf (g)(λ, −y), and therefore  λ) = φ(−λ, y). φ(y,

(3.3)

Now, assume that f 2,Rd = g2,Rd = 1, then by combining relations (2.8), (3.1) and (3.3), we deduce that   d Rd ×R

ln |(x, ω)||Vg (f )(x, ω)|2 |Vf (g)(x, ω)|2 dμ2d (x, ω)      d 1 ψ + ln 2 |Vg (f )(x, ω)|2 |Vf (g)(x, ω)|2 dμ2d (x, ω).  2 2 d Rd ×R

Hence by relation (1.3), we get   ln |(x, ω)||A(f )(x, ω)|4 dμ2d (x, ω) d Rd ×R      d 1 ψ + ln 2 |A(f )(x, ω)|4 dμ2d (x, ω).  2 2 d d  R ×R

2

 d is defined According to Shannon [12], the entropy of a probability density function ρ on Rd × R by

  E(ρ) = −

d Rd ×R

ln(ρ(x, ω))ρ(x, ω)dμ2d (x, ω),

whenever the integral on the right hand side is well defined. The entropy plays an important role in quantum mechanics and in signal theory, for a better understanding of its physical’s signification we refer the reader to [3]. Clearly the entropy represents an advantageous way to measure the decay

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of a function f , so that it was very interesting to localize the entropy of a probability measure and its Fourier transform. In this context, the first estimation has been given by Hirschman [9] and has been improved by Beckner [2] who used the Hausdorff-Young inequality to derive the following uncertainty inequality that is for every f ∈ L2 (Rd ) with f 2,Rd = 1, we have E(|f |2 ) + E(|f|2 )  d(1 − ln 2), whenever the left side is well defined. The aim of the following is to generalize the localization of the entropy to the STFT over the time-frequency plane and also to the radar ambiguity function. Theorem 3.2 — Let g be a window function and f ∈ L2 (Rd ), then    E(|Vg (f )|2 )  f 22,Rd g22,Rd d − ln f 22,Rd g22,Rd ,

(3.4)

and the inequality (3.4) is sharp. P ROOF : Assume that f 2,Rd = g2,Rd = 1 and following the idea of Lieb [11], then by relation (2.6) we deduce that  d , |Vg (f )(x, ω)|  Vg (f ) ∀(x, ω) ∈ Rd × R  d  f 2,Rd g2,Rd = 1. ∞,Rd ×R

(3.5)

In particular E(Vg (f ))  0 and therefore if the entropy E(Vg (f )) is infinite then the inequality (3.4) holds trivially. Suppose now that the entropy E(Vg (f )) is finite and let 0 < x < 1 and h be the function defined on ]2, 3] by h(p) = then ∀p ∈]2, 3[, h (p) =

xp − x2 , p−2

(p − 2)xp ln(x) − (xp − x2 )  0, (p − 2)2

and therefore h is increasing on ]2, 3], in particular ∀p ∈]2, 3], x2 ln(x) = lim

p→2+

hence 0

xp − x2 xp − x2  , p−2 p−2

x2 − xp  −x2 ln(x), p−2

(3.6)

d and by combining relations (3.5) and (3.6) we get, for every (x, ω) ∈ Rd × R 0

|Vg (f )(x, ω)|2 − |Vg (f )(x, ω)|p  −|Vg (f )(x, ω)|2 ln(|Vg (f )(x, ω)|). p−2

(3.7)

UNCERTAINTY PRINCIPLES

155

Let ϕ be the function defined on [2, +∞[ by   d 2 . |Vg (f )(x, ω)| dμ2d (x, ω) − p d Rd ×R

 

p

ϕ(p) =

According to Lieb [11], we know that for every 2  p < +∞ the STFT Vg (f ) belongs to  d ) and we have ×R

Lp (Rd

 d 2 f p2,Rd gp2,Rd . |Vg (f )(x, ω)| dμ2d (x, ω)  p d Rd ×R

 

p

(3.8)

Then, relation (3.8) impliesthatϕ(p)  0 for every p ∈ [2, +∞[ and by Plancherel theorem (2.8) dϕ  0 whenever this derivative is well defined. However, by we have ϕ(2) = 0. Therefore dp p=2+ using relation (3.7) and Lebesgue’s dominated convergence theorem we have    d p |Vg (f )(x, ω)| dμ2d (x, ω) dp Rd ×R d p=2+   2 |Vg (f )(x, ω)| − |Vg (f )(x, ω)|p dμ2d (x, ω) =− lim p−2  d p→2+ Rd ×R 1 = − E(|Vg (f )|2 ) 2 and consequently



dϕ dp

 p=2−

1 d = − E(|Vg (f )|2 ) + , 2 2

which gives E(|Vg (f )|2 )  d. d

2

d

2

Let f (x) = (4a) 4 e−a|x| and g(x) = (4b) 4 e−b|x| with a, b > 0 then f 2,Rd = g2,Rd = 1 and d according to relation (2.14) we have for every (x, ω) ∈ Rd × R d

|ω|2 (4ab) 2 −2ab |x|2 − 2(a+b) a+b e e . |Vg (f )(x, ω)| = (a + b)d

2

Therefore by a standard calculus and using relation (2.14), we get √

2 ab 2 . E(|Vg (f )| ) = d 1 − ln a+b In particular E(|Vg (f )|2 ) = d if, and only if a = b.

2

Corollaire 3.3 — Let f ∈ L2 (Rd ), then    E(|A(f )|2 )  f 42,Rd d − ln f 42,Rd ,

(3.9)

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H. LAMOUCHI AND S. OMRI

and the inequality (3.4) is sharp. In [1] Bonami, Demange and Jaming showed a Heisenberg type inequality for the radar ambiguity function with respect to the second dispersion and as noticed above the same inequality holds obviously for the STFT. The authors showed that given a window function g ∈ L2 (Rd ) then for every f ∈ L2 (Rd ) we have     2 2 |x| |Vg (f )(x, ω)| dμ2d (x, ω) d Rd ×R

d Rd ×R

|ω|2 |Vg (f )(x, ω)|2 dμ2d (x, ω)  d2 f 42,Rd g42,Rd

In what follows we shall use Theorem 3.2 to generalize the previous Heisenberg uncertainty principle for generalized dispersions. Theorem 3.4 — Let p and q be two positive real numbers. Then there exists a nonnegative constant Dp,q such that for every window function g and for every function f ∈ L2 (Rd ) we have  

 q   p+q |x| |Vg (f )(x, ω)| dμ2d (x, ω) 2

p

d Rd ×R

d Rd ×R

 p p+q |ω| |Vg (f )(x, ω)| dμ2d (x, ω) q

2

 Dp,q f 22,Rd g22,Rd , ⎛

⎛

⎞⎞ 2 2d−2 pqΓ( d 2 ) ⎠⎠ d d Γ( p )Γ( q ) −1 d(p+q)

pq ⎝d+ln⎝

where Dp,q =

(3.10)

d

. p e p q p+q Moreover, for p = q = 2 inequality (3.10) is sharp. q p+q

d P ROOF : Assume that f 2,Rd = g2,Rd = 1 and let ξt,p,q be the function defined on Rd × R by ξt,p,q (x, ω) =

2d−2 pqΓ( d2 )2 e− Γ( dp )Γ( dq )

|x|p +|ω|q t d

tp

+ dq

, so by basic calculus we see that

 d Rd ×R

ξt,p,q (x, ω)dμ2d (x, ω) = 1,

 d . Since the in particular dσt,p,q (x, ω) = ξt,p,q (x, ω)dμ2d (x, ω) is a probability measure on Rd × R function ϕ(t) = t ln(t) is convex over ]0, +∞[, hence according to Jensen’s inequality     |Vg (f )(x, ω)|2 |Vg (f )(x, ω)|2 ln dσt,p,q (x, ω)  0, ξt,p,q (x, ω) ξt,p,q (x, ω) d Rd ×R which implies in terms of entropy that for every positive real number t

  d 2d−2 pqΓ( d2 )2 1 + dq 2 p )+  ln(t E(|Vg (f )| )+ln (|x|p +|ω|q )|Vg (f )(x, ω)|2 dμ2d (x, ω), t d d  Γ( dp )Γ( dq ) R ×R

UNCERTAINTY PRINCIPLES

157

and by means of Theorem 3.2

  d Rd ×R

(|x|p + |ω|q )|Vg (f )(x, ω)|2 dμ2d (x, ω)  t d + ln

However the expression t(d + ln ⎛

⎛

pq ⎝d+ln⎝

t0 = e



⎞⎞ 2 2d−2 pqΓ( d 2 ) ⎠⎠ d d Γ( p )Γ( q ) −1 d(p+q)

2d−2 pqΓ( d2 )2

2d−2 pqΓ( d2 )2 Γ( dp )Γ( dq )



− ln(t

d d +q p

) .

d

− ln(t p

Γ( dp )Γ( dq )

+ dq

)) attains its upper bound at

, and consequently

  d Rd ×R

(|x|p + |ω|q )|Vg (f )(x, ω)|2 dμ2d (x, ω)  Cp,q ,

where

⎛

Cp,q =

d(p + q) e pq

⎛

pq ⎝d+ln⎝

⎞⎞ 2 2d−2 pqΓ( d 2 ) ⎠⎠ d d Γ( p )Γ( q ) −1 d(p+q)

.

Therefore for every window function g and for every function f ∈ L2 (Rd ), we get     p 2 |x| |Vg (f )(x, ω)| dμ2d (x, ω)+ |ω|q |Vg (f )(x, ω)|2 dμ2d (x, ω) d Rd ×R

d Rd ×R

 Cp,q f 22,Rd g22,Rd .

(3.11)

Now for every positive real number λ the dilates fλ and gλ belong to L2 (Rd ) and gλ is nonzero, then by relation (3.11), we have     p 2 |x| |Vgλ (fλ )(x, ω)| dμ2d (x, ω) + d Rd × R

d Rd ×R

|ω|q |Vgλ (fλ )(x, ω)|2 dμ2d (x, ω)

 Cp,q fλ 22,Rd gλ 22,Rd hence for every positive real number λ     −p p 2 q λ |x| |Vg (f )(x, ω)| dμ2d (x, ω) + λ d Rd ×R

d Rd ×R

|ω|q |Vg (f )(x, ω)|2 dμ2d (x, ω)

 Cp,q f 22,Rd g22,Rd . In particular, the inequality holds at the critical point 

1 p Rd ×R d |x|p |Vg (f )(x, ω)|2 dμ2d (x, ω) p+q  , λ= q Rd ×R d |ω|q |Vg (f )(x, ω)|2 dμ2d (x, ω)

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H. LAMOUCHI AND S. OMRI

which implies that    q   p+q p 2 |x| |Vg (f )(x, ω)| dμ2d (x, ω) d Rd ×R

 p p+q |ω| |Vg (f )(x, ω)| dμ2d (x, ω) 2

q

d Rd ×R

 Dp,q f 22,Rd g22,Rd , where

⎛

Dp,q = Cp,q

p

p p+q

q p+q

q p+q

⎛

pq ⎝d+ln⎝

=

d q p+q

p

e

⎞⎞ 2 2d−2 pqΓ( d 2 ) ⎠⎠ d )Γ( d ) Γ( p q −1 d(p+q)

.

p q p+q In the particular case when p = q = 2, we get xVg (f )(x, ω)2,Rd ×R d ωVg (f )(x, ω)2,Rd ×R d  df 22,Rd g22,Rd . We will show now that relation (3.12) is sharp, indeed let f (x) = 2 4 e− d

relation (2.14) we have   |x|2 |Vf (f )(x, ω)|2 dμ2d (x, ω) = d Rd ×R

1 (2π)d

 |x| e p

Rd

−2ab |x|2 a+b

|x|2 2

 dx

(3.12)

then, according to |ω|2

− 2(a+b)

d R

e

dω

=d and

 

1 |ω| |Vg (f )(x, ω)| dμ2d (x, ω) = (2π)d d Rd ×R 2

2

 e

−2ab |x|2 a+b

Rd

 dx

d R

|ω|2

− 2(a+b)

|ω|q e

dω

=d knowing that f 2,Rd = 1, we deduce that xVf (f )2,Rd ×R d ωVf (f )2,Rd ×R d = df 42,Rd .2 Corollary 3.5 — Let p and q be two positive real numbers then there exists a nonnegative constant Dp,q such that for every function f ∈ L2 (Rd ), we have  

 q   p+q |x| |A(f )(x, ω)| dμ2d (x, ω) p

d Rd ×R

2

d Rd ×R

2

|ω| |A(f )(x, ω)| dμ2d (x, ω) q

 Dp,q f 42,Rd .



p p+q

(3.13)

Moreover, for p = q = 2 inequality (3.10) is sharp. The Heisenberg uncertainty principle proved above says that the STFT and the radar ambiguity function cannot be concentrated near the origin in the time-frequency plane but it does not claim

UNCERTAINTY PRINCIPLES

159

whether the result remains true near several given points or more generally into a subset of the timefrequency plane with finite measure. In the following, we will prove through the local Price’s inequality, that in fact the so called property remains true. Theorem 3.6 — Let ξ, p be two positive real numbers such that 0 < ξ < d and p  1, then there is a nonnegative constant Mξ,p such that for every window function g, for every function f ∈ L2 (Rd )  d , we have and for every finite measurable subset Σ of Rd × R   |Vg (f )(x, ω)|p dμ2d (x, ω) Σ

2pd

1

p−

2pd

p−

2pd

f 2,R(d+ξ)(p+1) g2,R(d+ξ)(p+1) .  Mξ,p (μ2d (Σ)) (p+1) |(x, ω)|ξ Vg (f ) (d+ξ)(p+1) d d d d 2,R ×R

P ROOF : Without loss of generality we can assume that f 2,Rd = g2,Rd = 1, then for every positive real number s, we have Vg (f )p,Σ  Vg (f )1Bs p,Σ + Vg (f )1Bsc p,Σ ,  d of radius s. However, by H¨older’s inequality and relation (2.6) where Bs denotes the ball of Rd × R we get for every 0 < ξ < d   Vg (f )1Bs p,Σ =

1 |Vg (f )(x, ω)| 1Bs (x, ω)1Σ (x, ω)dμ2d (x, ω) p

d Rd ×R

 Vg (f )

 

p p+1

d ∞,Rd ×R

d Rd ×R

|Vg (f )(x, ω)|

p p+1

p

1 p 1Bs (x, ω)1Σ (x, ω)dμ2d (x, ω)

1

1

 μ2d (Σ) p(p+1) Vg (f )1Bs  p+1d

d 1,R ×R

 μ2d (Σ) 

1 p(p+1)

μ2d (Σ)

1

1

p+1 −ξ d  d |(x, ω)| 1Bs 

|(x, ω)|ξ Vg (f ) p+1d

d 2,R ×R

2,R ×R

1 p(p+1)

(2d Γ(d)(d − ξ))

1

1 2(p+1)

|(x, ω)|ξ Vg (f ) p+1d

d s 2,R ×R

d−ξ p+1

.

On the other hand, and again by H¨older’s inequality and relation (2.6), we deduce that Vg (f )1Bsc p,Σ  Vg (f )

p−1 p+1

 

d ∞,Rd ×R

 (μ2d (Σ))  (μ2d (Σ))

1 p(p+1)

 

d Rd ×R

d Rd ×R 1 p(p+1)

|Vg (f )(x, ω)|

2p p+1

1 1Bsc (x, ω)1Σ (x, ω)dμ2d (x, ω)

 1 p+1 |Vg (f )(x, ω)| 1Bsc (x, ω)dμ2d (x, ω) 2

2

− 2ξ s p+1 d  2,R ×R

|(x, ω)|ξ Vg (f ) p+1d

p

160

H. LAMOUCHI AND S. OMRI Hence,  

1 1 p 1  (μ2d (Σ)) p(p+1) |(x, ω)|ξ Vg (f ) p+1d |Vg (f )(x, ω)| dμ2d (x, ω) p

Σ

×

d−ξ

1

s p+1 (2d Γ(d)(d − ξ))

1 2(p+1)

d 2,R ×R



+ |(x, ω)|ξ Vg (f ) p+1d

d s 2,R ×R

2ξ − p+1

In particular the inequality holds for ⎛ ⎜ s0 = ⎝

2ξ|(x, ω)|ξ V

1 p+1 d 2,Rd ×R

g (f )

(2d Γ(d)(d

− ξ))

d−ξ

1 2(p+1)

⎞ p+1 d+ξ ⎟ ⎠

,

and therefore  

1 2d p 1  (μ2d (Σ)) p(p+1) |(x, ω)|ξ Vg (f ) (d+ξ)(p+1) |Vg (f )(x, ω)| dμ2d (x, ω) d 2,Rd ×R ⎞ ⎛ Σ d+ξ ⎠ × ⎝ ξ(d+2p+2) 2ξ ξ d−ξ ξ + (d+ξ)(p+1) (d+ξ)(p+1) (d+ξ)(p+1) d+ξ d+ξ 2 ξ Γ(d) (d − ξ) p

The proof is complete by applying the previous inequality to nonzero functions f, g ∈ L2 (Rd ).

g f and for every f 2,Rd g2,Rd 2

Corollary 3.7 — Let ξ, p be two positive real numbers such that 0 < ξ < d and p  1, then there is a nonnegative constant Mξ,p such that for every function f ∈ L2 (Rd ) and for every finite  d , we have measurable subset Σ of Rd × R  

Σ

|A(f )(x, ω)|p dμ2d (x, ω) 2pd

1

2p−

4pd

f 2,Rd(d+ξ)(p+1) .  Mξ,p (μ2d (Σ)) (p+1) |(x, ω)|ξ A(f ) (d+ξ)(p+1) d d 2,R ×R

ACKNOWLEDGEMENT The author wishes to thank the referee for its valuable comments. R EFERENCES 1. A. Bonami, B. Demange and P. Jaming, Hermite functions and uncertainty priciples for the Fourier and the widowed Fourier transforms, Rev. Mat. Iberoamericana., 19 (2003), 23-55.

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