International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 1 (2014), 1-10
http://www.etamaths.com

DONOHO-STARK UNCERTAINTY PRINCIPLE ASSOCIATED

WITH A SINGULAR SECOND-ORDER DIFFERENTIAL

OPERATOR

FETHI SOLTANI

Abstract. For a class of singular second-order differential operators ∆, we

prove a continuous-time principles for L1 theory and L2 theory, respectively.
Another version of continuous-time principle using L1 ∩ L2 theory is given.

1. Introduction

The classical uncertainty principle says that if a function f(t) is essentially zero

outside an interval of length δt and its Fourier transform f̂(w) is essentially zero
outside an interval of length δw, then

δt.δw ≥ 1;

a function and its Fourier transform cannot both be highly concentrated. The
uncertainty principle is widely known for its ”philosophical” applications: in quan-
tum mechanics, of course, it shows that a particle’s position and momentum cannot
be determined simultaneously [10]; in signal processing it establishes limits on the
extent to which the ”instantaneous frequency” of a signal can be measured [9].
However, it also has technical applications, for example in the theory of partial
differential equations [8].

Here we consider the second-order differential operator defined on ]0,∞[ by

∆u = u′′ +
A′

A
u′ + ρ2u,

where A is a nonnegative function satisfying certain conditions and ρ is a nonneg-
ative real number. This operator plays an important role in analysis. For example,
many special functions (orthogonal polynomials) are eigenfunctions of an operator
of ∆ type. The radial part of the Beltrami-Laplacian in a symmetric space is al-
so of ∆ type. Many aspects of such operators have been studied; we mention, in
chronological order, in 1979 Chébli [2]; in 1981 Trimèche [15]; in 1989 Zeuner [18];
in 1994 Xu [17]; in 1997 Trimèche [16]; in 1998 Nessibi et al. [13]. In particular, the
first two of these references investigate standard constructions of harmonic analy-
sis, such as translation operators, convolution product, and Fourier transform, in
connection with ∆.

2010 Mathematics Subject Classification. 42B10; 42B30; 33C45.
Key words and phrases. generalized Fourier transform; L1 uncertainty principle; L2 uncer-

tainty principles.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

1



2 FETHI SOLTANI

Many uncertainty principles have already been proved for the Sturm-Liouville
operarator ∆, namely by Rösler and Voit [14] who established an uncertainty princi-
ple for Hankel transforms. Bouattour and Trimèche [1] proved a Beurling’s theorem
for the Sturm-Liouville transform. Daher et al. [3, 4, 5, 6] give some related versions
of the uncertainty principle for the Sturm-Liouville transform (Titchmarsh’s the-
orem, Hardy’s theorem and Miyachi’s theorem). Ma [11, 12] proved a Heisenberg
uncertainty principle for the Sturm-Liouville transform.

Building on the ideas of Donoho and Stark [7] we show a continuous-time princi-
ple for the L1 theory. The analogous of this uncertainty principle in the L2 theory
is also given. We prove another versions of continuous-time principle for the L2

theory and for the L1 ∩L2 theory.
This paper is organized as follows. In Section 2 we recall some basic properties of

the Fourier transform F associated to ∆ (Plancherel theorem, inversion formula,...).
In Section 3 we prove a continuous-time principle for L1 theory. The last section of
this paper is devoted to show another versions of continuous-time principles using
L2 theory and L1 ∩L2 theory.

2. The operator ∆

We consider the second-order differential operator ∆ defined on ]0,∞[ by

∆u = u′′ +
A′

A
u′ + ρ2u,

where ρ is a nonnegative real number and

A(x) = x2α+1B(x), α > −1/2,

for B a positive, even, infinitely differentiable function on R such that B(0) = 1.
Moreover we assume that A and B satisfy the following conditions:

(i) A is increasing and lim
x→∞

A(x) = ∞.

(ii)
A′

A
is decreasing and lim

x→∞

A′(x)

A(x)
= 2ρ.

(iii) There exists a constant δ > 0 such that

A′(x)

A(x)
= 2ρ + D(x) exp(−δx) if ρ > 0,

A′(x)

A(x)
=

2α + 1

x
+ D(x) exp(−δx) if ρ = 0,

where D is an infinitely differentiable function on ]0,∞[, bounded and with bounded
derivatives on all intervals [x0,∞[, for x0 > 0. This operator was studied in [2, 13,
15], and the following results have been established:

(I) For all λ ∈ C, the equation{
∆u = −λ2u
u(0) = 1, u′(0) = 0

admits a unique solution, denoted by ϕλ, with the following properties:
ϕλ satisfies the product formula

ϕλ(x)ϕλ(y) =

∫ ∞
0

ϕλ(z)w(x,y,z)A(z)dz for x,y ≥ 0;



DONOHO-STARK UNCERTAINTY PRINCIPLE 3

where w(x,y, .) is a measurable positive function on [0,∞[, with support in [|x −
y|,x + y], satisfying ∫ ∞

0

w(x,y,z)A(z)dz = 1,

w(x,y,z) = w(y,x,z) for z ≥ 0,

w(x,y,z) = w(x,z,y) for z > 0;

for x ≥ 0, the function λ → ϕλ(x) is analytic on C;
for λ ∈ C, the function x → ϕλ(x) is even and infinitely differentiable on R;
for all λ,x ∈ R,

|ϕλ(x)| ≤ 1; (2.1)
for all λ,x > 0,

ϕλ(x) =
1√
B(x)

jα(λx) +
1√
A(x)

θλ(x),

where jα is defined by jα(0) = 1 and jα(s) = 2
αΓ(α + 1)s−αJα(s) if s 6= 0 (with

Jα the Bessel function of first kind), and the function θλ satisfies

|θλ(x)| ≤
c1

λα+
3
2

(∫ x
0

|Q(s)|ds
)

exp
(c2
λ

∫ x
0

|Q(s)|ds
)

with c1,c2 positive constants and Q the function defined on ]0,∞[ by

Q(x) =
1
4
−α2

x2
+

1

4

(A′(x)
A(x)

)2
+

1

2

(A′(x)
A(x)

)′
−ρ2.

(II) For nonzero λ ∈ C, the equation ∆u = −λ2u has a solution Φλ satisfying

Φλ(x) =
1√
A(x)

exp(iλx)V (x,λ),

with limx→∞V (x,λ) = 1. Consequently there exists a function (spectral function)

λ 7→ c(λ),

such that

ϕλ = c(λ)Φλ + c(−λ)Φ−λ for nonzero λ ∈ C.
Moreover there exist positive constants k1,k2,k3 such that

k1|λ|α+1/2 ≤ |c(λ)|−1 ≤ k2|λ|α+1/2

for all λ such that Imλ ≤ 0 and |λ| ≥ k3.
Notation 2.1. We denote by
µ the measure defined on [0,∞[ by dµ(x) := A(x)dx; and by Lp(µ), 1 ≤ p ≤∞,

the space of measurable functions f on [0,∞[, such that

‖f‖Lp(µ) :=
(∫ ∞

0

|f(x)|pdµ(x)
)1/p

< ∞, 1 ≤ p < ∞,

‖f‖L∞(µ) := ess sup
x∈[0,∞[

|f(x)| < ∞;

ν the measure defined on [0,∞[ by dν(λ) :=
dλ

2π|c(λ)|2
; and by Lp(ν), 1 ≤ p ≤∞,

the space of measurable functions f on [0,∞[, such that ‖f‖Lp(ν) < ∞.



4 FETHI SOLTANI

The Fourier transform associated with the operator ∆ is defined on L1(µ) by

F(f)(λ) :=
∫ ∞
0

ϕλ(x)f(x)dµ(x) for λ ∈ R.

Some of the properties of the Fourier transform F are collected bellow (see
[2, 13, 15, 16, 17]).

(a) L1 −L∞-boundedness. For all f ∈ L1(µ), F(f) ∈ L∞(ν) and
‖F(f)‖L∞(ν) ≤‖f‖L1(µ). (2.2)

(b) Inversion theorem. Let f ∈ L1(µ), such that F(f) ∈ L1(ν). Then

f(x) =

∫ ∞
0

ϕλ(x)F(f)(λ)dν(λ), a.e. x ∈ [0,∞[. (2.3)

(c) Plancherel theorem. The Dunkl transform F extends uniquely to an isometric
isomorphism of L2(µ) onto L2(ν). In particular,

‖f‖L2(µ) = ‖F(f)‖L2(ν). (2.4)
Let T be measurable set of [0,∞[. We introduce the time-limiting operator PT

by

PTf(t) :=

{
f(t), t ∈ T
0, t ∈ [0,∞[\T. (2.5)

This operator is bounded from Lp(µ), 1 ≤ p ≤∞ into itself and
‖PTf‖Lp(µ) ≤‖f‖Lp(µ), f ∈ Lp(µ). (2.6)

Let W be measurable set of [0,∞[. We introduce the partial sum operator SW
by

F(SWf) = F(f)1W . (2.7)
This operator is bounded from L2(µ) into itself and

‖SWf‖L2(µ) ≤‖f‖L2(µ), f ∈ L2(µ). (2.8)

Theorem 2.2. If ν(W) < ∞ and f ∈ L1(µ) or f ∈ L2(µ),

SWf(x) =

∫
W

ϕλ(x)F(f)(λ)dν(λ). (2.9)

Proof. If f ∈ L1(µ), then by (2.2),

‖F(f)1W‖L1(ν) =
∫
W

|F(f)(w)|dν(w) ≤ ν(W)‖f‖L1(µ),

and

‖F(f)1W‖L2(ν) =
(∫

W

|F(f)(w)|2dν(w)
)1/2

≤
√
ν(W)‖f‖L1(µ).

Thus Fk(f)1W ∈ L1(ν) ∩L2(ν) and by (2.7),

SWf = F−1
(
F(f)1W

)
.

This combined with (2.3) gives the result.
If f ∈ L2(µ), then by (2.4),

‖F(f)1W‖L1(ν) ≤
√
ν(W)‖f‖L2(µ),

and

‖F(f)1W‖L2(ν) ≤‖f‖L2(µ).



DONOHO-STARK UNCERTAINTY PRINCIPLE 5

Thus F(f)1W ∈ L1(ν) ∩L2(ν). This yields the desired result. �

3. An L1 uncertainty principle

Let T and W be measurable sets of [0,∞[. We say that a function f ∈ L1(µ) is
ε-concentrated to T if there is a measurable function g(t) vanishing outside T such
that ‖f −g‖L1(µ) ≤ ε‖f‖L1(µ).

If f is εT -concentrated on T in L
1(µ)-norm (g being the vanishing function) then

‖f −PTf‖L1(µ) =
∫
[0,∞[\T

|f(t)|dµ(t) ≤‖f −g‖L1(µ) ≤ εT‖f‖L1(µ)

and therefore f is εT -concentrated to T in L
1(µ)-norm if and only if ‖f−PTf‖L1(µ) ≤

εT‖f‖L1(µ).

Let B1(W) denote the set of functions g ∈ L1(µ) that are bandlimited to W (i.e.
g ∈ B1(W) implies SWg = g).

We say that f is ε-bandlimited to W in L1(µ)-norm if there is a g ∈ B1(W) with
‖f −g‖L1(µ) ≤ ε‖f‖L1(µ).

The space B1(W) satisfies the following property.
Lemma 3.1. Let T and W be measurable sets of [0,∞[. For g ∈ B1(W),

‖PTg‖L1(µ)
‖g‖L1(µ)

≤ µ(T)ν(W).

Proof. If µ(T) = ∞ or ν(W) = ∞, the inequality is clear. Assume that µ(T) < ∞
and ν(W) < ∞. For g ∈ B1(W), from Theorem 2.2,

g(t) =

∫
W

ϕw(t)F(g)(w)dν(w)

and by (2.1) and (2.2),

|g(t)| ≤ ν(W)‖g‖L1(µ).
Hence

‖PTg‖L1(µ) =
∫
T

|g(t)|dµ(t) ≤ µ(T)ν(W)‖g‖L1(µ).

Therefore, for g ∈ B1(W),
‖PTg‖L1(µ)
‖g‖L1(µ)

≤ µ(T)ν(W),

which yields the result. �

It is useful to have uncertainty principle for the L1(µ)-norm.
Theorem 3.2. Let T and W be measurable sets of [0,∞[ and f ∈ L1(µ). If f is
εT -concentrated to T and εW -bandlimited to W in L

1(µ)-norm, then

µ(T)ν(W) ≥
1 −εT −εW

1 + εW
.

Proof. Let f ∈ L1(µ). The triangle inequality gives
‖PTf‖L1(µ) ≥‖f‖L1(µ) −‖f −PTf‖L1(µ).

Since f is εT -concentrated to T in L
1(µ)-norm,

‖PTf‖L1(µ) ≥ (1 −εT )‖f‖L1(µ). (3.1)



6 FETHI SOLTANI

On the other hand, f is εW -bandlimited to W in L
1(µ)-norm, by definition there

is a g in B1(W) with ‖f −g‖L1(µ) ≤ εW‖f‖L1(µ). For this g and by (2.6), we have

‖PTg‖L1(µ) ≥ ‖PTf‖L1(µ) −‖PT (f −g)‖L1(µ)
≥ ‖PTf‖L1(µ) −εW‖f‖L1(µ)

and also

‖g‖L1(µ) ≤ (1 + εW )‖f‖L1(µ).
So that

‖PTg‖L1(µ)
‖g‖L1(µ)

≥
‖PTf‖L1(µ) −εW‖f‖L1(µ)

(1 + εW )‖f‖L1(µ)
.

Thus, by (3.1) we deduce

‖PTg‖L1(µ)
‖g‖L1(µ)

≥
1 −εT −εW

1 + εW
.

This combined with Lemma 3.1 proves Theorem 3.2. �

4. An L2 uncertainty principles

Let T and W be measurable sets of [0,∞[. We say that a function f ∈ L2(µ) is
ε-concentrated to T if there is a measurable function g(t) vanishing outside T such
that ‖f −g‖L2(µ) ≤ ε‖f‖L2(µ). Similarly, we say that F(f) is ε-concentrated to W
if there is a function h(w) vanishing outside W with ‖F(f) −h‖L2(ν) ≤ ε‖f‖L2(µ).

If f is εT -concentrated to T in L
2(µ)-norm (g being the vanishing function) then

‖f −PTf‖L2(µ) =
(∫

[0,∞[\T
|f(t)|2dµ(t)

)1/2
≤‖f −g‖L2(µ) ≤ εT‖f‖L2(µ)

and therefore f is εT -concentrated to T in L
2(µ)-norm if and only if ‖f−PTf‖L2(µ) ≤

εT‖f‖L2(µ).
From (2.7) it follows as for PT that F(f) is εW -concentrated to W in L2(ν)-norm

if and only if

‖F(f) −F(SWf)‖L2(ν) = ‖f −SWf‖L2(µ) ≤ εW‖f‖L2(µ).

Let B2(W) denote the set of functions g ∈ L2(µ) that are bandlimited to W (i.e.
g ∈ B2(W) implies SWg = g).

We say that f is ε-bandlimited to W in L2(µ)-norm if there is a g ∈ B2(W) with
‖f −g‖L2(µ) ≤ ε‖f‖L2(µ).

The space B2(W) satisfies the following property.
Lemma 4.1. Let T and W be measurable sets of [0,∞[. For g ∈ B2(W),

‖PTg‖L2(µ)
‖g‖L2(µ)

≤
√
µ(T)ν(W).

Proof. Assume that µ(T) < ∞ and ν(W) < ∞. For g ∈ B2(W), from (2.9),

g(t) =

∫
W

ϕw(t)F(g)(w)dν(w)

and by (2.1) and Hölder’s inequality,

|g(t)| ≤
√
ν(W)‖g‖L2(µ).



DONOHO-STARK UNCERTAINTY PRINCIPLE 7

Hence

‖PTg‖L2(µ) =
(∫

T

|g(t)|2dµ(t)
)1/2

≤
√
µ(T)ν(W)‖g‖L2(µ).

Therefore, for g ∈ B2(W),
‖PTg‖L2(µ)
‖g‖L2(µ)

≤
√
µ(T)ν(W) ,

which yields the result. �

It is useful to have uncertainty principle for the L2(µ)-norm.
Theorem 4.2. Let T and W be measurable sets of [0,∞[ and f ∈ L2(µ). If f is
εT -concentrated to T and εW -bandlimited to W in L

2(µ)-norm, then√
µ(T)ν(W) ≥

1 −εT −εW
1 + εW

.

Proof. Let f ∈ L2(µ). The triangle inequality gives
‖PTf‖L2(µ) ≥‖f‖L2(µ) −‖f −PTf‖L2(µ).

Since f is εT -concentrated to T in L
2(µ)-norm,

‖PTf‖L2(µ) ≥ (1 −εT )‖f‖L2(µ). (4.1)
On the other hand, f is εW -bandlimited to W in L

2(µ)-norm, by definition there
is a g in B2(W) with ‖f −g‖L2(µ) ≤ εW‖f‖L2(µ). For this g and by (2.6), we have

‖PTg‖L2(µ) ≥ ‖PTf‖L2(µ) −‖PT (f −g)‖L2(µ)
≥ ‖PTf‖L2(µ) −εW‖f‖L2(µ)

and also
‖g‖L2(µ) ≤ (1 + εW )‖f‖L2(µ).

So that
‖PTg‖L2(µ)
‖g‖L2(µ)

≥
‖PTf‖L2(µ) −εW‖f‖L2(µ)

(1 + εW )‖f‖L2(µ)
.

Thus, by (4.1) we deduce

‖PTg‖L2(µ)
‖g‖L2(µ)

≥
1 −εT −εW

1 + εW
.

This combined with Lemma 4.1 proves Theorem 4.2. �

Lemma 4.3. Let T and W be measurable sets of [0,∞[. For f ∈ L2(µ),
‖SWPTf‖L2(µ)
‖f‖L2(µ)

≤
√
µ(T)ν(W).

Proof. Assume that µ(T) < ∞ and ν(W) < ∞.
Let f ∈ L2(µ). From (2.5) and (2.9),

SWPTf(s) =

∫
W

ϕw(s)F(PTf)(w)dν(w)

=

∫
W

ϕw(s)

∫
T

ϕw(t)f(t)dµ(t)dν(w).

Since by (2.1),∫
W

∫
T

∣∣∣ϕw(s)ϕw(t)f(t)∣∣∣dµ(t)dν(w) ≤ ν(W)√µ(T)‖f‖L2(µ) < ∞



8 FETHI SOLTANI

by Fubini’s theorem,

SWPTf(s) =

∫
T

f(t)

∫
W

ϕw(s)ϕw(t)dν(w)dµ(t),

so that

SWPTf(s) =

∫
T

q(s,t)f(t)dµ(t), (4.2)

where

q(s,t) =

∫
W

ϕw(s)ϕw(t)dν(w), t ∈ T,s ∈ [0,∞[.

For t ∈ T , let

gt(s) = q(s,t) =

∫
W

ϕw(s)ϕw(t)dν(w).

Then the inversion formula (2.3) shows that

F(gt)(w) = 1Wϕw(t).
By Plancherel’s formula (2.4) it then follows∫ ∞

0

|q(s,t)|2dµ(s) =
∫ ∞
0

|gt(s)|2dµ(s) =
∫ ∞
0

|F(gt)(w)|2dν(w) ≤ ν(W).

By applying Hölder’s inequality to (4.2),

|SWPTf(s)|2 ≤‖f‖2L2(µ)
∫
T

|q(s,t)|2dµ(t).

Hence

‖SWPTf‖L2(µ) ≤‖f‖L2(µ)
(∫ ∞

0

∫
T

|q(s,t)|2dµ(t)dµ(s)
)1/2

.

By Fubini-Tonnelli’s theorem,

‖SWPTf‖L2(µ) ≤‖f‖L2(µ)
(∫

T

∫ ∞
0

|q(s,t)|2dµ(s)dµ(t)
)1/2

≤‖f‖L2(µ)
√
µ(T)ν(W).

Thus, the proof is complete. �

Another uncertainty principle for L2(µ)-norm is obtained.
Theorem 4.4. Let T and W be measurable sets of [0,∞[ and f ∈ L2(µ). If
f is εT -concentrated to T in L

2(µ)-norm and F(f) is εW -concentrated to W in
L2(ν)-norm, then √

µ(T)ν(W) ≥ 1 −εT −εW .
Proof. Let f ∈ L2(µ). From (2.8) it follows

‖f −SWPTf‖L2(µ) ≤ ‖f −SWf‖L2(µ) + ‖SWf −SWPTf‖L2(µ)
≤ εW‖f‖L2(µ) + ‖f −PTf‖L2(µ)
≤ (εT + εW )‖f‖L2(µ).

The triangle inequality gives

‖SWPTf‖L2(µ) ≥‖f‖L2(µ) −‖f −SWPTf‖L2(µ) ≥ (1 −εW −εT )‖f‖L2(µ).
It then follows that ‖SWPTf‖L2(µ) ≥ (1−εW −εT )‖f‖L2(µ). The Lemma 4.3 show
that √

µ(T)ν(W)‖f‖L2(µ) ≥ (1 −εT −εW )‖f‖L2(µ),
which gives the desired result. �



DONOHO-STARK UNCERTAINTY PRINCIPLE 9

An uncertainty principle for L1(µ) ∩L2(µ) theory is obtained.
Theorem 4.5. Let T and W be measurable sets of [0,∞[ and f ∈ L1(µ) ∩L2(µ).
If f is εT -concentrated to T in L

1(µ)-norm and F(f) is εW -concentrated to W in
L2(ν)-norm, then √

µ(T)ν(W) ≥ (1 −εT )(1 −εW ).
Proof. Assume that µ(T) < ∞ and ν(W) < ∞.

Let f ∈ L1(µ) ∩L2(µ). Since F(f) is εW -concentrated to W in L2(ν)-norm, then

‖f‖L2(µ) ≤ εW‖f‖L2(µ) +
(∫

W

|F(f)(w)|2dν(w)
)1/2

≤ εW‖f‖L2(µ) +
√
ν(W)‖F(f)‖L∞(ν).

Thus by (2.2),

(1 −εW )‖f‖L2(µ) ≤
√
ν(W)‖f‖L1(µ). (4.3)

On the other hand, since f is εT -concentrated to T in L
1(µ)-norm,

‖f‖L1(µ) ≤ εT‖f‖L1(µ) +
∫
T

|f(t)|dµ(t)

≤ εT‖f‖L1(µ) +
√
µ(T)‖f‖L2(µ).

Thus
(1 −εT )‖f‖L1(µ) ≤

√
µ(T)‖f‖L2(µ). (4.4)

Combining (4.3) and (4.4) we obtain the result of this theorem. �

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Department of Mathematics, Faculty of Science, Jazan University, P.O.Box 114,
Jazan, Kingdom of Saudi Arabia

Author partially supported by DGRST project 04/UR/15-02 and CMCU program 10G

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