The Journal of Fourier Analysis and Applications
Volume 3, Number 5, 1997
On a T h e o r e m
of lngham
S t 6 p h a n e Jaffard, M a r i u s Tucsnak, a n d E n r i q u e Z u a z u a
ABSTRACT
We prove a trigonometric inequality of lngham's type for nonharmonic Fourier series
when the gap condition between frequencies does not hold any more.
Properties proved for linear combinations of the (ei;~nt), w h e n L,, is a sequence of real numbers,
are often generalizations of the fact that the e i n t / V ~ - are an orthonormal basis of L 2 ([-Jr, zr ]). When
the set of the frequencies is no more Z but a sequence of real (or complex) numbers, the very first
generalization is to investigate under which conditions the (e i)~nt) are a Riesz basis of L 2 ( [ - r r , zr]);
this problem has been settled by Hrushev et al. [4]. Because this criterion is not widely known, we
recall it in the simple case where the sequence (;~,) is real. If v 6 L ~ ( ~ ) , the Hilbert transform of
v is defined by
-Jr
x - t
4-
v(t)dt .
Denote by ~,oo 4- C the space of functions which are the sum of the Hilbert transform of a function
inf { l l f - g l l ~ } . I f A = (Xn)n~zisa
o f L ° ° ( R ) and of a constant, and let d i s t ( f , L°° 4- C) =
g~Z~+C
sequence of real numbers, let
,~(x) = 2
f0" ,,.t
1 4- I~-n -
-ax.
The (e iznx) constitute a Riesz basis of L2([0, a]) if and only i f d i s t ( o t ~ L °° + C) < zr/2.
This condition is, however, too strict to be satisfied in many applications (particularly we have
in mind control theory, where the ~-n naturally appear as eigenvalues of elliptic operators, see [3, 6]
and references therein).
One can relax the basis condition in two directions; either drop independency, and the natural
generalization is given by Frames, a notion introduced in the seminal paper of Duffin and Schaeffer [2]
where frames of complex exponentials were studied; or, in the opposite direction, the completeness
requirement can be dropped: One keeps only the "Riesz basis-type" condition of independency,
namely,
i'_IZa,,e'"'' V.
dt> C
la,,] 2 .
(0.1)
7r
Ingham in [5] determined conditions on the Zn under which such an inequality holds. The problem
was completely settled by Kahane in [7]. It is easy to see that a gap condition is needed for (0.1) to
hold: if~.n is increasing,
33>0:~.n+l-;,n
>~-
Math Subject Classifications. 42A63, 42A75, 49J20.
Keywords and Phrases. Nonharmonic Fourier series, spline functions, Riesz basis, control of PDEs.
1997 CRC Press LLC
ISSN 1069-5869
Stdphane Jaffard, Marius Tucsnak, and Enrique Zuazua
578
(The necessity is clear by taking Ln+l and ,kn arbitrarily close, and a,,+l = -an.) However, a similar
inequality is required in cases where the gap condition holds no more. An example (which is actually
the initial motivation of this paper, see [6]) is given by the set of frequencies Za U Zb when a/b ¢ Q.
In this example, no more than two frequencies can be arbitrarily close. This assumption is clearly
implicit in the following theorem, which is the main result of this paper.
Theorem 1.
Let ~.n be a strictly increasing sequence satisfying
Ln+2 - )-n > 8 > 0 .
(0.2)
Let A = {n : )-,,+1 - ).n < 3/2} and B = {n : n • A and n - 1 fg A}. For any T > 6-v/-()/8, there
exists C > 0 such that
~ a n e ix"` 2at
C~
(la, I2 + la,+ll 2) I,~,+t - ;.,,I 2 + [a, + a , + l l 2 + C ~
nEA
la, I2
nEB
p
(0.3)
Remarks
•
The apparently complicated form taken by this lower bound is nevertheless optimal; indeed,
we will prove in Proposition 1 that the two terms of (0.3) are actually equivalent.
•
Theorem 1 still holds under the weaker hypothesis that the Xn are distinct and 3 N such
that (0.2) holds only for ]nl > N. The proof is exactly the same as the proof given by Ball
and Slemrod for their generalization of Ingham's result in [1] (see also [3]).
•
The constant 6~/-6 is probably not optimal (however, it clearly has to be larger than 2~r).
Actually, the best possible time constant in (0.3) is probably related to the lower density of
the sequence (,kn), and might be obtained as an extension of the techniques introduced by
Kahane in [7].
The constant C does not depend on the sequence (Ln), but only on 3. We actually prove
the existence of a universal constant C' such that C = C'/3.
•
The proof roughly follows the idea of Ingham, with the choice of a more sophisticated kernel.
P r o o f o f T h e o r e m 1.
then
Let f (t) = y~ a,,e i;~.t, k(t) E L I (~)l..)L°°(R) and K (~.) -= f+o~ k(t)e-i)'t dt
k ( t )l f (t )12 dt =oo
k(t )anei;~.t-d"£me-i;'mt dt =
•-oo
(0.4)
= ~'~-~an-d-£mK ()~m - ;~n) •
I f k ( t ) < 0 outside [ - A , A],
f_ke~k(t)lf(t)12dt
00
f_'
k(t)lf(tyl2dt
A
C f_a I f ( t ) 1 2 d t ;
A
and we are thus reduced to estimating (0.4). We will make the following hypotheses on k(t):
k(t) is even,
(0.5)
k(t) <_0 outside [ - A , A ] ,
(0.6)
K(u) = 0 if lul >_ 3 / 2 ,
(0.7)
the maximum of IK(u)l is attained at 0 only,
(0.8)
::IC > 0 such that K(u) = K(0)(1 - Clxl 2 + olxl 2) .
(0.9)
On a Theorem of lngham
579
Let us assume for the moment that such a function exists. If [~.n-~-n- 1[ > 8/2 and lXn -~-n+ 1 I >-~
8/2, the frequency ,kn appears in one term only in (0.4), which is lan 12K(0). Otherwise, ~-n appears
coupled with ~-n+l or Ln-I in a sum of three terms of the form
[an
12K (0) + [an+l[2 K (0) + 2"R.e(anaj----n-n~) K ()~n - ~-n+l )
(0.10)
(or the same expression replacing n + 1 by n - 1). Note that if this second possibility holds, there are
no terms of the form 2T42 (an ~ )
K (~-n - ~-,1-1) or 2 7 ~ e ( a n + 2 ~ K (~.n+2 -- ~.n+l) because the gap
condition (0.2) implies that if 1~,, - ~-,,+l [ < 5/2, then }L,, - ~.n-l [ > 8/2 and [~.n+2 - kn+l[ > 8/2.
Stated otherwise, if two consecutive frequencies are coupled (in the sense that last term in (0.10)
exists), these two frequencies are not coupled with others.
Thus, (0.4) is a sum either of terms [anlZK(O) or of the type given by (0.10). Of course, we
have to deal with the second type only.
We choose a constant d small enough.
•
•
If [,kn - An+l[ > d, [K(L,, - L,,+l)l < r/K(0) with an r/ < 1 (we use the hypothesis that
IK (~.)1 reaches its maximum only at 0); and (0.10) is equivalent to [an 12 + lan+ll 2.
If I~-n - ~-n+ll > d, using the last hypothesis on K, (0.10) is equal to
K(0) (tan 12 + [a,,+l 12 + 2~e(a,,a,---?~)(1 - CIX,1 - ~-,,+112)) + o (ana-d-~--~lXn - Zn+l 12)
= g ( 0 ) (la,, + a,,+l 12 (1 - C[~.,, - X,,+l [2) + C (fan[ 2) + [an+it 2) I~-n - Xn+t[ 2)
+ o (a,la-'-n'~lZn - ~-,,+ 112)
which is equivalent to
I~.,, - ~.,,+l 12 (la, Iz) + la,+ll z) + [a,, + a,+ml z .
We still have to prove the existence of a function k satisfying the hypotheses (0.5) to (0.9). Its
Fourier transform K will be constructed as follows. Let X ( = X o) = 1[_ U2.1/2l and
Xn=X*...*X
T
n convolutions
(X n is thus a piecewise polynomial of degree n). We will choose
I ( )"
K = Xs + - - ~ X 5
,
where the constant A will be fixed later.
The functions X and hence Z 5 are even, so that K and k are also even, hence (0.5); Z is
supported in [ - I / 2 , 1/2], so that X 5 is supported in [ - 3 , 3], hence (0.7) with 8 = 6. Since
(X) = f 1/2 e i t U d u _ sin(t/2) ,
,/-1/2
t
thus
~(t) =
,
~-
(,_
which is negative outside [ - A , A], hence (0.6).
We still have to pick A such that the last two hypotheses hold. We first translate K to bring
its support on [0, 6], and we have to check (0.9) in the neighborhood o f x = 3. The functions X m
(after the same translation) are the classical B-sptines of order m on regular knots usually denoted
by N~n (t). In particular
St~phane Jaffard, Marius Tucsnak, and Enrique Zuazua
580
•
N~n is supported by [0, m + 1],
•
Yk E {0 . . . . . m}, N~n is a polynomial of degree m on [k, k + 1],
•
N ~ E C m-I (~),
•
•
the graph of N~n is symmetric with respect to the axis t = (m + I)/2.
If N m = N~(t - i), the N m are obtained by the following recursion formula (see [8])
N m (t) =
In- 1
( t - i)Nm-l(t) + (i + m + 1 - t)N~+
1 (t)
m
Starting with N/°(t) = 1[i,i+ II, a tedious calculation (or running a small computer program, or finding
the right reference...) yields
NS(t) = t5/120
Ng(t) = ( - 5 t 5 + 3Ot4 - 60t 3 + 60t 2 - 30t + 6)/120
N5(t) (10t 5 -- 120t 4 --I-540t 3 - 1140t 2 + 1170t - 474)/120
if t 6 [0, 1]
if t a [1,2]
if t 6 1 2 , 3 ] ;
In particular, N05(3) = 11/20. The graph of N05 being symmetric with respect to t = 3 we have to
check the last two hypotheses only on [0, 3]. Let us determine under which conditions the last one
holds. On the interval [2, 3],
(NS)"(t) = (5t 3 - 36t 2 + 81t - 5 7 ) / 3 ,
so that (N05)"(3) = - 1 ; and (NS)(4)(t) = 10t - 24 so that (N5)(4)(3) = 6. The second derivative
1 t ~t5~Pt
of No5 + A--r~,,0)
takes the value --1 + 6/A 2 at t = 3, and (0.9) thus holds i f A > ~/-6.
We still have to check hypothesis (0.8) when A > ~/-6. Let us first check that it holds in the
"limit case" A = 4'-6. Let
1
F(t) = N5(t) +-~ (N5)"(t) .
•
F is increasing on [0, 1] because N~ and (N05)" both are, and F(1) = 13/360.
•
On [1, 2], (NSo)" = - 5 t 3 / 6 + 3t 2 - 3t + 1 which is increasing on [1, (6 + ~/-6)/5] and
decreasing on [(6 + q'6)/5, 2]. Note that (N05) " ( 1 ) = 1/6, (N05)" ( ( 6 +
•
~/-6)/5) < 1/2,
and (NoS)'' (2) = 1/3. Since No5 increases from 1/120 to 13/60 on the interval, [F(t)I <
3/10 on [1,2].
We now check that F(t) is increasing on [2, 3]. Since
(N5)'(t) = --fi-t - 3 (5t3 - 33t2 + 6 3 t - 3 9 )
and (/705) (3) (t) = (t - 3)(5t - 9), thus
F(t) = (t ~3) 2 (5t2 - 18t + 19)
which is nonnegative on [2, 3]; and F(3) = 23/60.
Thus, (0.8) holds for A = ~/'6.
Let us show by a perturbation argument that (0.8) also holds for A > ~
Vr6. Let
arbitrarily close to
On a Theoremof lngham
581
F(t) - FE(t) = O(~), so that, for ~ small enough, FE(3) will be larger than sup[0.21FE(t); FE is
increasing on [2, 3] because F and - (N5) " are both increasing, so that (0.8) holds for F~, if E is
small enough. The theorem is proved for 8 = 6. The general case follows by replacing the sequence
~-n by CLn, which amounts to change T into T/C.
[]
One problem that did not appear when the gap condition held is the determination if the order
of magnitude of the lower bound in (0.3) is the correct one. It is actually the case, as shown by the
following proposition which states that, instead of the lower bound, an equivalence actually holds
in (0.3).
Proposition 1.
Let )~n be a sequence satisfying the same assumptions as in Theorem 1. For any T > O, there
exists C > 0 such that
f lz
T
an eight 12d t < C E
(10,,I 2 + l a n + l [ 2 ) [ ) . n + l - ~.n12 + lan + a n + l [ 2 + C E l a n ] 2 .
tlEA
n~B
Note that there is no hypothesis on the time T here.
Proof of Proposition 1. The proof follows the same idea as the proof of Theorem 1, but this
time we use K(u) = X 5, hence k(t) = ~
. Since k(t) > 0 on [ - T , T] for T < 2zr,
f_TT l E anei~'nt 2 dt < C f t-~ k(t) E aneiXnt 2 dt
O0
= EEan'~mK(Lm--Ln),
(0.1 1)
which contains only terms of the form [an [2K(0) (which pose no problem) or of the form
[an[2K(0) + lan+lI2 K(O) + 2T~e (ana-'~) K(Zn - )~n+l)
= x(o) (la. + a.+~ 12) + 2he (a.~-ZZZ)(X(X. -X.+~) -- X(O))
<_ K(O)(]an +an+ll2)+Cl)~n--)~n+ll2(]anl 2-t-jan+l]2) ;
hence the proposition for T < 2~r, since (0.1 1) is also clearly bounded by C ~
follows for arbitrary large values of T by splitting the integral.
[]
la,,I2. The result
References
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control systems. Comm. in Pure and App. Math., 37, 555-587.
[2] Duffin, R.J. and Schaeffer, A.C. (1952). A class of nonharrnonic Fourier series. Trans. Am. Math. Soc., 44,
141-154.
[3] Haraux,A., Quelques m6thods et r6sultats r6cents en tMorie de la contr61abilit6 exacte. Preprint.
[4] Hrushev, S.V., Nikol'skii N.K. and Pavlov, B.S. Unconditional bases of exponentials and reproducing kernels, in
ComplexAnalysis and Spectral Theory, Lecture Notes in Math, 864, 214---335.
[5] Irlgham,A.E. (1936). Some trigonometrical inequalities with applications to the theory of series. Math. Zeischrift,
41,367-379.
[6] Jaffard, S., Tucsnak, M. and Zuazua, E. (1997). Singular internal stabilization of the wave equation. To appear
in the Journal of Differential Equations.
[7] Kahane,J.P. (1962). Pseudo-p6riodicit6 et series de Fourier lacunaires. Annales scientifiques de I'E.N.S.T., 76.
[8] N~irnberger,G. (1989). Approximation by Spline Functions. Springer Verlag, New York.
582
Stdphane Jaffard, Marius Tucsnak, and Enriqtte Zuazua
Received February 24, 1997
Ddpartement de Math6mathiques, Universiti~Paris XII, France and CMLA, ENS Cachan
CMAP, Ecole Polytechnique and Universit6de Versailles
Departamento de MatemfiticaAplicada, UniversidadComplutense, Madrid, Spain
Partially supported by grants PB93-1203 of the DGICYT (Spain) and CHRX-CT94-04771 of the UE.