c 1998 Society for Industrial and Applied Mathematics
°
SIAM J. MATH. ANAL.
Vol. 29, No. 5, pp. 1129–1139, September 1998
005
CONVERGENCE AND DIVERGENCE OF DECREASING
REARRANGED FOURIER SERIES∗
ANTONIO CÓRDOBA† AND PABLO FERNÁNDEZ†
Abstract. In a number of useful applications, e.g., data compression, the appropriate partial
sums of the Fourier series are formed by taking into consideration the size of the coefficients rather
than the size of the frequencies involved. The purpose of this paper is to show the limitations of that
method of summation. We use several results from the number theory to construct counterexamples
to Lp -convergence for p < 2. We also show how to obtain positive results if we combine the two
points of view, i.e., cutting on frequencies and the size of coefficients at the same time. This can be
considered as a kind of uncertainty principle for Fourier sums.
Key words. partial Fourier sums, Lp -convergence, Gaussian sums
AMS subject classifications. 42A20, 11L03
PII. S0036141097320705
1. Introduction. For any function, f ∈L1 (T), we can construct its Fourier series
f (x) ∼
∞
X
fˆ(k)e2πikx .
k=−∞
The traditional way of reconstructing the function from its Fourier coefficients is to
consider the partial sums
X
fˆ(k)e2πikx .
SN f (x) =
|k|≤N
It is well known that there is convergence in norm. If f ∈ Lp (T), for 1 < p < ∞, then
SN f → f in Lp (T) as N → ∞;
and since Carleson [1], it is also known that we have almost everywhere convergence.
However, taking into account the interpretation of the Fourier coefficients as, for
example, the x-ray diffraction pattern of a periodic electron density, it seems to be
more natural to pay attention to the coefficients that give us more information, that
is, those of bigger magnitude, and to reconstruct the function ordering the Fourier
coefficients in decreasing order. The same comments also apply if we are interested in
the application of the Fourier series to signal processing algorithms. The mathematical
expression of this fact leads us to consider, for each λ > 0, partial sums
X
fˆ(k)e2πikx
S̃λ f (x) =
|fˆ(k)|>λ
and their limit when λ → 0+ .
∗ Received by the editors April 30, 1997; accepted for publication (in revised form) August 25,
1997; published electronically April 14, 1998.
http://www.siam.org/journals/sima/29-5/32070.html
† Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049-Cantoblanco, Madrid,
Spain ([email protected], [email protected]).
1129
1130
ANTONIO CÓRDOBA AND PABLO FERNÁNDEZ
In a recent paper [2], Körner answered in the negative a question asked by Carleson and Coifman, proving the existence of a function f ∈ L2 (T) such that
X
2πik x ˆ
f (k)e
lim sup
= ∞ for almost every x ∈ T.
λ→0+
|fˆ(k)|≥λ
Körner’s proof is based on an ingenious modification of a construction due to Olevskii
for the Haar system, and it also uses a probabilistic lemma of Salem and Zygmund.
Using a different method, we show in this paper that Lp -convergence, for p < 2,
also fails for the partial sums S̃λ . More concretely, we have the following theorem.
Theorem 1.
a) If we define a maximal operator
S̃ ∗ f (x) = sup S̃λ f (x) ,
λ>0
then, for all 1 ≤ p < 2, there is a function f ∈ Lp (T) (explicitly constructed) such
that
kS̃ ∗ f kp = ∞.
b) For each p < 2, there exists a function f ∈ Lp such that lim supλ→0+ kS̃λ f kp
= ∞.
Our arguments are of a number theory nature, and we use the Farey dissection
of the interval [0, 1) and the prime number theorem in the proof.
This divergence phenomenon suggests that S̃λ f is not the proper sum to be taken.
One may argue that one reason for the failure of S̃λ f to converge is because we have
not taken into account the uncertainty principle in the following way: it does not
make sense to impose restrictions upon the size of |fˆ(k)| and not upon |k| itself.
Let us consider the modified partial sums
X
δ
f (x) =
fˆ(k)e2πikx , δ > 0.
SN
|k| ≤ N
|fˆ(k)| ≥ N −δ
Then we have the following theorem.
Theorem 2.
a) If δ < 12 , then for every p < 2 there is a function f ∈ Lp (T) such that
δ
f kp = ∞.
k lim sup SN
N →∞
b) If δ ≥ 12 , then
δ
f kp < ∞ for every f ∈ Lp (T), 1 < p ≤ 2.
sup kSN
N
c) If δ > 12 , then we have
δ
f − f kp = 0
lim kSN
N →∞
for every f ∈ Lp (T), 1 < p ≤ 2.
d) If δ = 12 , then for every p < 2 there exists f ∈ Lp (T) such that
δ
f − f kp > 0.
lim sup kSN
N →∞
DECREASING REARRANGED FOURIER SERIES
1131
More generally, given a decreasing function φ, one can consider partial sums
X
φ
f (x) =
SN
fˆ(k)e2πikx .
|k| ≤ N
|fˆ(k)| ≥ φ(N )
φ
depends upon the condition
Our construction shows that the behavior of SN
N · φ2 (N ) = o(1).
Finally, we would like to say that we believe more important than the actual results
presented in this paper, are the methods of construction of the examples. They
illustrate, yet again, the connection between Fourier series and number theory.
2. A trigonometrical sum estimate. Take N large enough and consider
X
j
∗
2πip x e
PN (x) = max PN (x) = max .
1≤j≤N
1≤j≤N p prime
N <p≤N +j
Lemma 1.
kPN∗ kr ≥ C N 3/4−1/2r log−1−1/r (N ), for any 1 < r < 2.
Proof. First, we take the primes q,
have the Farey intervals
Ia/q =
√
N ≤q<
√
2 N . For each a, (a, q) = 1, we
1
a
a
1
− 2, + 2
q
8q
q
8q
.
It is easy to see that these intervals are disjoint. Let us consider the set
q−1
[
[
EN =
√
Ia/q .
a=1
q prime√
N ≤ q < 2N
Then,
kPN∗ krr =
Z
1
0
r
(PN∗ (x)) dx ≥
X
=
√
q prime√
N ≤ q < 2N
Z
r
EN
(PN∗ (x)) dx
q−1 Z
X
a=1
Ia/q
r
(PN∗ (x)) dx.
We have the following fact.
√
Fact 1. If |x − y| ≤ 8q12 , with q ≥ N , then PN∗ (x) ≥ CPN∗ (y). (The proof
follows by summation in parts.)
1132
ANTONIO CÓRDOBA AND PABLO FERNÁNDEZ
Using this fact, we obtain
kPN∗ krr
q−1 Z
X
X
≥C
√
a=1
q prime√
N ≤ q < 2N
1
q 2 a=1 √
dx (PN∗ (a/q))
q−1 X
X
≥C
r
Ia/q
q prime√
N ≤ q < 2N
X
p prime
N < p < 2N
r
2πip a/q e
r
q−1
q−1
X
X
1 X X
2πip a/q =C
e
q 2 a=1 r=1
p prime
q
prime
√
√
N < p < 2N
N ≤ q < 2N
p ≡ r(q)
r
q−1 q−1
X
1 X X 2πir a/q
=C
e
[π(2N,
q,
r)
−
π(N,
q,
r)]
,
q2
√
a=1 r=1
q prime√
N ≤ q < 2N
where π(x, a, b), with (a, b) = 1, counts the number of primes less than or equal to x
in the arithmetic progression b, b + a, b + 2a, b + 3a, . . .. Let us rename the difference
of π’s in our expression as a coefficient bN,q,r , such that bN,q,r ≥ 0. Then, using the
inequality
1/r X
1/2
X
≥
|aj |2
, if r ≤ 2,
|aj |r
we can write
X
kPN∗ krr ≥ C
√
q prime√
N ≤ q < 2N
X
=C
√
q prime√
N ≤ q < 2N

r/2
q−1
q−1

X
X
X
1 
2
2πi a (r−s)/q
b
+
b
b
e
(q
−
1)
N,q,r N,q,s
N,r,q

q2 
r=1
a=1
X
=C
√

2 r/2
q−1 q−1

1 X X 2πir a/q
e
b
N,r,q

q 2 a=1 r=1
q prime√
N ≤ q < 2N
r6=s

r/2
q−1


X
X
1
2
b
−
b
b
.
(q
−
1)
N,q,r N,q,s
N,q,r

q2 
r=1
r6=s
Since the inequality
N
N
X
a2j
j=1
holds for ak ≥ 0, we have
kPN∗ krr
≥C
X
√ q prime√
N ≤ q < 2N
1
q2
(q−1
X
r=1
−
X
aj ak ≥
j6=k
)r/2
b2N,q,r
N
X
a2j
j=1
C
≥
N
X
√ q prime√
N ≤ q < 2N
(q−1
X
r=1
)r/2
b2N,q,r
.
DECREASING REARRANGED FOURIER SERIES
1133
Pq−1 2
Now we want to estimate the size of the interior sum,
r=1 bN,q,r . In general, it
is quite difficult to obtain lower bounds for the size of the bN,q,r ’s, even assuming
the generalized Riemann hypothesis, especially if we want to make these estimates
uniform in q and r (in the considered range). Fortunately, we are dealing with the sum
of the squares and this makes things easier. By Cauchy’s inequality and Chebyshev’s
theorem, we have
(q − 1)
q−1
X
b2N,q,r
≥
Ãq−1
X
r=1
!2
r=1
N2
.
log2 (N )
r/2
bN,q,r
>C
Therefore,
kPN∗ krr ≥
≥
C
N
X
√
q prime√
N ≤ q < 2N
N2
(q − 1) log2 (N )
√
√
Nr
C
#{primes q / N ≤ q < 2N }
r
r/4
N N
log (N )
≥ C N 3r/4−1/2 log−r−1 (N )
3. Basic construction. Take α > 0 (to be determined) and consider the functions
k+1
2
∞
−1
X
1 X
cos(2π n x),
f1 (x) = 1 + 2
2kα
k=0
n=2k
{z
}
|
f 1k (x)
2
k+1
2
∞
−1
X
1 X
an cos(2π n x),
f (x) = 1 + 2
2kα
k=0
n=2k
{z
}
|
f2k (x)
(
where an =
1+
1
n
if n prime,
1−
1
2k
if not.
We can evaluate the Lp norm, for p > 1, of the functions f2k using the well-known
estimates for the Lp -norm of the Dirichlet kernel,
°
° °
°
kf2k kp ≤ °f2k − f21k °p + °f21k °p ≤ C 2k (1−1/p−α) .
Therefore,
kf kp ≤ 1 + 2
∞
X
kf2k kp ≤ 1 + C
k=0
As a result, f ∈ Lp (T) whenever α > 1 − p1 .
∞
X
k=0
2k(1−1/p−α) .
1134
ANTONIO CÓRDOBA AND PABLO FERNÁNDEZ
For each k = 0, 1, 2 . . . and for some j, 2k < j < 2k+1 (we will see later how to
choose j), we construct the sequence
1
1
& 0 as k → ∞.
ak,j = kα 1 +
2
j
The operator S̃ak,j keeps all the frequencies up to 2k ; and from the next dyadic block,
it keeps only some prime frequencies (those less than j):
X e2πi ν x 1 X 2πi ν x X ˆ
2πi ν x
e
+
f (ν)e
|S̃ak,j f (x)| ≥ kα −
.
2 ν prime
ν ν prime
|ν|≤2k
|
2k ≤ ν ≤ j
{z
} 2k ≤ ν ≤ j
{z
}
|
I
II
Now, for each x ∈ T, we can choose j = j(x) in such a way that the first term equals
2−kα P2∗k (x) (see the previous section for the definition of PN∗ ). On the other hand,
both terms I and II are O(1) as k → ∞, so for every x ∈ T, we have
1
supS̃λ f (x) ≥ kα P2∗k (x) − O(1).
2
λ>0
It follows that
kS̃ ∗ f kp ≥
1
kP ∗k kp − O(1).
2kα 2
Recalling our basic lemma, with N = 2k , we obtain
kS̃ ∗ f kp ≥ Cp 2k(3/4−1/2p−α) k −1−1/p − O(1).
So the Lp norm diverges when α <
an α with
1−
3
4
1
− 2p
. Therefore, for each 1 ≤ p < 2, we can find
3
1
1
<α< −
p
4 2p
such that the function f constructed above satisfies
°
°
°
°
kf kp < ∞ and °S̃ ∗ f ° = ∞.
p
This completes the proof of Theorem 1a).
In order to prove part b), we need an extra argument. Let us begin with the
well-known estimate (see [3], Khintchin inequality)
p
1/p

Z 1 Z 1 2n+1
X−1

rk (t)e2πikx dx dt
∼ 2n/2 ,
0
0 k=2n
where {rk (t)} denotes the Rademacher system of orthonormal functions. We use it
to “construct,” for each p < 2, a polynomial
Pn (x) =
2n+1
X−1
k=2n
ak e2πikx , where ak is either 0 or 1,
DECREASING REARRANGED FOURIER SERIES
1135
with kPn kp ≥ Cp 2n/2 , for some Cp > 0. Next, we consider, for some α > 0 to be
determined, the function
Qn (x) =
2n+1
X−1
2πikx
bk e
, where bk =
k=2n
2−nα
2−nα − 2−n
if ak = 1,
if ak = 0.
Then,
kQn kp ≤ Cp 2n(1−1/p−α) + Cp0 2−n/p + Cp00 2−n/2 ,
thus the function
f (x) =
∞
X
Qn (x)
n=1
satisfies
1
kf kp < ∞ if α > 1 − .
p
On the other hand,
X
fˆ(k)e2πi k x =
|fˆ(k)|≥2−nα
and so
X
fˆ(k)e2πi k x + 2−nα Pn (x),
|k|≤2n
°
°
°
°
X
°
°
ˆ(k)e2πi k x ° ≥ 2−nα kPn k − O(1)
°
f
p
°
°
°
°|fˆ(k)|≥2−nα
p
≥ Cp 2n(1/2−α) − O(1).
Consequently, for each p < 2, we can find an α, 1 − 1/p < α < 1/2, such that
kf kp < ∞ and lim sup kS̃λ f kp = ∞.
λ→0+
This proves part b) of Theorem 1.
4. The modified partial sums. Now we consider, for each δ > 0, the modified
partial sums
X
δ
f (x) =
fˆ(k)e2πikx .
SN
|k| ≤ N
|fˆ(k)| ≥ N −δ
Case δ ≥ 12 . For r ≤ 2, let us compare these operators with the following partial
sums:
Z 1
1/r
δ
δ
f − S N f kr =
|SN
f (x) − SN f (x)|r dx
kSN
Z  1
=

0 
0
X
|k| ≤ N
|fˆ(k)| < N −δ
r 1/r

2πi k x ˆ

f (k)e
dx
1136
ANTONIO CÓRDOBA AND PABLO FERNÁNDEZ
 1
≤

0 

Z

=

X
|k| ≤ N
|fˆ(k)| < N −δ
X
2 1/2

fˆ(k)e2πi k x dx

1/2

|fˆ(k)|2 

|k| ≤ N
|fˆ(k)| < N −δ
¡
1/2
≤ C N N −2δ
≤ C N 1/2−δ .
Therefore,
δ
f − SN f kr = O(1) as N → ∞.
kSN
If f ∈ Lr , the partial sums SN f tend to f in the Lr -norm. Therefore, applying the
triangular inequality, we obtain
°
° δ
°SN f − f ° = O(1)
as N → ∞.
r
This proves part b) of Theorem 2. Part c) is quite easy now, because if δ > 12 , we
get an o(1) as N → ∞, that is, the function is recovered, in norm, when we sum its
Fourier series in this way.
Case δ < 12 . We have to make a slight modification of the function f defined in
section 3. Let us begin with
fk (x) =
2k+1
X−1
1
2k/2
an e2πinx ,
n=2k
(
with an =
Thus,
f (x) =
X
1−
1−
1
b2k/2δ c
1
b2k/2δ c
+
1
n1/2δ
if n is prime,
if not.
fk (x) is in Lp for any p < 2.
k
Next, we translate these frequencies to the right:
gk (x) = e2πix{b2
=
2k/2
c−2k }
fk (x)
b2k/2δ c+2k
X
al+2k −b2k/2δ c e2πilx .
l=b2k/2δ c
P
gk (x) is, of course, in all Lp , p < 2. Now, take the sequence
Ã
!
1
1
1
= k/2δ δ 1 − k/2δ + ¡
1/2δ .
b2
c
b2
c
j + 2k − b2k/2δ c
The function g(x) =
ak,j
1
k/2δ
k
If we estimate the sums
X
a−δ
k,j
|ν| ≤
|ĝ(ν)| ≥ ak,j
ĝ(ν)e2πiνx ,
1137
DECREASING REARRANGED FOURIER SERIES
it is easy to see that we are just summing
X
ĝ(ν)e2πiνx ,
|ĝ(ν)|≥ak,j
and we have seen that these sums diverge in norm as k → ∞ with the adequate choice
of j.
Case δ = 12 . We can use the same arguments with the Rademacher functions used
in the proof of part b) of Theorem 1 to construct, for each p < 2, a polynomial Pn ,
n
with coefficients 0 or 1, such that kPn kp ≥ Cp 2 2 for some Cp > 0; and a polynomial
Qn (putting α = 1/2 in the definition of its coefficients) in such a way that
kQn kp ≤ Cp0 2−n/2 · 2n(1−1/p) .
Therefore,
f (x) =
∞
X
Qn (x) satisfies kf kp ≤
Cp0
n=1
On the other hand,
X
°
°
°
°
°
°
°
°
fˆ(k)e2πi k x =
X
|k| ≤ 2n
|fˆ(k)| ≥ 2−n/2
2−n(1/p−1/2) < ∞.
n=1
|k| ≤ 2n
|fˆ(k)| ≥ 2−n/2
and so
∞
X
X
fˆ(k)e2πi k x + 2−n/2 Pn (x),
|k|≤2n−1
°
°
°
°
°
∞
°X
°
°
°
°
2πi
k
x
−n/2
− f°
≥
2
kP
k
−
Q
fˆ(k)e
°
n p
k°
°
°
°
°
k=n
p
°
p
Cp
≥ √ − o(1)
2
as n → ∞.
5. Final remarks. With the use of Gaussian sums, one may produce explicit
examples of functions f ∈ Lp (T), p < 43 , such that
X
fˆ(k)e2πik x = ∞ for almost every x ∈ T.
lim sup λ→0+ ˆ
|f (k)|≥λ
The construction is as follows. Take
fk (x) =
22k+2
X−1
an cos(2πnx)
22k
with an =
2−k/2+k − 2−3k ,
if
2−k/2+k − 2−3k (22k+2 − |n|)−1/2 , if
Clearly,
kfk kp ∼ 2k(+3/2−2/p) .
n 6= s2 ,
n = s2 .
1138
ANTONIO CÓRDOBA AND PABLO FERNÁNDEZ
For each p < 4/3, taking small enough, we have that
f (x) =
∞
X
fk (x)
∈ Lp .
k=1
Next, we will use the following facts.
Fact 2. If (P, Q) = 1, with M
2 ≤ Q ≤ M and Q 6≡ 2 mod 4, then
M +k
p
X 2πij 2 P/Q ≥ C Q.
e
sup 1≤k≤M j=M
1
M2 ,
then
M
+k
X
2 e2πij x ≥ C
sup 1≤k≤M j=M
Fact 3. If |x − y| ≤
M
+k
X
2 e2πij y .
sup 1≤k≤M j=M
Fact 4. For each x ∈ [0, 1), let us consider the sequence of convergents {Pn /Qn }
of its continuous fraction expansion. Then, for all irrational x, there are infinite Qn ,
which are not congruent with 2 mod 4. Consider the function f defined by
X
X
fk (x) =
fˆ(ν) e2πiνx .
f (x) =
ν
k
Take an irrational x and its sequence {Qn } of denominators not congruent with 2
mod 4. It determines dyadic blocks,
2n−1 ≤ Qn ≤ 2n .
We introduce the coefficients
an,j =
1
2n/2−n
−
23n
1
p
,
2n+2
2
− |j|
where n identifies the dyadic block previously selected and j, 22n < j < 22n+2 , is
chosen so that
X
p
k(j) 2πiν 2 P /Q n
n
e
≥ C Qn ,
ν=2n
√
where k(j) is the first integer less or equal j. Then,
X
X
k(j)
2
1
2πis x ˆ(ν)e2πiνx ≥
e
f
2n/2−n + O(1)
s=2n
|fˆ(ν)|≥a
n,j
≥
Hence,
C
2n/2−n
X
fˆ(ν)e2πiνx = ∞
sup λ→0+ ˆ
|f (ν)|≥λ
p
Qn + O(1) ≥ C 2n + O(1).
for almost all x ∈ [0, 1).
DECREASING REARRANGED FOURIER SERIES
1139
As we stated in the introduction, Körner [2] obtained almost everywhere divergence
for functions f in the class L2 (T).
REFERENCES
[1] L. Carleson, On the convergence and growth of partial sums of Fourier series, Acta Math.,
116 (1966), pp. 135–157.
[2] T.W. Körner, Divergence of decreasing rearranged Fourier series, Ann. Math., 144 (1996),
pp. 167–180.
[3] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 1988.