On $L^{1}$-Convergence of Fourier Series Under $MVBV$ Condition
arXiv:0704.1865v1  [math.CA]  14 Apr 2007
On L1-Convergence of Fourier Series Under
MVBV Condition
Dansheng Yu,∗Ping Zhou†and Songping Zhou‡
Abstract
Let f ∈ L2π be a real-valued even function with its Fourier series a02 +
an cosnx,
and let Sn (f, x) , n ≥ 1, be the n-th partial sum of the Fourier series. It is well-known
that if the nonnegative sequence {an} is decreasing and lim
an = 0, then
‖f − Sn(f)‖L = 0 if and only if lim
an logn = 0.
We weaken the monotone condition in this classical result to the so-called mean value
bounded variation (MVBV ) condition. The generalization of the above classical result
in real-valued function space is presented as a special case of the main result in this
paper which gives the L1-convergence of a function f ∈ L2π in complex space. We also
give results on L1-approximation of a function f ∈ L2π under the MVBV condition.
2000 Mathematics Subject Classification: 42A25, 41A50.
Keywords: complex trigonometric series, L1 convergence, monotonicity, mean value
bounded variation.
1 Introduction
Let L2π be the space of all complex-valued integrable functions f(x) of period 2π
equipped with the norm
‖f‖L =
|f(x)|dx.
∗Supported in part by NSERC RCD grant of St. Francis Xavier University and in part by
AARMS of Canada.
†Supported by NSERC of Canada.
‡The third author’s research is done as a W. F. James Chair Professor of St. Francis Xavier
University. His research is also supported in part by NSF of China under grant number 10471130.
http://arxiv.org/abs/0704.1865v1
Denote the Fourier series of f ∈ L2π by
f̂(k)eikx, (1)
and its partial sum Sn(f, x) by
f̂(k)eikx.
When f(x) ∈ L2π is a real valued even function, then the Fourier series of f has the
ak cos kx, (2)
correspondingly, its partial sum Sn(f, x) is
ak cos kx.
The following two classical convergence results can be found in many monographs
(see [1] and [9], for example):
Result One: If a nonnegative sequence {bn}∞n=1 is decreasing and limn→∞ bn = 0,
then the series
bn sinnx converges uniformly if and only if lim
nbn = 0.
Result Two: Let f ∈ L2π be an even function and (2) be its Fourier series. If
the sequence {an}∞n=0 is nonnegative, decreasing, and limn→∞ an = 0, then
‖f − Sn(f)‖L = 0 if and only if lim
an logn = 0.
These results have been generalized by weakening the monotone conditions of
the coefficient sequences. They have also been generalized to the complex valued
function spaces. The most recent generalizations of Result One can be found in [8]
where the monotonic condition is finally weakened to the MVBV condition (Mean
Value Bounded Variation condition, see Corollary 2 in Section 2 for definition), and
it is proved to be the weakest possible condition we can have to replace the monotone
condition in Result One. The process of generalizing Result Two can be found in many
papers, for example, see [2] - [7]. In this paper, we will weaken the monotone condition
in Result Two (and all its later generalized conditions, see [8] for the relations between
these conditions), to theMV BV condition in the complex valued function spaces (see
Definition 1 in Section 2) in Theorem 1, and give the generalization in real valued
function spaces as a special case of Theorem 1 in Corollary 2. Like the important role
that theMVBV condition plays in generalizing Result One, although we are not able
to prove it here, we propose that Theorem 1 in Section 2 is the ultimate generalization
of Result Two, i.e. the MVBV condition is also the weakest possible condition we
can have to replace the monotone condition in Result Two. We also discuss, under
the MVBV condition, the L1-approximation rate of a function f ∈ L2π in the last
section.
Throughout this paper, we always use C(x) to indicate a positive constant de-
pending upon x only, and use C to indicates an absolute positive constant. They
may have different values in different occurrences.
2 L1 convergence
In this section, we first give the definition ofMV BV condition, or the classMVBV S,
and then prove our main result on L1-convergence of the Fourier series of a complex
valued function f(x) ∈ L2π whose coefficients form a sequence in the class MVBV S.
Definition 1. Let c := {cn}∞n=0 be a sequence of complex numbers satisfying
cn ∈ K(θ1) := {z : | arg z| ≤ θ1} for some θ1 ∈ [0, π/2) and all n = 0, 1, 2, · · · . If
there is a number λ ≥ 2 such that
|∆ck| :=
|ck+1 − ck| ≤ C(c)
k=[λ−1m]
holds for all m = 1, 2, · · · , then we say that the sequence c is a Mean Value Bounded
Variation Sequence, i.e., c ∈ MVBVS, in complex sense, or the sequence c satisfies
the MVBV condition.
Our main result of this paper is:
Theorem 1. Let f(x) ∈ L2π be a complex-valued function. If the Fourier coef-
ficients f̂(n) of f satisfy that {f̂(n)}+∞n=0 ∈ MVBV S and
lim sup
|∆f̂(k)−∆f̂(−k)| log k = 0, (3)
where
∆f̂(k) = f̂(k + 1)− f̂(k), ∆f̂(−k) = f̂(−k − 1)− f̂(−k), k ≥ 0.
‖f − Sn(f)‖L = 0
if and only if
f̂(n) log |n| = 0.
In order to prove Theorem 1, we present the following four lemmas.
Lemma 1. Let {cn} ∈MVBV S, then for any given 1 < µ < 2, we have
|∆ck| log k = O
[λ−1n]≤k≤[λn]
|ck| log k
, n→ ∞,
where the implicit constant depends only on the sequence {cn} and λ.
For sufficiently large n, the lemma can be derived directly from the conditions
that 1 < µ < 2 and {cn} ∈ MVBV S.
Lemma 2. Let {f̂(n)} ∈ K(θ0) for some θ0 ∈ [0, π/2), then
∣f̂(n+ k)
∣ = O (‖f − Sn(f)‖L)
for all n = 1, 2, · · · , where the implicit constant depends only on θ0.
Proof. Write
φ±n(x) :=
ei(k∓n)x − e−i(k±n)x
It follows from a well-known inequality (e.g. see Theorem 2.5 in [6])
sin kx
|φ±n(x)| ≤ 6
Hence
(f(x)− Sn(f, x))φ±n(x)dx
≤ ‖f − Sn(f)‖L,
and therefore
f̂(n + k)
= O(‖f − Sn(f)‖L).
Now as {f̂(n)} ∈ K(θ0) for some θ0 ∈ [0, π/2) and for all n ≥ 1, we have
∣f̂(n + k)
∣ ≤ C(θ0)
Ref̂(n + k)
≤ C(θ0)
f̂(n+ k)
= O(‖f − Sn(f)‖L).
Lemma 3. ([5]). Write
Dk(x) : =
sin((2k + 1)x/2)
2 sin(x/2)
D∗k(x) : =
cos(x/2)−cos((2k+1)x/2)
2 sin(x/2)
|x| ≤ 1/n,
− cos((2k+1)x/2)
2 sin(x/2)
1/n ≤ |x| ≤ π,
Ek(x) : = Dk(x) + iD
k(x).
For k = n, n + 1, · · · , 2n, we have
Ek(±x)−Ek−1(±x) = e±ikx, (4)
Ek(x) + Ek(−x) = 2Dk(x), (5)
‖Ek‖L + ‖Dk‖L = O(log k). (6)
Lemma 4. Let {f̂(n)} ∈MVBV S. If lim
‖f − Sn(f)‖L = 0, then
f̂(n) log n = 0.
Proof. By the definition of MV BV S, we derive that for k = n, n + 1, · · · , 2n,
|f̂(2n)| ≤
|∆f̂(j)|+ |f̂(k)|
|∆f̂(j)|+ |f̂(k)|
j=[λ−1k]
|f̂(j)|
+ |f̂(k)|.
Therefore, it follows that from the fact that
log n ≤ C (λ)
[(λ+1)−2n]
j=[λ]+1
we have
|f̂(2n)| logn ≤ C(λ)|f̂(2n)|
[(λ+1)−2n]
j=[λ]+1
≤ C(λ)
[(λ+1)−2n]
j=[λ]+1
[λ(n+j)]
k=[λ−1(n+j)]
|f̂(k)|+ |f̂(n+ j)|
[(λ+1)−2n]
j=[λ]+1
[λ(n+j)]
k=[λ−1(n+j)]
|f̂(k)|
+C(λ)
[(λ+1)−2n]
|f̂(n+ j)|
= : I1 + I2. (7)
By applying Lemma 2, we see that
I2 ≤ C(λ, θ0)‖f − Sn(f)‖L. (8)
We calculate I1 as follows (note that we may add more repeated terms in the right
hand side of every inequality below):
[(λ+1)−2n]
j=[λ]+1
[λn]+[λj]+1
k=[λ−1n]+[λ−1j]
|f̂(k)|
≤ C(λ)
[(λ+1)−2n]
j=[λ]+1
[(λ+1)2]
[λn]+1
k=[λ−1n]
∣f̂ (m[λ−1j] + k)
≤ C(λ)
[(λ+1)2]
[(λ+1)−2n]
j=[λ]+1
[λn]−[λ−1n]+1
∣f̂ ([λ−1n] +m[λ−1j] + k)
≤ C(λ)
[(λ+1)2]
[λn]−[λ−1n]+1
(λ(λ+1)2)
∣f̂ ([λ−1n] + k + j)
[(λ+1)2]
[λn]−[λ−1n]+1
∥f − S[λ−1n]+k(f)
(by Lemma 2)
[λn]−[λ−1n]+1
∥f − S[λ−1n]+k(f)
. (9)
Finally, by combining (7) - (9) and the condition
‖f − Sn(f)‖L = 0,
we get
f̂(2n) logn = 0.
A similar argument yields that
|f̂(2n + 1)| logn = 0.
This proves Lemma 4.
We now come to the proof of Theorem 1.
Proof of Theorem 1. Sufficiency. Given ε > 0, by (3), there is a 1 < µ < 2
such that
∣∆f̂(k)−∆f̂(−k)
∣ log k ≤ ε (10)
holds for sufficiently large n > 0. Let
τµn,n(f, x) :=
[µn]− n
[µn]−1
Sk(f, x)
be the Vallée Poussin sum of order n of f . Then we have
‖f − τµn,n(f)‖L = 0. (11)
By (4), (5), and applying Abel transformation, we get
τµn,n(f, x)− Sn(f, x)
[µn]− n
k=n+1
([µn]− k)
f̂(k)eikx + f̂(−k)e−ikx
[µn]− n
([µn]− k)
2∆f̂(k)Dk(x)− (∆f̂(k)−∆f̂(−k))Ek(−x)
[µn]− n
[µn]−1
f̂(k + 1)Ek(x)− f̂(−k − 1)Ek(−x)
f̂(n)En(x) + f̂(−n)En(−x)
. (12)
Thus, by (6) and Lemma 1, we have
‖f − Sn(f)‖L
≤ ‖f − τµn,n(f)‖L + ‖τµn,n(f)− Sn(f)‖L
= ‖f − τµn,n(f)‖L +O
∣∆f̂(k)
∣ log k
∣∆f̂(k)−∆f̂(−k)
∣ log k
n≤|k|≤[µn]
|f̂(k)| log |k|
= ‖f − τµn,n(f)‖L +O
[λ−1n]≤|k|≤[λn]
|f̂(k)| log |k|
∣∆f̂(k)−∆f̂(−k)
∣ log k
 , (13)
lim sup
‖f − Sn(f)‖L ≤ ε
follows from (10), (11) and the condition that
f̂(n) log |n| = 0.
This implies that
‖f − Sn(f)‖L = 0.
Necessity. Since {f̂(n)} ∈MV BV S, by applying Lemma 4, we have
f̂(n) log n = 0. (14)
In order to prove lim
f̂(n) log |n| = 0, by applying (12) and (6), we see that for any
given µ, 1 < µ < 2,
‖f̂(−n)En(−x)‖L
≤ ‖τµn,n(f)− Sn(f)‖L
[µn]− n
[µn]−1
f̂(−k − 1)Ek(−x)
|∆f̂(k)−∆f̂(−k)| log k +
∣∆f̂(k)
∣ log k
n≤k≤[µn]
|f̂(k)| log k
. (15)
It is not difficult to see that
[µn]−1
f̂(−k − 1)Ek(−x)
= I +O
n max
n<k≤[µn]
|f̂(−k)|
where
n−1≤|x|≤π
2 sin(x/2)
[µn]−1
f̂(−k − 1)e
i(2k+1)x
Since the trigonometric function system is orthonormal, we have
n−1≤|x|≤π
[µn]−1
f̂(−k − 1)e
i(2k+1)x
sin2(x/2)
k=n+1
|f̂(−k)|2
n max
n≤k≤[µn]
|f̂(−k)|
which yields that
[µn]− n
[µn]−1
f̂(−k − 1)Ek(−x)
n<k≤[µn]
|f̂(−k)|
. (16)
By combining (11), (14) - (16), with Lemma 1 and the condition
‖f − Sn(f)‖L = 0,
and the fact (since f ∈ L2π) that
f̂(−n) = 0,
we have for n→ ∞,
‖f̂(−n)En(−x)‖L ≤
|∆f̂(k)−∆f̂(−k)| log k
+‖τµn,n(f)− Sn(f)‖L
[λ−1n]≤k≤[λn]
|f̂(k)| log k
n<k≤[µn]
|f̂(−k)|
|∆f̂(k)−∆f̂(−k)| log k + o(1). (17)
On the other hand, we have
‖f̂(−n)En(−x)‖L ≥ |f̂(−n)|‖Dn(x)‖L ≥
|f̂(−n)| logn. (18)
Hence, from (17), (18), and (10), we have that
|f̂(−n)| logn ≤
|∆f̂(k)−∆f̂(−k)| log k ≤ ε
holds for sufficiently large n, which, together with (14), completes the proof of neces-
sity.
In view of Lemma 1, we can see that the condition (3) in Theorem 1 can be
replaced by the following condition
lim sup
|∆f̂(−k)| log k = 0,
and the proof of the result is easier. Therefore we have a corollary to Theorem 1.
Corollary 1. Let f(x) ∈ L2π be a complex valued function. If both {f̂(n)}+∞n=0 ∈
MVBV S and {f̂(−n)}+∞n=0 ∈MVBV S, then
‖f − Sn(f)‖L = 0
if and only if
f̂(n) log |n| = 0.
If f(x) is a real valued function, then its Fourier coefficients f̂(n) and f̂(−n) are
a pair of conjugate complex numbers. Consequently, {f̂(n)}+∞n=0 ∈ MVBV S if and
only if {f̂(−n)}+∞n=0 ∈ MVBV S. Thus, we have the following generalization of the
classical result (cf. Result Two in the introduction):
Corollary 2. Let f(x) ∈ L2π be a real valued even function and (2) be its Fourier
series. If A = {an}+∞n=0 ∈MV BV S in real sense, i.e. {an} is a nonnegative sequence,
and there is a number λ ≥ 2 such that
|∆ak| ≤ C(A)
k=[λ−1m]
for all n = 1, 2, . . . , then
‖f − Sn(f)‖L = 0
if and only if
an logn = 0.
3 L1 Approximation
Let En(f)L be the best approximation of a complex valued function f ∈ L2π by
trigonometric polynomials of degree n in L1 norm, that is,
En(f)L := inf
We establish the corresponding L1−approximation theorem in a similar way to
Theorem 1:
Theorem 2. Let f(x) ∈ L2π be a complex valued function, {ψn} a decreasing
sequence tending to zero with
ψn ∼ ψ2n, (19)
i.e., there exist positive constants C1 and C2, such that C1ψn ≤ ψ2n ≤ C2ψn. If both
{f̂(n)}+∞n=0 ∈MV BV S and {f̂(−n)}+∞n=0 ∈MVBV S, then
‖f − Sn(f)‖L = O(ψn) (20)
if and only if
En(f)L = O(ψn) and f̂(n) log |n| = O(ψ|n|). (21)
Proof. Under the condition of Theorem 2, we see from (13) in the proof of
Theorem 1 that
‖f − Sn(f)‖L ≤ ‖f − τµn,n(f)‖L +O
[λ−1n]≤|k|≤[λn]
|f̂(k)| log |k|
≤ C(µ)En(f) +O
[λ−1n]≤|k|≤[λn]
|f̂(k)| log |k|
thus (20) holds if (19) and (21) hold. Now if (20) holds, then
En(f)L = O(ψn)
‖f − τµn,n(f)‖L = O(ψn).
From (7) - (9) in the proof of Lemma 4 and condition (19), we have
|f̂(n)| logn ≤ C(λ)
[λn]−[λ−1n]+1
∥f − S[λ−1n]+j(f)
+C(λ)‖f − Sn(f)‖L
= O(ψn).
Since {f̂(−n)}+∞n=0 ∈ MVBV S, by a similar argument to (22), we also have
∣f̂(−n)
∣ logn = O(ψn).
This completes the proof of Theorem 2.
In particular, if we take
ψn :=
(n+ 1)r
f (r),
n + 1
where r is a positive integer, and ω(f, t)L is the modulus of continuity of f in L
norm, i.e.
ω(f, t)L := max
0≤h≤t
‖f(x+ h)− f(x)‖L.
By Theorem 2 and the Jackson theorem (e.g. see [6] or [9]) in L1−space, we imme-
diately have
Corollary 3. Let f(x) ∈ L2π be a complex valued function. If both {f̂(n)}+∞n=0 ∈
MVBV S and {f̂(−n)}+∞n=0 ∈MVBV S hold, then
‖f − Sn(f)‖L = O
(n+ 1)r
f (r),
n + 1
if and only if
f̂(n) log |n| = O
(n+ 1)r
f (r),
n + 1
This corollary generalizes the corresponding results in [5] and [2].
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Dan Sheng Yu and Song Ping Zhou:
Institute of Mathematics
Zhejiang Sci-Tech University
Xiasha Economic Development Area
Hangzhou, Zhejiang 310018 China
Department of Mathematics, Statitics & Computer Science
St. Francis Xavier University
Antigonish, Nova Scotia, Canada B2G 2W5
Email: dsyu@zjip.com (D. S. Yu)
szhou@zjip.com (S. P. Zhou)
Ping Zhou:
Department of Mathematics, Statitics & Computer Science
St. Francis Xavier University
Antigonish, Nova Scotia, Canada B2G 2W5
Email: pzhou@stfx.ca
