Journal of Inequalities in Pure and
Applied Mathematics
ON THE ABSOLUTE CONVERGENCE OFVSMALL GAPS
FOURIER SERIES OF FUNCTIONS OF ϕ BV
volume 6, issue 4, article 94,
2005.
R. G. VYAS
Department of Mathematics
Faculty of Science
The Maharaja Sayajirao University of Baroda
Vadodara-390002, Gujarat, India.
Received 06 July, 2005;
accepted 29 July, 2005.
Communicated by: L. Leindler
EMail: drrgvyas@yahoo.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
P
ink x be its Fourier
b
Let f be a 2π periodic function in L1 [0, 2π] and ∞
k=−∞ f(nk )e
series with ‘small’ gaps nk+1 − nk ≥ q ≥P1. Here we obtain a sufficiency
b k ) |β (0< β ≤2) if f is of
condition for the convergence of the series k∈Z | f(n
ϕ ∧ BV locally. We also obtain beautiful interconnections between the types of
lacunarity in Fourier series and the localness of the hypothesis to be satisfied by
the generic function allows us to interpolate results concerning lacunary Fourier
series and non-lacunary Fourier series.
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
R. G. Vyas
2000 Mathematics Subject Classification: 42A16, 42A28, 26A45.
Key words: Fourier series with small gaps, Absolute convergence of Fourier series
and ϕ∧-bounded variation.
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Contents
3
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1.
Introduction
Let f be a 2π periodic function in L1 [0, 2π] and fb(n), n ∈ Z, be its Fourier
coefficients. The series
X
(1.1)
fb(nk )eink x ,
k∈Z
wherein {nk }∞
1 is a strictly increasing sequence of natural numbers and n−k =
−nk , for all k, satisfies an inequality
(1.2)
(nk+1 − nk ) ≥ q ≥ 1 for all
k = 0, 1, 2, ...,
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
R. G. Vyas
is called the Fourier series of f with ‘small’ gaps.
Obviously, if nk = k, for all k, (i.e. nk+1 − nk = q = 1, for all k), then we
get non-lacunary Fourier series and if {nk } is such that
(1.3)
(nk+1 − nk ) → ∞ as
k→∞
then (1.1) is said to be the lacunary Fourier series.
In 1982 M. Schramm and D. Waterman [3] have introduced the class ϕ ∧
BV (I) of functions of ϕ∧-bounded variation over I and have studied sufficiency conditions for the absolute convergence of Fourier series of functions of
∧BV (p) and ϕ ∧ BV .
Definition 1.1. Given a nonnegative convex function ϕ, defined on [0, ∞) such
that ϕ(x)
→ 0 as x → 0, for some constant d ≥ 2, ϕ(2x) ≤ dϕ(x)
x
V for all
x ≥ 0 and given a sequence of non-decreasing positive real numbers = {λm }
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V
P
(m = 1, 2, . . .) V
such that m λ1m diverges we say that f ∈ ϕ BV (that is f is
a function of ϕ -bounded variation over (I)) if
VΛϕ (f, I) = sup {VΛϕ ({Im }, f, I)} < ∞,
{Im }
where
VΛϕ ({Im }, f, I) =
X ϕ |f (bm ) − f (am )|
m
λm
!
,
and {Im } is a sequence of non-overlapping subintervals Im = [am , bm ] ⊂ I =
[a, b].
Definition 1.2. For p ≥ 1, the p-integral modulus of continuity ω (p) (δ, f, I) of
f over I is defined as
ω (p) (δ, f, I) = sup k(Th f − f )(x)kp,I ,
0≤h≤δ
where Th f (x) = f (x + h) for all x and k(·)kp,I = k(·)χI kp in which χI is the
characteristic function of I and k(·)kp denotes the Lp -norm. p = ∞ gives the
modulus of continuity ω(δ, f, I).
By applying the Wiener-Ingham result [1, Vol. I, p. 222] for the finite
trigonometric sums with ‘small’ gap (1.2) we have already
the suffi studied
β
P
ciency conditions for the convergence of the series k∈Z fb(nk ) (0 < β ≤ 2)
V
for the functions of BV and ∧BV (p) in terms of the modulus ofVcontinuity
[6]. Here we obtain a sufficiency condition if function f is of ϕ BV . We
prove the following theorem.
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
R. G. Vyas
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Theorem 1.1. Let f ∈ L[−π, π] possess a Fourier series
V with ‘small’ gaps
2π
(1.2) and I be a subinterval of length δ1 > q . If f ∈ ϕ BV (I), 1 ≤ p < 2r,
1 ≤ r < ∞, and
β


2r−p  r1 ,  2
1
((2−p)s+p)
∞
, f, I
X
  −1  ω

nk

 ϕ 
P
k

nk 1

 < ∞,
j=1 λj
k=1
where
(1.4)
1
r
+
1
s
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
= 1, then
β
X b
f (nk ) < ∞
(0 < β ≤ 2).
R. G. Vyas
k∈Z
Theorem 1.1 with β = 1 is a ‘small’ gaps analogue of the Schramm and
Waterman result [3, Theorem 2]. Observe that the interval I considered in the
theorem for the gap condition (1.2) is of length > 2π
, so that when nk = k,
q
for all k, I is of length 2π. Hence for non-lacunary Fourier series (equality
throughout in (1.2)) the theorem with β = 1 gives the Schramm and Waterman
result [3, Theorem 2] as a particular case.
We need the following lemmas to prove the theorem.
2
Lemma 1.2 ([2, Lemma 2]). Let f and I be as in Theorem 1.1. If f ∈ L (I)
then
2
X −1
b
(1.5)
f (nk ) ≤ Aδ |I| k f k22,I ,
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k∈Z
where Aδ depends only on δ.
J. Ineq. Pure and Appl. Math. 6(4) Art. 94, 2005
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Lemma 1.3. If |nk | > p then for t ∈ N one has
Z π
p
π
sin2t |nk | hdh ≥ t+1 .
2 p
0
Proof. Obvious.
Lemma 1.4 (Stechkin, refer to [5]). If un ≥ 0 for n ∈ N, un 6= 0 and a
function F (u) is concave, increasing, and F (0) = 0, then
∞
∞
X
X
un + un+1 + · · ·
F (un ) ≤ 2
F
.
n
1
1
Proof of Theorem 1.1. Let I = x0 − δ21 , x0 + δ21 for some x0 and δ2 be such
that 0 < 2π
< δ2 < δ1 . Put δ3 = δ1 − δ2 and J = x0 − δ22 , x0 + δ22 . Suppose
q
integers T and j satisfy
(1.6)
|nT | >
4π
δ3
and 0 ≤ j ≤
δ3 |nT |
.
4π
f ∈ ϕ ∧ BV (I) implies
|f (x)| ≤ |f (a)| + |f (x) − f (a)| ≤ |f (a)| + Cϕ−1 (V∧ϕ (f, I)) for all x ∈ I.
Since f is bounded over I, we have f ∈ L2 (I), so that (1.5) holds and f ∈
L2 [−π, π]. If we put fj = (T2jh f − T(2j−1)h f ) then fj ∈ L2 (I) and the Fourier
series of fj also possess gaps (1.2). Hence by Lemma 1.2 we get
2
X n
h
k
2
2
ˆ
=
O
kf
k
(1.7)
f
(n
)
sin
k j 2,J
2
k∈Z
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
R. G. Vyas
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because
ink (2j− 12 h)
fˆj (nk ) = 2ifˆ(nk )e
sin
nk h
2
.
Integrating both the sides of (1.7) over (0, nπT ) with respect to h and using
Lemma 1.3, we get
(1.8)
RnT
Z
∞
2
X
ˆ
=
f (nk ) = O(nT )
0
|nk |≥nT
Since 2 =
π
nT
kfj k22,J dh.
(2−p)s+p
s
+ pr , by using Hölder’s inequality, we get from (1.8)
Z
B = |fj (x)|2 dx
J
Z
≤
(2−p)s+p
|fj (x)|
1s Z
≤
r1
|fj (x)| dx
dx
J
1/r
Ωh,J
p
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
R. G. Vyas
Title Page
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J
Z
r1
p
|fj (x)| dx
,
J
where Ωh,J = (ω (2−p)s+p (h, f, J))2r−p . Thus
Z
r
(1.9)
B ≤ Ωh,J |fj (x)|p dx.
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Since f is bounded over I, there exists some positive constant M ≥ 21 such
that |f (x)| ≤ M for all x ∈ I. Dividing f by the positive constant M alters
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ωp (h, f, J) by the same constant M and ϕ(2 |f (x)|) ≤ dϕ(|f (x)|) for all x, we
may assume that |f (x)| ≤ 1 for all x ∈ I. Hence from (1.9) we get
Z
r
B ≤ Ωh,J |fj (x)| dx.
J
Since ϕ(2x) ≤ dϕ(x), we have ϕ(ax) ≤ dlog2 a ϕ(x), so
r
R
|fj (x)| dx
B
log2 CΩh,J
J
ϕ
ϕ
≤d
δ2
δ2
R
|fj (x)| dx
log2 d
J
= CΩh,J ϕ
δ2
R
|f (x)| dx
log2 d−1
J Rj
= CΩh,J
Ωh,J ϕ
1dx
J
R
ϕ |fj (x)| dx
J
≤ CΩh,J
(by Jensen’s inequality for integrals)
δ2
Z
ϕ |fj (x)| dx .
= CΩh,J
J
Multiplying both the sides of the equation by λ1j and then taking the summation
over j = 1 to nT (T ∈ N) we get


! !
r
Z X
nT
Ω
B
ϕ
|f
(x)|
h,J
j
≤ C P 
dx .
(1.10)
ϕ
nT
1
δ2
λ
j
J
j=1
j=1
λj
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
R. G. Vyas
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Observe that for x in J, h in (0, nπT ) and for each j of the summation the points
x + 2jh and x + (2j − 1)h lie in I; moreover f ∈ ϕ ∧ BV (I) implies
nT
X
ϕ |fj (x)|
j=1
λj
= O(1).
Therefore, it follows from (1.10) that


 r1 
Ω1/n ,I


B = O ϕ−1  P T   .
nT
1
j=1
λj
R. G. Vyas
Substituting back the value of B in the equation (1.8), we get

 r1 

∞
X Ω1/n ,I

2

RnT =
fˆ(nk ) = O ϕ−1  P T   .
nT
1
|nk |≥nT
Thus
RnT
j=1
λj
2r−p  r1 
1
(2−p)s+p
, f, I
 −1  ω
nT
 

= O ϕ 
 
PnT
.
1


j=1
λj
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
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2
Finally, Lemma 1.4 with uk = fˆ(nk ) (k ∈ Z) and F (u) = uβ/2 gives
∞
∞ β
2 X
X
ˆ
ˆ
F f (nk )
f (nk ) = 2
|k|=1
k=1
≤4
∞
X
F
k=1
=4
Rnk
k
β2
∞ X
Rn
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
k
k
k=1


= O(1) 
∞
X
k=1
"
 ϕ−1
(ω
( 1 , f, I))2r−p
Pnnkk 1
j=1 λj
(2−p)s+p

!# r1 ,  β2

k  .
R. G. Vyas
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This proves the theorem.
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References
[1] A. ZYMUND, Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cambridge, 1979 (reprint).
[2] J.R. PATADIA AND R.G. VYAS, Fourier series with small gaps and functions of generalized variations, J. Math. Analy. and Appl., 182(1) (1994),
113–126.
[3] M. SCHRAMM ANDVD. WATERMAN,
V Absolute convergence of Fourier
(p)
series of functions of BV and Φ BV, Acta. Math. Hungar, 40 (1982),
273–276.
[4] N.K. BARRY, A Treatise on Trigonometric Series, Pergamon, New York,
1964.
[5] N.V. PATEL AND V.M. SHAH, A note on the absolute convergence of lacunary Fourier series, Proc. Amer. Math. Soc., 93 (1985), 433–439.
[6] R.G. VYAS, On the Absolute convergence of small gaps Fourier series of
functions of ∧BV(p) , J. Inequal. Pure and Appl. Math., 6(1) (2005), Art. 23,
1–6. [ONLINE: http://jipam.vu.edu.au/article.php?sid=
492]
On The Absolute Convergence
Of Small Gaps Fourier
V Series Of
Functions Of ϕ BV
R. G. Vyas
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