Journal of Inequalities in Pure and
Applied Mathematics
ON THE ABSOLUTE CONVERGENCE
OF SMALL GAPS FOURIER
V
(p)
SERIES OF FUNCTIONS OF BV
volume 6, issue 1, article 23,
2005.
R.G. VYAS
Department of Mathematics
Faculty of Science
The Maharaja Sayajirao University of Baroda
Vadodara-390002, Gujarat, India.
Received 18 October, 2004;
accepted 29 November, 2004.
Communicated by: L. Leindler
EMail: drrgvyas@yahoo.com
Abstract
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c 2000 Victoria University
ISSN (electronic): 1443-5756
196-04
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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 ≥ 1. Here we have obtained
V sufficiency
conditions for the absolute convergence of such series if f is of BV (p) locally.
We have also obtained a beautiful interconnection between lacunary and nonlacunary Fourier series.
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
2000 Mathematics Subject Classification: 42Axx.
Key words: FourierVseries with small gaps, Absolute convergence of Fourier series
and p- -bounded variation.
R.G. Vyas
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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, satisfy 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 BV (p)
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.
By applying the Wiener-Ingham result [1, Vol. I, p. 222] for the finite
trigonometric sums with small gap (1.2) we have studied the sufficiency condiβ
P
tion for the convergence of the series k∈Z fb(nk ) (0 < β ≤ 2) in terms of
V
BV and the modulus of continuity [2, Theorem 3]. Here we have generalized
this
and we have also obtained a sufficiency condition
if function f is of
V result
V
BVV(p) . In 1980 Shiba [4] generalized the class BV . He introduced the
class BV (p) .
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Definition
positive real
V1.1. Given an interval I, a sequence
Pof non-decreasing
1
numbers = {λm } (m = 1, 2, . . .) such that m λm diverges and 1 ≤ p < ∞
V
V
we say that f ∈ BV (p) (that is f is a function of p − -bounded variation
over (I)) if
VΛp (f, I) = sup {VΛp ({Im }, f, I)} < ∞,
{Im }
where
VΛp ({Im }, f, I) =
X |f (bm ) − f (am )|p
m
! p1
λm
,
and {Im } is a sequence of non-overlapping subintervals Im = [am , bm ] ⊂ I =
[a, b].
V
Note that, if p = 1, one gets the class BV (I); if λm ≡ 1 for all m, one gets
the class BV (p) ; if p = 1 and λm ≡ m for all m, one gets the class Harmonic
BV (I). if p = 1 and λm ≡ 1 for all m, one gets the class BV (I).
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 ,
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
R.G. Vyas
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0≤h≤δ
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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).
We prove the following theorems.
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Theorem 1.1. Let f ∈ L[−π, π] possess a Fourier V
series with ‘small’ gaps
2π
(1.2) and I be a subinterval of length δ1 > q . If f ∈ BV (I) and
 β2
ω( n1k , f, I)
  < ∞,
Pnk 1 k
k=1
j=1 λj
∞
X

then
X
(1.4)
β
<∞
fb(nk )
(0 < β ≤ 2).
k∈Z
P k 1
Since {λj } is non-decreasing, one gets nj=1
≥ λnnk and hence our earlier
λj
k
theorem [2, Theorem 3] follows from Theorem 1.1.
Theorem 1.1 with β = 1 and λn ≡ 1 shows that the Fourier series of f
with ‘small’ gaps condition (1.2) (respectively (1.3)) converges absolutely if
the hypothesis of the Stechkin theorem [5, Vol. II, p. 196] is satisfied only in a
subinterval of [0, 2π] of length > 2π
(respectively of arbitrary positive length).
q
V
Theorem 1.2. Let f and I be as in Theorem 1.1. If f ∈ BV (p) (I), 1 ≤ p <
2r, 1 < r < ∞ and
∞
X
k=1
where
1
r
+
1
s

 ω

((2−p)s+p)
k
P
1
, f, I
nk
nk
j=1
= 1, then (1.4) holds.
2−p/r  β2
r1
1
λj

 < ∞,
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
R.G. Vyas
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Theorem 1.2 with β = 1 is a ‘small’ gaps analogue of the Schramm and
Waterman result [3, Theorem 1].
We need the following lemmas to prove the theorems.
Lemma 1.3 ([2, Lemma 4]). Let f and I be as in Theorem 1.1. If f ∈ L2 (I)
then
X
2
fb(nk ) ≤ Aδ |I|−1 kf k22,I ,
(1.5)
k∈Z
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
where Aδ depends only on δ.
Lemma 1.4. If |nk | > p then for t ∈ N one has
π
p
Z
sin2t |nk | h dh ≥
0
R.G. Vyas
π
2t+1 p
.
Title Page
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Proof. Obvious.
Lemma 1.5 (Stechkin, refer to [6]). If un ≥ 0 for n ∈ N, un 6= 0 and a
function F (u) is concave, increasing, and F (0) = 0, then
∞
X
F (un ) ≤ 2
1
Lemma 1.6. If f ∈
∞
X
1
V
BV
(p)
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F
un + un+1 + · · ·
n
.
(I) implies f is bounded over I.
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Proof. Observe that
p
p
|f (x)| ≤ 2
≤ 2p
|f (b) − f (x)|p
|f (x) − f (a)|p
+ λ2
|f (a)| + λ1
λ1
λ2
p
|f (a)| + λ2 V∧p (f, I)
p
Hence the lemma follows.
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)
4π
|nT | >
δ3
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
R.G. Vyas
δ3 |nT |
and 0 ≤ j ≤
.
4π
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V
Since f ∈ BV (I) implies f is bounded over I by Lemma 1.6 (for p = 1),
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 possesses
gaps (1.2). Hence by Lemma 1.3 we get
X
2
nk h
2
2
ˆ
(1.7)
f (nk ) sin
= O kfj k2,J
2
k∈Z
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because
ink (2j− 12 h)
fˆj (nk ) = 2ifˆ(nk )e
sin
nk h
2
.
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Integrating both the sides of (1.7) over (0, nπT ) with respect to h and using
Lemma 1.4, we get
Z π
∞
X
2
nT
fˆ(nk ) = O(nT )
(1.8)
k fj k22,J dh.
0
|nk |≥nT
Multiplying both the sides of the equation by λ1j and then taking summation
over j, we get



! ∞
Z π
X |fj |2
X 1
X
2
nT


 dh.
(1.9)
fˆ(nk )  = O(nT )
λ
λ
j
j
0
j
j
|nk |≥nT
1,J
Now, since x ∈ J and h ∈ (0, nπT ) we have |fj (x)| = O(ω( n1T , f, I)), for each
V
P |f (x)|
j of the summation; since x ∈ J and f ∈ BV (I) we have j jλj = O(1)
because for each j the points x + 2jh and x + (2j − 1)h lie in I for h ∈ (0, nπT )
and x ∈ J ⊂ I. Therefore
!
!
X
X |fj (x)|2
1
|fj (x)|
=O ω
, f, I
λ
n
λj
j
T
j
j
1
=O ω
, f, I
.
nT
It follows now from (1.9) that
RnT =
X
|nk ≥nT
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
R.G. Vyas
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fˆ(nk )
2

 1
ω nT , f, I
= O  PnT 1  .
j=1 λj
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2
Finally, Lemma 1.5 with uk = fˆ(nk ) (k ∈ Z) and F (u) = uβ/2 gives
∞
X
fˆ(nk )
β
=2
|k|=1
∞
X
fˆ(nk )
Rnk
k
β/2
F
2
k=1
≤4
≤4
∞
X
F
k=1
∞ X
k=1

= O(1) 
Rnk
k
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
∞
X
ω( n1k , f, I)
k=1
1
k
k(Σnj=1
)
λj
!(β/2) 
R.G. Vyas
.
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This proves the theorem.
Contents
Proof of Theorem 1.2. Since f ∈ BV (p) (I), Lemma 1.6 implies f is bounded
over I. Therefore f ∈ L2 (I), and hence (1.5) holds so that f ∈ L2 [−π, π].
Using the notations and procedure of Theorem 1.1 we get (1.9). Since 2 =
(2−p)s+p
+ pr , by using Hölder’s inequality, we get from (1.9)
s
V
Z
2
Z
|fj (x)| dx ≤
J
(2−p)s+p
|fj (x)|
1s Z
|fj (x)| dx
dx
J
1/r
J
Z
≤ Ωh,J
J
|fj (x)|p dx
p
r1
r1
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where Ωh,J = (ω (2−p)s+p (h, f, J))2r−p .
This together with (1.9) implies, putting
B=
X
2
fb(nk ) sin2
k∈Z
nk h
2
that
B≤
1/r
Ωh,J
Z
,
r1
p
|fj (x)| dx
.
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
J
Thus
Z
r
B ≤ Ωh,J
p
|fj (x)| dx .
R.G. Vyas
J
Now multiplying both the sides of the equation by λ1j and then taking the summation over j = 1 to nT (T ∈ N) we get
R P
|fj (x)|p
Ωh,J J
dx
j
λj
P 1
Br ≤
.
j λj
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Therefore
B≤
Ω
Ph,J1
j λj
! r1
Z
J
p
X |fj (x)|
j
λj
! r1
!
dx
.
Substituting back the value of B and then integrating both the sides of the equa-
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tion with respect to h over (0, nπT ), we get
(1.10)
X
k∈Z
2
Z
π/nT
fb(nk )
sin
2
0
|nk | h
2
 r1

Ω1/n ,J
= O  P T 
1
dh
π/nT
Z
0
j λj
X |fj (x)|p
Z
J
λj
j
! r1
!
dx
dh.
Observe that for x in J, h in (0, nπT ) and for each j of the summation the points
V
x + 2jh and x + (2j − 1)h lie in I; moreover,f ∈ BV (p) (I) implies
R.G. Vyas
X |fj (x)|p
j
= O(1).
λj
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Therefore, it follows from (1.10) and Lemma 1.4 that
RnT ≡
X
2
fb(nk )
=O
Ω1/n ,I
PnT T 1

ω
RnT = O 
Contents
! r1
.
j=1 λj
|nk |≥nT
Thus
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
(2−p)s+p
P
1
nT
, f, I
r1
nT 1
j=1 λj
2−p/r 
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
.
Now proceeding as in the proof of Theorem 1.1, the theorem is proved using
Lemma 1.5.
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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. Anal. and Appl., 182(1) (1994),
113–126.
[3] M. SCHRAM AND V
D. WATERMAN,
V Absolute convergence of Fourier se(p)
ries of functions of BV and Φ BV , Acta. Math. Hungar, 40 (1982),
273–276.
On the Absolute Convergence
of Small Gaps Fourier
V Series of
Functions of BV (p)
R.G. Vyas
[4] M. SHIBA,
On the absolute convergence of Fourier series of functions of
V
(p)
class BV , Sci. Rep. Fukushima Univ., 30 (1980), 7–10.
[5] N.K. BARRY, A Treatise on Trigonometric Series,
York,1964.
Pergamon, New
[6] 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.
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