J I P A

advertisement
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
Contents
JJ
J
II
I
Home Page
Go Back
Close
c 2000 Victoria University
ISSN (electronic): 1443-5756
196-04
Quit
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
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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) .
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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
Title Page
Contents
JJ
J
II
I
Go Back
0≤h≤δ
Close
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.
Quit
Page 4 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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
Contents
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)
JJ
J
II
I
Go Back
F
un + un+1 + · · ·
n
.
(I) implies f is bounded over I.
Close
Quit
Page 6 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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
Title Page
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
Contents
JJ
J
II
I
Go Back
Close
Quit
because
ink (2j− 12 h)
fˆj (nk ) = 2ifˆ(nk )e
sin
nk h
2
.
Page 7 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
fˆ(nk )
2

 1
ω nT , f, I
= O  PnT 1  .
j=1 λj
Page 8 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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
.
Title Page
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
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 12
,
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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
Title Page
Contents
JJ
J
II
I
Go Back
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-
Close
Quit
Page 10 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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
Title Page
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 
JJ
J
II
I
Go Back
Close

.
Now proceeding as in the proof of Theorem 1.1, the theorem is proved using
Lemma 1.5.
Quit
Page 11 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
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.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 12
J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005
http://jipam.vu.edu.au
Download