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volume 6, issue 1, article 23, 2005.

Received 18 October, 2004;

accepted 29 November, 2004.

Communicated by:L. Leindler

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Journal of Inequalities in Pure and Applied Mathematics

ON THE ABSOLUTE CONVERGENCE OF SMALL GAPS FOURIER SERIES OF FUNCTIONS OF V

BV^(p)

R.G. VYAS

Department of Mathematics Faculty of Science

The Maharaja Sayajirao University of Baroda Vadodara-390002, Gujarat, India.

EMail:drrgvyas@yahoo.com

2000c Victoria University ISSN (electronic): 1443-5756 196-04

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On the Absolute Convergence of Small Gaps Fourier Series of

Functions ofV BV^(p) R.G. Vyas

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J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005

http://jipam.vu.edu.au

Abstract Letfbe a2πperiodic function inL¹[0,2π]andP∞

k=−∞f(nb _k)eⁱⁿ^k^xbe its Fourier series with ‘small’ gapsn_k+1−n_k≥q≥1. Here we have obtained sufficiency conditions for the absolute convergence of such series iffis ofV

BV^(p)locally.

We have also obtained a beautiful interconnection between lacunary and non- lacunary Fourier series.

2000 Mathematics Subject Classification:42Axx.

Key words: Fourier series with small gaps, Absolute convergence of Fourier series and p-V

-bounded variation.

1. Introduction

Let f be a 2π periodic function in L¹[0,2π] andfb(n), n ∈ Z, be its Fourier coefficients. The series

(1.1) X

k∈Z

fb(n_k)eⁱⁿ^k^x,

wherein{n_k}^∞₁ is a strictly increasing sequence of natural numbers andn−k =

−nk, for allk, satisfy an inequality

(1.2) (n_k+1−n_k)≥q ≥1 for all k = 0,1,2, . . . , is called the Fourier series off with ‘small’ gaps.

Obviously, ifn_k =k, for allk, (i.e. n_k+1−n_k = q = 1, for allk), then we get non-lacunary Fourier series and if{nk}is such that

(1.3) (n_k+1−n_k)→ ∞ 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 condition for the convergence of the series P

k∈Z

fb(n_k)

β

(0 < β ≤ 2)in terms of VBV and the modulus of continuity [2, Theorem 3]. Here we have generalized this result and we have also obtained a sufficiency condition if functionf is of VBV^(p). In 1980 Shiba [4] generalized the class V

BV. He introduced the classV

BV^(p).

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Definition 1.1. Given an intervalI, a sequence of non-decreasing positive real numbersV

={λm}(m= 1,2, . . .)such thatP

m 1

λm diverges and1≤p <∞ we say thatf ∈ V

BV^(p) (that is f is a function ofp−V

-bounded variation over (I)) if

VΛp(f, I) = sup

{I_m}

{VΛp({Im}, f, I)}<∞, where

VΛp({Im}, f, I) = X

m

|f(b_m)−f(a_m)|^p λ_m

!¹_p ,

and{I_m}is a sequence of non-overlapping subintervalsI_m = [a_m, b_m]⊂ I = [a, b].

Note that, ifp= 1, one gets the classV

BV(I); ifλm ≡1for allm, one gets the class BV^(p); if p = 1andλ_m ≡ m for allm, one gets the class Harmonic BV(I). ifp= 1andλ_m ≡1for allm, one gets the classBV(I).

Definition 1.2. Forp≥1, thep−integral modulus of continuityω^(p)(δ, f, I)of f overIis defined as

ω^(p)(δ, f, I) = sup

0≤h≤δ

k(T_hf −f)(x)k_p,I,

whereT_hf(x) = f(x+h)for allxandk(·)k_p,I = k(·)χ_Ik_p in which χ_I is the characteristic function of I andk(·)k_p denotes theL^p-norm. p = ∞gives the modulus of continuityω(δ, f, I).

We prove the following theorems.

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Theorem 1.1. Let f ∈ L[−π, π] possess a Fourier series with ‘small’ gaps (1.2) andIbe a subinterval of lengthδ1 > ^2π_q . Iff ∈V

BV(I)and

∞

X

k=1



 ω(_n¹

k, f, I) k

Pn_k j=1

1 λj





β 2

<∞,

then

(1.4) X

k∈Z

fb(n_k)

β

<∞ (0< β≤2).

Since{λ_j}is non-decreasing, one getsPnk

j=1 1 λj ≥ _λⁿ^k

nk and hence our earlier theorem [2, Theorem 3] follows from Theorem1.1.

Theorem 1.1 with β = 1 and λ_n ≡ 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_q (respectively of arbitrary positive length).

Theorem 1.2. Letf andI be as in Theorem1.1. Iff ∈ V

BV^(p)(I),1 ≤p <

2r,1< r <∞and

∞

X

k=1







ω^((2−p)s+p)

1

nk, f, I2−p/r

k Pnk

j=1

1 λj

¹_r







β 2

<∞,

where ¹_r +¹_s = 1,then (1.4) holds.

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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]). Letf andI be as in Theorem1.1. Iff ∈ L²(I) then

(1.5) X

k∈Z

f(nb _k)

2 ≤A_δ|I|⁻¹kfk²_2,I,

whereA_δ depends only onδ.

Lemma 1.4. If|n_k|> pthen fort ∈None has

Z ^π_p

0

sin^2t|n_k|h dh≥ π 2^t+1p. Proof. Obvious.

Lemma 1.5 (Stechkin, refer to [6]). If u_n ≥ 0 for n ∈ N, u_n 6= 0 and a functionF(u)is concave, increasing, andF(0) = 0, then

∞

X

1

F(u_n)≤2

∞

X

1

F

u_n+u_n+1+· · · n

.

Lemma 1.6. Iff ∈V

BV^(p)(I)impliesf is bounded overI.

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Proof. Observe that

|f(x)|^p ≤2^p

|f(a)|^p+λ₁|f(x)−f(a)|^p

λ₁ +λ₂|f(b)−f(x)|^p λ₂

≤2^p |f(a)|^p+λ₂V∧_p(f, I) Hence the lemma follows.

Proof of Theorem1.1. Let I =

x₀−^δ₂¹, x₀+^δ₂¹

for some x₀ and δ₂ be such that0< ^2π_q < δ₂ < δ₁. Putδ₃ =δ₁−δ₂ andJ =

x₀− ^δ₂², x₀+ ^δ₂²

. Suppose integersT andj satisfy

(1.6) |n_T|> 4π

δ₃ and 0≤j ≤ δ₃|n_T| 4π . Since f ∈ V

BV(I) impliesf is bounded over I by Lemma1.6 (for p = 1), we have f ∈ L²(I), so that (1.5) holds and f ∈ L²[−π, π]. If we put f_j = (T_2jhf−T(2j−1)hf)thenf_j ∈L²(I)and the Fourier series off_j also possesses gaps (1.2). Hence by Lemma1.3we get

(1.7) X

k∈Z

fˆ(n_k)

2

sin² n_kh

2

=O

kf_jk²_2,J

because

fˆ_j(n_k) = 2if(nˆ _k)eⁱⁿ^k^(2j−¹²^h)sin n_kh

2

.

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Integrating both the sides of (1.7) over (0,_n^π

T) with respect to h and using Lemma1.4, we get

(1.8)

∞

X

|n_k|≥n_T

fˆ(n_k)

2

=O(n_T) Z ^π

nT

0

kf_j k²_2,J dh.

Multiplying both the sides of the equation by _λ¹

j and then taking summation overj, we get

(1.9) X

j

1 λ_j

!



∞

X

|n_k|≥n_T

fˆ(n_k)

2



=O(n_T) Z ^π

nT

0





X

j

|f_j|² λ_j

1,J



dh.

Now, sincex ∈J andh ∈ (0,_n^π

T)we have|f_j(x)| =O(ω(_n¹

T, f, I)), for each j of the summation; sincex∈J andf ∈V

BV(I)we haveP

j

|fj(x)|

λj =O(1) because for eachjthe pointsx+ 2jhandx+ (2j−1)hlie inI forh∈(0,_n^π

T) andx∈J ⊂I. Therefore

X

j

|f_j(x)|² λ_j

!

=O

ω 1

n_T, f, I

X

j

|f_j(x)|

λ_j

!

=O

ω 1

n_T, f, I

.

It follows now from (1.9) that R_n_T = X

|nk≥nT

fˆ(n_k)

2

=O



 ω

1 nT, f, I

Pn_T

j=1 1 λj



.

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Finally, Lemma1.5withuk=

f(nˆ k)

2

(k∈Z)andF(u) =u^β/2 gives

∞

X

|k|=1

fˆ(n_k)

β

= 2

∞

X

k=1

F

fˆ(n_k)

2

≤4

∞

X

k=1

F R_n_k

k

≤4

∞

X

k=1

R_n_k k

β/2

=O(1)





∞

X

k=1

ω(_n¹

k, f, I) k(Σⁿ_j=1^k _λ¹

j)

!(β/2)

.

This proves the theorem.

Proof of Theorem1.2. Sincef ∈V

BV^(p)(I), Lemma1.6impliesfis 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

s +^p_r, by using Hölder’s inequality, we get from (1.9) Z

J

|f_j(x)|²dx≤ Z

J

|f_j(x)|^(2−p)s+pdx ¹_s Z

J

|f_j(x)|^pdx ¹_r

≤Ω^1/r_h,J Z

J

|f_j(x)|^pdx ¹_r

,

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whereΩ_h,J = (ω^(2−p)s+p(h, f, J))^2r−p. This together with (1.9) implies, putting

B =X

k∈Z

fb(n_k)

2

sin² n_kh

2

,

that

B ≤Ω^1/r_h,J Z

J

|f_j(x)|^pdx ¹_r

. Thus

B^r ≤Ω_h,J Z

J

|f_j(x)|^pdx

. Now multiplying both the sides of the equation by _λ¹

j and then taking the summation overj = 1tonT (T ∈N)we get

B^r ≤

Ω_h,J R

J

P

j

|f_j(x)|^p λj

dx P

j 1 λj

.

Therefore

B ≤ Ω_h,J P

j 1 λj

!¹_r Z

J

X

j

|f_j(x)|^p λ_j

! dx

!¹_r .

Substituting back the value ofB and then integrating both the sides of the equa-

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tion with respect tohover(0,_n^π

T), we get (1.10) X

k∈Z

f(nb _k)

2Z π/nT

0

sin²

|n_k|h 2

dh

=O





Ω_1/n_T_,J P

j 1 λj





1 r

Z π/nT

0

Z

J

X

j

|f_j(x)|^p λj

! dx

!¹_r dh.

Observe that forxinJ,hin(0,_n^π

T)and for eachjof the summation the points x+ 2jhandx+ (2j−1)hlie inI; moreover,f ∈V

BV^(p)(I)implies X

j

|f_j(x)|^p

λ_j =O(1).

Therefore, it follows from (1.10) and Lemma1.4that R_n_T ≡ X

|n_k|≥n_T

fb(n_k)

2

=O Ω_1/n_T_,I PnT

j=1 1 λj

!¹_r .

Thus

R_n_T =O







ω^(2−p)s+p

1

nT, f, I^2−p/r Pn_T

j=1 1 λj

¹_r





.

Now proceeding as in the proof of Theorem 1.1, the theorem is proved using Lemma1.5.

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References

[1] A. ZYMUND, Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cam- bridge, 1979 (reprint).

[2] J.R. PATADIA AND R.G. VYAS, Fourier series with small gaps and func- tions of generalized variations, J. Math. Anal. and Appl., 182(1) (1994), 113–126.

[3] M. SCHRAM ANDD. WATERMAN, Absolute convergence of Fourier series of functions ofV

BV^(p)andΦV

BV, Acta. Math. Hungar, 40 (1982), 273–276.

[4] M. SHIBA, On the absolute convergence of Fourier series of functions of classV

BV^(p), Sci. Rep. Fukushima Univ., 30 (1980), 7–10.

[5] N.K. BARRY, A Treatise on Trigonometric Series, Pergamon, New York,1964.

[6] N.V. PATEL ANDV.M. SHAH, A note on the absolute convergence of la- cunary Fourier series, Proc. Amer. Math. Soc., 93 (1985), 433–439.