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volume 6, issue 4, article 94, 2005.

Received 06 July, 2005;

accepted 29 July, 2005.

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 OFϕV

BV

R. G. VYAS

Department of Mathematics Faculty of Science

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

EMail:drrgvyas@yahoo.com

c

2000Victoria University ISSN (electronic): 1443-5756 205-05

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On The Absolute Convergence Of Small Gaps Fourier Series Of

Functions OfϕV BV R. G. Vyas

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J. Ineq. Pure and Appl. Math. 6(4) Art. 94, 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’ gaps n_k+1−n_k ≥ q ≥ 1. Here we obtain a sufficiency condition for the convergence of the seriesP

k∈Z|fb(n_k)|^β(0< β≤2) iff is of 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.

2000 Mathematics Subject Classification:42A16, 42A28, 26A45.

Key words: Fourier series with small gaps, Absolute convergence of Fourier series andϕ∧-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, satisfies 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.

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)andϕ∧BV.

Definition 1.1. Given a nonnegative convex functionϕ, defined on[0,∞)such that ^ϕ(x)_x → 0 as x → 0, for some constant d ≥ 2, ϕ(2x) ≤ dϕ(x) for all x≥0and given a sequence of non-decreasing positive real numbersV

={λ_m}

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(m = 1,2, . . .)such thatP

m 1

λm diverges we say thatf ∈ϕV

BV (that isfis a function ofϕV

-bounded variation over (I)) if V_Λ_ϕ(f, I) = sup

{Im}

{V_Λ_ϕ({I_m}, f, I)}<∞, where

V_Λ_ϕ({I_m}, f, I) = X

m

ϕ|f(b_m)−f(a_m)|

λ_m

! ,

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

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).

By applying the Wiener-Ingham result [1, Vol. I, p. 222] for the finite trigonometric sums with ‘small’ gap (1.2) we have already studied the sufficiency conditions for the convergence of the seriesP

k∈Z

f(nb k)

β

(0< β ≤2) for the functions of V

BV and ∧BV^(p) in terms of the modulus of continuity [6]. Here we obtain a sufficiency condition if function f is of ϕV

BV. We prove the following theorem.

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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 ∈ϕV

BV(I),1≤p <2r, 1≤r <∞, and

∞

X

k=1











ϕ⁻¹







ω^((2−p)s+p)

1

nk, f, I2r−p

Pnk

j=1 1 λj













1 r

, k







β 2

<∞,

where ¹_r +¹_s = 1, then

(1.4) X

k∈Z

fb(n_k)

β

<∞ (0< β≤2).

Theorem 1.1 with β = 1 is a ‘small’ gaps analogue of the Schramm and Waterman result [3, Theorem 2]. Observe that the intervalI considered in the theorem for the gap condition (1.2) is of length > ^2π_q , so that when n_k = k, for all k, I is of length 2π. Hence for non-lacunary Fourier series (equality throughout in (1.2)) the theorem withβ = 1gives the Schramm and Waterman result [3, Theorem 2] as a particular case.

We need the following lemmas to prove the theorem.

Lemma 1.2 ([2, Lemma 2]). Letf andI be as in Theorem1.1. Iff ∈ L²(I) then

(1.5) X

k∈Z

fb(nk)

2

≤Aδ|I|⁻¹ kf k²_2,I, whereAδ depends only onδ.

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Lemma 1.3. If|n_k|> pthen fort ∈None has Z ^π_p

0

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

Lemma 1.4 (Stechkin, refer to [5]). 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

.

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 . f ∈ϕ∧BV(I)implies

|f(x)| ≤ |f(a)|+|f(x)−f(a)| ≤ |f(a)|+Cϕ⁻¹(V∧ϕ(f, I)) for allx∈I.

Since f is bounded over I, we have f ∈ L²(I), so that (1.5) holds and f ∈ L²[−π, π]. If we putf_j = (T_2jhf −T(2j−1)hf)thenf_j ∈L²(I)and the Fourier series off_j also possess gaps (1.2). Hence by Lemma1.2we get

(1.7) X

k∈Z

fˆ(n_k)

2

sin² n_kh

2

=O

kf_jk²_2,J

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because

fˆj(nk) = 2if(nˆ k)eⁱⁿ^k^(2j−¹²^h)sin n_kh

2

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

T) with respect to h and using Lemma1.3, we get

(1.8) Rn_T =

∞

X

|n_k|≥n_T

f(nˆ k)

2

=O(nT) Z ^π

nT

0

kfjk²_2,J dh.

Since2 = ^(2−p)s+p_s + ^p_r, by using Hölder’s inequality, we get from (1.8) B =

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

,

whereΩ_h,J = (ω^(2−p)s+p(h, f, J))^2r−p. Thus

(1.9) B^r≤Ω_h,J

Z

J

|f_j(x)|^pdx.

Since f is bounded over I, there exists some positive constant M ≥ ¹₂ such that |f(x)| ≤ M for all x ∈ I. Dividing f by the positive constant M alters

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ω_p(h, f, J)by the same constantM andϕ(2|f(x)|)≤dϕ(|f(x)|)for allx, we may assume that|f(x)| ≤1for allx∈I. Hence from (1.9) we get

B^r ≤Ω_h,J Z

J

|f_j(x)|dx.

Sinceϕ(2x)≤dϕ(x), we haveϕ(ax)≤d^log²^aϕ(x), so ϕ

B^r δ₂

≤d^log²^CΩ^h,Jϕ R

J|fj(x)|dx δ₂

=CΩ^log_h,J²^dϕ R

J|f_j(x)|dx δ₂

=CΩ^log_h,J²^d−1Ω_h,Jϕ R

J|f_j(x)|dx R

J1dx

≤CΩh,J

R

Jϕ|f_j(x)|dx δ₂

(by Jensen’s inequality for integrals)

=CΩh,J

Z

J

ϕ|fj(x)|dx

.

Multiplying both the sides of the equation by _λ¹

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

(1.10) ϕ

B^r δ₂

≤C





Ω_h,J PnT

j=1

1 λj



 Z

J nT

X

j=1

ϕ|f_j(x)|

λ_j

! dx

! .

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Observe that forxinJ,hin(0,_n^π

T)and for eachjof the summation the points x+ 2jhandx+ (2j−1)hlie inI; moreoverf ∈ϕ∧BV(I)implies

nT

X

j=1

ϕ|f_j(x)|

λj

=O(1).

Therefore, it follows from (1.10) that

B =O









ϕ⁻¹





Ω_1/n_T_,I PnT

j=1

1 λj









1 r





.

Substituting back the value ofBin the equation (1.8), we get

R_n_T =

∞

X

|n_k|≥n_T

fˆ(n_k)

2

=O









ϕ⁻¹





Ω_1/n_T_,I PnT

j=1

1 λj









1 r





.

Thus

RnT =O











ϕ⁻¹







ω^(2−p)s+p

1

nT, f, I^2r−p PnT

j=1

1 λj













1 r





 .

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Finally, Lemma1.4withuk=

f(nˆ k)

2

(k∈Z)andF(u) =u^β/2 gives

∞

X

|k|=1

f(nˆ _k)

β

= 2

∞

X

k=1

F

f(nˆ _k)

2

≤4

∞

X

k=1

F R_n_k

k

= 4

∞

X

k=1

R_n_k k

^β₂

=O(1)







∞

X

k=1





"

ϕ⁻¹ (ω^(2−p)s+p(_n¹

k, f, I))^2r−p Pn_k

j=1 1 λj

!#¹_r, k





β

2



.

This proves the theorem.

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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. Analy. and Appl., 182(1) (1994), 113–126.

[3] M. SCHRAMM AND D. WATERMAN, Absolute convergence of Fourier series of functions ofV

BV^(p)andΦV

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 ANDV.M. SHAH, A note on the absolute convergence of la- cunary 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]