(1)http://jipam.vu.edu.au/ Volume 6, Issue 4, Article 94, 2005 ON THE ABSOLUTE CONVERGENCE OF SMALL GAPS FOURIER SERIES OF FUNCTIONS OF ϕV BV R

http://jipam.vu.edu.au/

Volume 6, Issue 4, Article 94, 2005

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

BV

R. G. VYAS

DEPARTMENT OFMATHEMATICS

FACULTY OFSCIENCE

THEMAHARAJASAYAJIRAOUNIVERSITY OFBARODA

VADODARA-390002, GUJARAT, INDIA. drrgvyas@yahoo.com

Received 06 July, 2005; accepted 29 July, 2005 Communicated by L. Leindler

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

k=−∞fb(nk)e^inkxbe its Fourier series with ‘small’ gapsnk+1−nk ≥ q ≥ 1. Here we obtain a sufficiency condition for the convergence of the seriesP

k∈Z |fb(nk)|^β(0< β≤2) iff is ofϕ∧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.

Key words and phrases: Fourier series with small gaps, Absolute convergence of Fourier series andϕ∧-bounded variation.

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

1. INTRODUCTION

Letf be a2πperiodic function inL¹[0,2π]andf(n),b n ∈Z, be its Fourier coefficients. The series

(1.1) X

k∈Z

f(nb _k)e^inkx,

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

(1.3) (n_k+1−n_k)→ ∞ as k→ ∞

then (1.1) is said to be the lacunary Fourier series.

ISSN (electronic): 1443-5756

c 2005 Victoria University. All rights reserved.

205-05

In 1982 M. Schramm and D. Waterman [3] have introduced the classϕ ∧BV(I) of func- tions ofϕ∧-bounded variation overI 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}(m = 1,2, . . .)such thatP

m 1

λm diverges we say thatf ∈ϕV

BV (that isf is a function ofϕV

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

{Im}

{V_Λϕ({I_m}, f, I)}<∞, where

V_Λϕ({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. 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

0≤h≤δ

k(T_hf−f)(x)k_p,I,

whereT_hf(x) = f(x+h)for all x andk(·)k_p,I = k(·)χ_Ik_p in which χ_I is the characteristic function ofIandk(·)k_pdenotes 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

fb(nk)

β

(0< β ≤2)for the functions ofV

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

BV. We prove the following theorem.

Theorem 1.1. Let f ∈ L[−π, π] possess a Fourier series with ‘small’ gaps (1.2) and I be 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

n_k, 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 withβ= 1is 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 length2π. 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.

Lemma 1.2 ([2, Lemma 2]). Letf andI be as in Theorem 1.1. Iff ∈L²(I)then

(1.5) X

k∈Z

f(nb _k)

2

≤A_δ|I|⁻¹ kf k²_2,I,

whereA_δdepends only onδ.

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 function F(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 Theorem 1.1. LetI =

x₀−^δ₂¹, x₀+^δ₂¹

for some x₀ and δ₂ be such that 0 < ^2π_q <

δ₂ < δ₁. Putδ₃ =δ₁−δ₂andJ =

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.

Sincef is bounded overI, we havef ∈L²(I), so that (1.5) holds andf ∈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 possess gaps (1.2).

Hence by Lemma 1.2 we get

(1.7) X

k∈Z

f(nˆ _k)

2

sin² n_kh

2

=O

kf_jk²_2,J

because

fˆ_j(n_k) = 2if(nˆ _k)e^{ink(2j−12h)}sin n_kh

2

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

T)with respect tohand using Lemma 1.3, we get

(1.8) R_nT =

∞

X

|n_k|≥n_T

fˆ(n_k)

2

=O(n_T) Z _nT^π

0

kf_jk²_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.

Sincef is bounded overI, there exists some positive constantM ≥ ¹₂ such that|f(x)| ≤ M for allx∈I. Dividingf by the positive constantM altersω_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^log2aϕ(x), so ϕ

B^r δ₂

≤d^log2CΩh,Jϕ R

J|f_j(x)|dx δ₂

=CΩ^log_h,J^2dϕ R

J|f_j(x)|dx δ₂

=CΩ^log_h,J^2d−1Ω_h,Jϕ R

J|f_j(x)|dx R

J1dx

≤CΩ_h,J R

Jϕ|f_j(x)|dx δ₂

(by Jensen’s inequality for integrals)

=CΩ_h,J Z

J

ϕ|f_j(x)|dx

.

Multiplying both the sides of the equation by _λ¹

j and then taking the summation overj = 1to n_T (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

! .

Observe that forx inJ, hin(0,_n^π

T)and for each j of the summation the pointsx+ 2jhand x+ (2j−1)hlie inI; moreoverf ∈ϕ∧BV(I)implies

nT

X

j=1

ϕ|f_j(x)|

λ_j =O(1).

Therefore, it follows from (1.10) that

B =O









ϕ⁻¹





Ω_1/nT,I PnT

j=1

1 λj









1 r



.

Substituting back the value ofB in the equation (1.8), we get

R_nT =

∞

X

|n_k|≥n_T

f(nˆ _k)

2

=O









ϕ⁻¹





Ω_1/nT,I PnT

j=1

1 λj









1 r





.

Thus

R_nT =O













ϕ⁻¹







ω^(2−p)s+p 1

nT, f, I

2r−p

PnT

j=1

1 λj













1 r







 .

Finally, Lemma 1.4 withu_k =

fˆ(n_k)

2

(k ∈Z)andF(u) = u^β/2gives

∞

X

|k|=1

fˆ(n_k)

β

= 2

∞

X

k=1

F

fˆ(n_k)

2

≤4

∞

X

k=1

F R_nk

k

= 4

∞

X

k=1

R_nk 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.

REFERENCES

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

[2] J.R. PATADIAANDR.G. VYAS, Fourier series with small gaps and functions of generalized varia- tions, J. Math. Analy. and Appl., 182(1) (1994), 113–126.

[3] M. SCHRAMM AND D. WATERMAN, Absolute convergence of Fourier series of functions of VBV^(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. PATELANDV.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]