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

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

Volume 6, Issue 1, Article 23, 2005

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

BV^(p)

R.G. VYAS

DEPARTMENT OFMATHEMATICS, FACULTY OFSCIENCE,

THEMAHARAJASAYAJIRAOUNIVERSITY OFBARODA, VADODARA-390002, GUJARAT, INDIA.

[email protected]

Received 18 October, 2004; accepted 29 November, 2004 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 have obtained sufficiency conditions for the absolute convergence of such series iff is ofVBV^(p) locally. We have also obtained a beautiful interconnection between lacunary and non-lacunary Fourier series.

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

-bounded variation.

2000 Mathematics Subject Classification. 42Axx.

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, 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, ifnk =k, for allk, (i.e. nk+1−nk =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.

196-04

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 V

BV 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 ofV

BV^(p). In 1980 Shiba [4] generalized the class V

BV. He introduced the classV

BV^(p).

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

= {λ_m}(m= 1,2, . . .)such thatP

m 1

λm diverges and1≤p <∞we say thatf ∈V

BV^(p)(that isf is a function ofp−V

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

{I_m}

{V_Λp({I_m}, f, I)}<∞,

where

V_Λp({I_m}, 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, if p = 1, one gets the class V

BV(I); if λm ≡ 1 for all m, one gets the class BV^(p); if p = 1 and λ_m ≡ m for all m, one gets the class Harmonic BV(I). if p = 1and λ_m ≡1for allm, one gets the classBV(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

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).

We prove the following theorems.

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 ∈V

BV(I)and

∞

X

k=1



 ω(_n¹

k, f, I) k

Pnk

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 gets Pnk

j=1 1

λj ≥ _λ^nk

nk and hence our earlier theorem [2, Theorem 3] follows from Theorem 1.1.

Theorem 1.1 with β = 1 andλ_n ≡ 1shows that the Fourier series of f with ‘small’ gaps condition (1.2) (respectively (1.3)) converges absolutely if the hypothesis of the Stechkin theo- rem [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 Theorem 1.1. 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.

Theorem 1.2 withβ= 1is 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 Theorem 1.1. Iff ∈L²(I)then

(1.5) X

k∈Z

fb(n_k)

2

≤A_δ|I|⁻¹kfk²_2,I,

whereA_δdepends 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 un ≥ 0 for n ∈ N, un 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

. Lemma 1.6. Iff ∈V

BV^(p)(I)impliesf is bounded overI. 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 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|nT| 4π . Sincef ∈V

BV(I)impliesfis bounded overIby Lemma 1.6 (forp= 1), we havef ∈L²(I), so that (1.5) holds andf ∈L²[−π, π]. If we putfj = (T2jhf −T(2j−1)hf)thenfj ∈L²(I)and the Fourier series offj also possesses gaps (1.2). Hence by Lemma 1.3 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 nkh

2

.

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

T)with respect tohand using Lemma 1.4, we get (1.8)

∞

X

|n_k|≥nT

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|≥nT

f(nˆ _k)

2



=O(n_T) Z ^π

nT

0





X

j

|f_j|² λ_j

1,J



dh.

Now, since x ∈ J and h ∈ (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 eachj the pointsx+ 2jhandx+ (2j−1)hlie inI forh ∈(0,_n^π

T)andx∈J ⊂I. Therefore X

j

|f_j(x)|² λj

!

=O

ω 1

nT

, f, I

X

j

|f_j(x)|

λj

!

=O

ω 1

n_T, f, I

. It follows now from (1.9) that

R_nT = X

|n_k≥n_T

fˆ(n_k)

2

=O



 ω

1

n_T, f, I PnT

j=1 1 λj



.

Finally, Lemma 1.5 withu_k =

fˆ(n_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_nk

k

≤4

∞

X

k=1

R_nk 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 Theorem 1.2. Sincef ∈V

BV^(p)(I), Lemma 1.6 impliesf is bounded overI. There- foref ∈L²(I), and hence (1.5) holds so thatf ∈L²[−π, π]. Using the notations and procedure of Theorem 1.1 we get (1.9). Since2 = ^(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

, 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 = 1 tonT (T ∈N)we get

B^r ≤ Ωh,J

R

J

P

j

|fj(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 ofBand then integrating both the sides of the equation with respect tohover(0,_n^π

T), we get

(1.10) X

k∈Z

fb(n_k)

2Z π/n_T 0

sin²

|n_k|h 2

dh

=O





Ω_1/nT,J P

j 1 λj





1 r

Z π/nT

0

Z

J

X

j

|f_j(x)|^p λ_j

! dx

!¹_r dh.

Observe that forx inJ, hin(0,_n^π

T)and for each j of the summation the pointsx+ 2jhand x+ (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 Lemma 1.4 that

R_nT ≡ X

|n_k|≥n_T

fb(n_k)

2

=O Ω_1/nT,I PnT

j=1 1 λj

!¹_r .

Thus

R_nT =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 Lemma 1.5.

REFERENCES

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

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[3] M. SCHRAM AND D. WATERMAN, Absolute convergence of Fourier series of functions of VBV^(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. PATELANDV.M. SHAH, A note on the absolute convergence of lacunary Fourier series, Proc.

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