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On the absolute convergence of a lacunary Fourier series
JOURNAL
OF MATHEMATICAL
ANALYSIS
On the Absolute
AND
65, 391-398 (1978)
APPLICATIONS
Convergence
of a Lacunary Fourier Series
J. R. PATADIA llefiavtment
of Mathematics, Faculty of Science, The M.S. University
of Baroda,
Bauoda-390 002, Gzljarat State, India Submitted by R. P. Boas
1 Let cos nlzx + b,, sin nkx) be the Fourier gaps
(ni,
numbers
, nk :.l),
series of a 2r-periodic where
(nk)
(k
E
N)
(1.1)
function f EL[--rr, n] with an infinity of is a strictly increasing sequence of natural
such that (ri~.+~- nk) + co
Steckin has proved the following gap condition.
as
/2 -
co.
theorem for the Fourier
(1.2) series without
any
THEOREM [I; Vol. 11,~. 1961. 1ff OYa g iven increasing sequence {nfi} of natural numbers we haze
then for the Fourier series off E L2[-rr,
n] we obtain
il (I a,, I + I bnkI) < ~0,
(1.3)
where W”‘(l/Tz, ,f)
=
sup O
[(j-" if(x + h) -f@ - h)12~$'2/ . -w
Observe that with nk = k for all k, SteCkin’s theorem reduces to the theorem of S&s2 [I; Vol. II, p. 1551.
391 0022-247X/78/0652-0391$02.00/0 Copyright 0 1978 by Academic Press, Inc. A11 rights of reproduction in any form reserved.
392
J. R. PATADIA
In this paper we propose to prove that if the sequence (Q} satisfies (1.2) and if (1.1) is the Fourier series off then (1.3) holds, that is, (1.1) converges absolutely, even when the hypothesis in the SteEkin’s theorem is satisfied only in a subinterval of [-r, z-1. We then generalize our result by considering the higher order differences in Theorem 2. Let x0 be a fixed point and 0 < 6 < n. Let I = {x: j x - s,, i < Sj be a subinterval of [-rr, rr] and put
W e prove the following theorems. THEOREM
1. 1f
% I_____ (~‘Yllnk ,fY c k6/2
< *
(0
(1.5)
k=l
and if (1.2) holds then for the Fourier series (1.1) off sL2(1) we have
where ~(~)(l/n, THEOREM
,f) is us in (1.4).
2.
Theorem 1 holds if (1.5) is replaced by the more general condition
where wj2)(l/nk ,f)
=
sup
[(J, 1 i.
(-l)“-’
(4, f (x + (2j - I) hf
dxjl;p/
.
04h41/7Ll,
THEOREM
3.
Theorem 1 holds ;f (1.5) is replaced by the condition
where EL::(f) is the best approximation given by
off(x)
in which T%,<(x) is a trigonometric polynomial
in the space L2 over the interval I
of order not higher than n,. .
CONVERGENCE
OF A FOURIER
SERIES
393
2 We need the following lemmas. Lemma 1 is a special case of a very general theorem due to Paley and Wiener [3, Theorem XLII’]. Lemma 2 is due to SteCkin [4, Lemma 21. The inequality (2.2) of Lemma 3 is a simple consequence of the more general lemma quoted by Kennedy [2, Lemma 11. LEMMA
(1.1) then
I.
f
If f E L2(J), EL2[--a, r].
where J is an interval,
and a. (nli+l - nk) --t co in
LEMMA 2. If u, >, 0 (n E N), u, S+ 0 and let the function F(u) be concave, increasing, and such that F(0) = 0. Then
LEMMA
3. Let f EL2(I);
n, = 0, n, = -nek
(k < 0);
CnO = 0, C711c =
B(arLiG- ib,> (k > 0) and C,,, = (?,-, (k < 0). If (nk+l - nl;) > %7W,
i
for all k,
I C,, I2 < 86-l 1 If(x I
dx;
(2.1)
(2.2)
C
I G, I2< C(a)(J2Ylln, ,f))”
(2.3)
C
I C,, I2< WI (d%ln, ,f)Y,
(2.4)
lnk12n,, OY more generally
Ing >n,, and
(2.5) where C(6) is some constant depending on 6. Proof of Lemma 3.
We have
(2.6)
394
J. R. PATADIA
and if we put $(r, x) = f C,,r’“k’ --p.
exp(in,x)
(0
< 1)
for all real x, then its existence is assured by (2.6) and we obviously
(2.7) get
cos nhx + bnk sin nkx) rnk. Since feL2(1) and (1.2) holds, therefore fgL2[-n, hence by a known theorem [5, p. 871 it follows that f(x)
= L2 -
lir$t 4(r, x)
n] by Lemma
1; and
(1x1 G4*
(2.9)
Now, on account of (2.1), (2.6), (2.7), and (2.9), we can apply Lemma in [2] to obtain the inequality (2.2). Now put Ax> =f(x
(2.8)
+ h) -f(x
-
1 quoted
(2.10)
4
and Czk = 2iC,, sin n,h.
(2.11)
Then 1 Czk 1 < 2 1Cnk j and hence by (2.6) we have gC;Jrf”“:
(2.12)
(O
Put g(r, x) = f Czkkne exp(in,x) -cc
(0 < Y < 1).
We then get the identity g(r, x) = +(r, x + h) - 4(~, x together with (2.9) and (2.10) we obtain g(x) = L2 -
liII;t g(r, x)
this
(2.14)
(I x I G n).
It follows from (2.1), (2.12), (2.13) (2.14), and the inequality
g I Ck I2G 8q
h), and from
(2.13)
(2.2) that
I &>I” &
(2.15)
Hence by (2.10) and (2.11) we get 4 ‘f 1 C,, I2 sin2 1nk / h < 86-l sI lf(x -m
+ h) -f(z
- h)j2 dx.
(2.16)
CONVERGENCE
OF A FOURIER
395
SERIES
Integrating both the sides of (2.16) with respect to h over (0, r/n,) (p E N), we get
If(x+h)-f(x- A)/2 d”).
(2.17)
We see that if j nk j > np then nlmp
sin2 ] nk j h dh = m
1
s0
(Ink/lnph o sin2 t dt s [ln~l/n,l7T
sin2 t dt
.i0
1 %?
[I nk (1
+
hbl
[I nk
l/%1>
. +
using 1 < [ / nk l/n,] < / nk i/n, < 1 + [I nk l/nJ, where [ ] denotes the integral part. Therefore, from (2.17), using w(2yh/n,f) < C(h) . wy1/?2, f) (A > 0, C(h) is a constant depending on A), we get 2 < 86-l % (w’~‘(+z, ,f))2 < 86-l $
C(W’~‘(l/n, ,f))2
from which (2.3) follows. Further, if we put
m = j$o(-I>“-j(;)A, + (2j - I)h)
(2.10)’
and Cck = Cn, exp(--in,&)
(exp(2in,h) - 1)’
= 2V,, exp(-&2,/z) (- 1)” exp(iZ(n,h - a/2)) sinz n,h,
(2.11)’
396
J. R. PATADIA
then / Czk 1 < 2z / Cmk ( and hence by (2.6) we have (2.12)’ Put g(r, x) = 2 Czkr’nk’ exp(in,x) --co
(2.13)’
(0 < r < 1).
We shall then get the identity g(r, x) = i
(- l)l-j
13 I#@, x + (2j - 1) h).
GO
This together with (2.9) and (2.10)’ gives g(x) = L2 -
l&tit g(r, X)
(2.14)’
(I x I < 4.
It follows from (2.1), (2.12)‘, (2.13)‘, (2.14)‘, and the inequality
(2.2) that
g I CZk I2d W’ ‘I[ Ig(4l”ax. Further,
as in the foregoing nlnp s0
proof of (2.3), analogously
(2.15)’
we get
1 Iin*, x dx sinzzI flk I h dh 3 x l +[I ‘,[, n, ,,n,l soa sin21 1
Using this along with (2.3), (2.4) is proved. Finally, let
(2.15)’ and proceeding
Tn,(x) = be any trigonometric
21-l
polynomial
Tnp(r, x) =
1
m=-nn
21-3 *21-2”‘T’T
analogously
1
as in the proof of
(2.18)
am exp(imx)
of order not higher than nz, (p E N). If we write
C a,rl”l m=-78,
exp(imx)
(0
< 1)
(2.19)
CONVERGENCE
OF A FOURIER
for all real X, then since T”, EI?[-,, T&v)
= L2 -
397
SERIES
~1 it obviously
lirmtit Tn,(r, x)
follows that (I x I ,< 4
(2.20)
Put g(x) = fb-4 -
(2.21)
T&)
and if
k + n,s and
= Cns- an,,
if
k = n,
and
kl
zzz0,
if
k fn,
and
kl >s,
= Csn,
if
k = n,
and
kI >n,.
A, = -ak,
Then, for /k j >n,, / ,4,1 =0 Hence by (2.6) we have
if k fn,
41 C 1A, IY~~I < co --3c
1k 1
and /A,
(0
(2.22)
= j Cn, I if k = n, .
(2.23)
< 1)
and if we put g(r, x) = f A,rlrcl exp(ikx) -B
(0
< 1)
(2.24)
for all real X, then its existence is assured by (2.23) and we get the identity g(r, X) = $(Y, X) - Tn,(y, x). This together with (2.9) (2.20), and (2.21) implies g(x) = L’ -
liff{t
g(r, x)
(I * I G q).
(2.25)
Now, on account of (2.1) (2.23), (2.24), and (2.25) we can apply the Lemma quoted in [2] to obtain
f I A, I2< 86-l s, I g(x)[” dx. --m Therefore
1
(2.26)
we get (2.27)
Since (2.27) holds for arbitrary trigonometric polynomial of order not higher than n, , we get the inequality (2.5). This completes the proof of Lemma 3. Proof 409/65/2-11
of Theorem
1.
Let
n, = 0, nk = -nMk
(k < 0); CnO = 0, C,* =
398
J. R. PATADIA
S(%k- ib,,) (k > 0) and C,* = Cn-, (k < 0). We assume throughout, without loss of generality, that (2.1) holds. In view of (1.2) this can be achieved, if necessary, by adding to f(x) a polynomial in exp(in,x), a process which affects neither the hypothesis nor the conclusion of the theorem. Then, putting m, = YL+, I Cnk I2 in th e inequality (2.3) of Lemma 3, we get r;a” < C(w(a)(lIn, ,f))“, where C is some constant depending on 6. Now, applying the Lemma 2 with uk = / C,& I2 (k E 2) and F(u) = uej2, we obtain using (2.29)
= g1 (rnklv’2
G c g1 ((~Yll%
,f)Y-P’“)
< 00
on account of (1.5). Therefore (1.6) holds and this completes the proof of Theorem 1. Applying the inequalities (2.4) and (2.5) instead of the inequality (2.3) in the proof of Theorem I and proceeding analogously we get Theorems 2 and 3, respectively. ACKNOWLEDGMENTS The author is thankful tion of this paper.
to Professor V. M. Shah for his help and guidance in the prepara-
REFERENCES 1. N. K. BARY, York, 1964.
“A
Treatise
on Trigonometric
Series,”
Vols.
I and II,
Pergamon,
New
2. P. B. KENNEDY, Fourier series with gaps, Quart. J. Math. Oxford Ser. 2 7 (1956), 224-230. 3. R. E. A. C. PALEY AND N. WIENER, “Fourier Transforms in the Complex Domain,” Vol. XIX, Amer. Math. Sot. Colloq. Publ., 1934. 4. S. B. STECKIN, On the absolute convergence of orthogonal series, I, Amer. Math. Sot. Trunsl. Ser. 1 3 (1962), 271-280. 5. A. ZYGMUND, “Trigonometrical Series,” Warsaw, 1934.
OF MATHEMATICAL
ANALYSIS
On the Absolute
AND
65, 391-398 (1978)
APPLICATIONS
Convergence
of a Lacunary Fourier Series
J. R. PATADIA llefiavtment
of Mathematics, Faculty of Science, The M.S. University
of Baroda,
Bauoda-390 002, Gzljarat State, India Submitted by R. P. Boas
1 Let cos nlzx + b,, sin nkx) be the Fourier gaps
(ni,
numbers
, nk :.l),
series of a 2r-periodic where
(nk)
(k
E
N)
(1.1)
function f EL[--rr, n] with an infinity of is a strictly increasing sequence of natural
such that (ri~.+~- nk) + co
Steckin has proved the following gap condition.
as
/2 -
co.
theorem for the Fourier
(1.2) series without
any
THEOREM [I; Vol. 11,~. 1961. 1ff OYa g iven increasing sequence {nfi} of natural numbers we haze
then for the Fourier series off E L2[-rr,
n] we obtain
il (I a,, I + I bnkI) < ~0,
(1.3)
where W”‘(l/Tz, ,f)
=
sup O
[(j-" if(x + h) -f@ - h)12~$'2/ . -w
Observe that with nk = k for all k, SteCkin’s theorem reduces to the theorem of S&s2 [I; Vol. II, p. 1551.
391 0022-247X/78/0652-0391$02.00/0 Copyright 0 1978 by Academic Press, Inc. A11 rights of reproduction in any form reserved.
392
J. R. PATADIA
In this paper we propose to prove that if the sequence (Q} satisfies (1.2) and if (1.1) is the Fourier series off then (1.3) holds, that is, (1.1) converges absolutely, even when the hypothesis in the SteEkin’s theorem is satisfied only in a subinterval of [-r, z-1. We then generalize our result by considering the higher order differences in Theorem 2. Let x0 be a fixed point and 0 < 6 < n. Let I = {x: j x - s,, i < Sj be a subinterval of [-rr, rr] and put
W e prove the following theorems. THEOREM
1. 1f
% I_____ (~‘Yllnk ,fY c k6/2
< *
(0
(1.5)
k=l
and if (1.2) holds then for the Fourier series (1.1) off sL2(1) we have
where ~(~)(l/n, THEOREM
,f) is us in (1.4).
2.
Theorem 1 holds if (1.5) is replaced by the more general condition
where wj2)(l/nk ,f)
=
sup
[(J, 1 i.
(-l)“-’
(4, f (x + (2j - I) hf
dxjl;p/
.
04h41/7Ll,
THEOREM
3.
Theorem 1 holds ;f (1.5) is replaced by the condition
where EL::(f) is the best approximation given by
off(x)
in which T%,<(x) is a trigonometric polynomial
in the space L2 over the interval I
of order not higher than n,. .
CONVERGENCE
OF A FOURIER
SERIES
393
2 We need the following lemmas. Lemma 1 is a special case of a very general theorem due to Paley and Wiener [3, Theorem XLII’]. Lemma 2 is due to SteCkin [4, Lemma 21. The inequality (2.2) of Lemma 3 is a simple consequence of the more general lemma quoted by Kennedy [2, Lemma 11. LEMMA
(1.1) then
I.
f
If f E L2(J), EL2[--a, r].
where J is an interval,
and a. (nli+l - nk) --t co in
LEMMA 2. If u, >, 0 (n E N), u, S+ 0 and let the function F(u) be concave, increasing, and such that F(0) = 0. Then
LEMMA
3. Let f EL2(I);
n, = 0, n, = -nek
(k < 0);
CnO = 0, C711c =
B(arLiG- ib,> (k > 0) and C,,, = (?,-, (k < 0). If (nk+l - nl;) > %7W,
i
for all k,
I C,, I2 < 86-l 1 If(x I
dx;
(2.1)
(2.2)
C
I G, I2< C(a)(J2Ylln, ,f))”
(2.3)
C
I C,, I2< WI (d%ln, ,f)Y,
(2.4)
lnk12n,, OY more generally
Ing >n,, and
(2.5) where C(6) is some constant depending on 6. Proof of Lemma 3.
We have
(2.6)
394
J. R. PATADIA
and if we put $(r, x) = f C,,r’“k’ --p.
exp(in,x)
(0
< 1)
for all real x, then its existence is assured by (2.6) and we obviously
(2.7) get
cos nhx + bnk sin nkx) rnk. Since feL2(1) and (1.2) holds, therefore fgL2[-n, hence by a known theorem [5, p. 871 it follows that f(x)
= L2 -
lir$t 4(r, x)
n] by Lemma
1; and
(1x1 G4*
(2.9)
Now, on account of (2.1), (2.6), (2.7), and (2.9), we can apply Lemma in [2] to obtain the inequality (2.2). Now put Ax> =f(x
(2.8)
+ h) -f(x
-
1 quoted
(2.10)
4
and Czk = 2iC,, sin n,h.
(2.11)
Then 1 Czk 1 < 2 1Cnk j and hence by (2.6) we have gC;Jrf”“:
(2.12)
(O
Put g(r, x) = f Czkkne exp(in,x) -cc
(0 < Y < 1).
We then get the identity g(r, x) = +(r, x + h) - 4(~, x together with (2.9) and (2.10) we obtain g(x) = L2 -
liII;t g(r, x)
this
(2.14)
(I x I G n).
It follows from (2.1), (2.12), (2.13) (2.14), and the inequality
g I Ck I2G 8q
h), and from
(2.13)
(2.2) that
I &>I” &
(2.15)
Hence by (2.10) and (2.11) we get 4 ‘f 1 C,, I2 sin2 1nk / h < 86-l sI lf(x -m
+ h) -f(z
- h)j2 dx.
(2.16)
CONVERGENCE
OF A FOURIER
395
SERIES
Integrating both the sides of (2.16) with respect to h over (0, r/n,) (p E N), we get
If(x+h)-f(x- A)/2 d”).
(2.17)
We see that if j nk j > np then nlmp
sin2 ] nk j h dh = m
1
s0
(Ink/lnph o sin2 t dt s [ln~l/n,l7T
sin2 t dt
.i0
1 %?
[I nk (1
+
hbl
[I nk
l/%1>
. +
using 1 < [ / nk l/n,] < / nk i/n, < 1 + [I nk l/nJ, where [ ] denotes the integral part. Therefore, from (2.17), using w(2yh/n,f) < C(h) . wy1/?2, f) (A > 0, C(h) is a constant depending on A), we get 2 < 86-l % (w’~‘(+z, ,f))2 < 86-l $
C(W’~‘(l/n, ,f))2
from which (2.3) follows. Further, if we put
m = j$o(-I>“-j(;)A, + (2j - I)h)
(2.10)’
and Cck = Cn, exp(--in,&)
(exp(2in,h) - 1)’
= 2V,, exp(-&2,/z) (- 1)” exp(iZ(n,h - a/2)) sinz n,h,
(2.11)’
396
J. R. PATADIA
then / Czk 1 < 2z / Cmk ( and hence by (2.6) we have (2.12)’ Put g(r, x) = 2 Czkr’nk’ exp(in,x) --co
(2.13)’
(0 < r < 1).
We shall then get the identity g(r, x) = i
(- l)l-j
13 I#@, x + (2j - 1) h).
GO
This together with (2.9) and (2.10)’ gives g(x) = L2 -
l&tit g(r, X)
(2.14)’
(I x I < 4.
It follows from (2.1), (2.12)‘, (2.13)‘, (2.14)‘, and the inequality
(2.2) that
g I CZk I2d W’ ‘I[ Ig(4l”ax. Further,
as in the foregoing nlnp s0
proof of (2.3), analogously
(2.15)’
we get
1 Iin*, x dx sinzzI flk I h dh 3 x l +[I ‘,[, n, ,,n,l soa sin21 1
Using this along with (2.3), (2.4) is proved. Finally, let
(2.15)’ and proceeding
Tn,(x) = be any trigonometric
21-l
polynomial
Tnp(r, x) =
1
m=-nn
21-3 *21-2”‘T’T
analogously
1
as in the proof of
(2.18)
am exp(imx)
of order not higher than nz, (p E N). If we write
C a,rl”l m=-78,
exp(imx)
(0
< 1)
(2.19)
CONVERGENCE
OF A FOURIER
for all real X, then since T”, EI?[-,, T&v)
= L2 -
397
SERIES
~1 it obviously
lirmtit Tn,(r, x)
follows that (I x I ,< 4
(2.20)
Put g(x) = fb-4 -
(2.21)
T&)
and if
k + n,s and
= Cns- an,,
if
k = n,
and
kl
zzz0,
if
k fn,
and
kl >s,
= Csn,
if
k = n,
and
kI >n,.
A, = -ak,
Then, for /k j >n,, / ,4,1 =0 Hence by (2.6) we have
if k fn,
41 C 1A, IY~~I < co --3c
1k 1
and /A,
(0
(2.22)
= j Cn, I if k = n, .
(2.23)
< 1)
and if we put g(r, x) = f A,rlrcl exp(ikx) -B
(0
< 1)
(2.24)
for all real X, then its existence is assured by (2.23) and we get the identity g(r, X) = $(Y, X) - Tn,(y, x). This together with (2.9) (2.20), and (2.21) implies g(x) = L’ -
liff{t
g(r, x)
(I * I G q).
(2.25)
Now, on account of (2.1) (2.23), (2.24), and (2.25) we can apply the Lemma quoted in [2] to obtain
f I A, I2< 86-l s, I g(x)[” dx. --m Therefore
1
(2.26)
we get (2.27)
Since (2.27) holds for arbitrary trigonometric polynomial of order not higher than n, , we get the inequality (2.5). This completes the proof of Lemma 3. Proof 409/65/2-11
of Theorem
1.
Let
n, = 0, nk = -nMk
(k < 0); CnO = 0, C,* =
398
J. R. PATADIA
S(%k- ib,,) (k > 0) and C,* = Cn-, (k < 0). We assume throughout, without loss of generality, that (2.1) holds. In view of (1.2) this can be achieved, if necessary, by adding to f(x) a polynomial in exp(in,x), a process which affects neither the hypothesis nor the conclusion of the theorem. Then, putting m, = YL+, I Cnk I2 in th e inequality (2.3) of Lemma 3, we get r;a” < C(w(a)(lIn, ,f))“, where C is some constant depending on 6. Now, applying the Lemma 2 with uk = / C,& I2 (k E 2) and F(u) = uej2, we obtain using (2.29)
= g1 (rnklv’2
G c g1 ((~Yll%
,f)Y-P’“)
< 00
on account of (1.5). Therefore (1.6) holds and this completes the proof of Theorem 1. Applying the inequalities (2.4) and (2.5) instead of the inequality (2.3) in the proof of Theorem I and proceeding analogously we get Theorems 2 and 3, respectively. ACKNOWLEDGMENTS The author is thankful tion of this paper.
to Professor V. M. Shah for his help and guidance in the prepara-
REFERENCES 1. N. K. BARY, York, 1964.
“A
Treatise
on Trigonometric
Series,”
Vols.
I and II,
Pergamon,
New
2. P. B. KENNEDY, Fourier series with gaps, Quart. J. Math. Oxford Ser. 2 7 (1956), 224-230. 3. R. E. A. C. PALEY AND N. WIENER, “Fourier Transforms in the Complex Domain,” Vol. XIX, Amer. Math. Sot. Colloq. Publ., 1934. 4. S. B. STECKIN, On the absolute convergence of orthogonal series, I, Amer. Math. Sot. Trunsl. Ser. 1 3 (1962), 271-280. 5. A. ZYGMUND, “Trigonometrical Series,” Warsaw, 1934.