On the convergence of a lacunary trigonometrical series

C OMPOSITIO M ATHEMATICA

F U C HENG H SIANG

On the convergence of a lacunary trigonometrical series

Compositio Mathematica, tome 15 (1962-1964), p. 28-33

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Series

by

Fu

Cheng Hsiang

1. A

lacunary trigonometrical

^{series is a} series for which the

terms different from zero are very sparse. This kind of

trigonomet-

rical series _{may be}

put

^{in the form:}

We here _assume, for

simplicity,

^{that the constant term of the}

series also vanishes. A series S _Ci is said to possess ^a gap

(u, v)

if _c, ⁼ 0 for u i ^{v. It} is known that

[3,

p.

251, § 10.81]

i f

^{a series ~}

Ci

possesses

inlinitely

many gaps

(Mkl m’ k )

^{such that}

M’k/Mk &#x3E; U

&#x3E; 1

f or

some p and is summable

(C, 1) to

^sum

^S,

then

S.

^{and also}

Smk

^{converges to S.}

2.

Suppose

that the above

trigonometrical

series is the Fourier series of an

integrable

^function

f(x).

^This series possesses in-

finitely

many gaps

(nv, nV+l)

^{such that}

n,,,/n,,

&#x3E; A &#x3E; 1 for some

03BB.

Write,

^{for a}

given point

xo ,

Since

holds for almost all _x, the Fourier series of an

integrable

^func-

tion is therefore almost

everywhere

^summable

(C, 1) by Fejér’s

theorem. From this

fact,

^{we draw}

immediately

^the

following

^well-

known

KOLMOGOROFF’S THEOREM

[1, 2]. Il

the Fourier series

o f

^an

integrable function f (x )

possesses

in f initely

many gaps

(nv, nV+l)

such that

n"+l/n"

&#x3E; ,u &#x3E; 1, ^{then the}

partial sum Sny

^{converges almost}

everywhere

^to

1(x).

We can also conclude

that,

^{at a}

given point

xo ^at which

q*(t) =0(1) (t

^-

+0),

^if

nV+l/nv

&#x3E; Â &#x3E; 1, then

Snv(aeo)

^-

^f(xo)

as v 00.

29 3. We write

In this note, ^we

replace

^{the condition}

cp*(t)

⁼

0(l) by

^the

weaker

conditioncpl (t)

⁼

0(1) (t -&#x3E; + 0).

^We

develop Kolmogoroff’s

theorem into the

following

manner.

THEOREM.

Il

^the

lacunary

Fourier series

of

^an

integrable f unction f(x)

possesses

in f initely

many gaps

( n,, , nV+l)

^{such that}

nV+l/nV

&#x3E; Â&#x3E; 1, and

i f,

^{at a}

given point

xo,

( i ) CPl(t)

^{= 0}

(1) (t

^-

+0)

^and

when ⁰

t q f or

^some ~, then

Snv(xo)

^~

^f(x0)

^{as v} --&#x3E; oo.

4.

Now,

^{we are in a}

position

^to prove the theorem.

Take,

^for

instance,

n" ⁼ n, n"+l ^{= m} and dénote

respectively by Dn(t)

and

Kn(t)

Dirichlet’s and

Fejér’s

kernels. Then

Then,

^{from the}

identity

and in virtue of the

special property

^{of the}

lacunary

^Fourier

séries,

^{we obtain}

Write

Intégration ^{by parts gives}

say. We are

going

^{to estimate} the orders of the

integrals I1

^and

I. respectively.

^{We write}

say. Since

( 2

^sin

t/2)-I-t-1

^is

bounded,

^we

obtain, by

^Riemann-

Lebesgue’s ^theorem,

as m - oo

by

De la Vallée Poussin’s test

[8,

p.

88, f 2.8]

^{for the}

convergence of Fourier series at a

given point by

the condition

(ii) and ~2 (t)

^{= 0}

(1)

^{as 1 - + 0.}

Similarly,

31

as n - oo

by

^{the same test.} It follows that

as n - oo. In

estimating

^{the order of}

1",

^{let us write}

say. We have

say.

Now,

Considering

^that

for 0 ^t

n/2

and oc &#x3E; 0, ^we

get

Therefore,

Moreover,

^{we have}

Last, by

the second mean value

theorem,

^{we obtain}

’where A is an absolute constant. From the above

analysis,

^it

follows that

For a

given a

&#x3E; 0, ^{we can choose a ô so} small that

by

the condition

(i).

^After

fixing

^{ô, we take a}

sufficiently large

88

integer n

which makes

11.1, 1141

^and

(m-n)-16-S

^{all less than e.}

Thus,

^{we obtain}

finally

Since e is an

arbitrary

^small

quantity, letting

^{it tend to zero,}

we

get

This proves the theorem.

REFERENCES.

A. KOLMOGOROFF

[1] ^Une contribution à l’étude de la convergence des séries de Fourier, ^Fund.

Math., ⁵ (1924), ^96-97.

J. MARCINKIEWICZ

[2] ^{A new} proof ^{of a} theorem of Fourier series, Journ. London Math. Soc., 8

(1933), ^279.

A. ZYGMUND

[8] Trigonometrical Series, ^Warsaw (1985).

National Taiwan University, Taipei, ^China.