Divergence of the Fourier series of continuous functions with a restriction on the fractality of their graphs Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 3, No. 2, 2017

DIVERGENCE OF THE FOURIER SERIES OF CONTINUOUS FUNCTIONS WITH A RESTRICTION ON THE FRACTALITY

OF THEIR GRAPHS1

Maxim L. Gridnev

Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia, [email protected]

Abstract: We consider certain classes of functions with a restriction on the fractality of their graphs. Modifying Lebesgue's example, we construct continuous functions from these classes whose Fourier series diverge at one point, i.e. the Fourier series of continuous functions from this classes do not converge everywhere.

Key words: Trigonometric Fourier series, Fractality, Divergence at one point, ^ntinuous functions.

Let f be a 2n-periodic integrable function, and let

— + ^(flfc cos kx + bk sin kx), (1)

k=l

where

1 rn 1 rn

1 r 1 r

— / f (t) cos kt dt, bk = — / f(t) sinkt dt, n J-n n J-n

ak

n J-n n

be the trigonometric Fourier series of the function f. Denote by Sn(f,x) the nth partial sum of (1). It is known (see [1, Ch. 1, Sect. 39]) that if f has bounded variation on the period (f € BV), then its Fourier series converges everywhere on R, and if, in addition, f is continuous on R, then the Fourier series converges to f uniformly on R. Salem [2] (see also [1, Ch. 4, Sect. 5]) considered the classes BVp of functions of bounded p-variation and proved that if f € BVp, then the Fourier series of f also converges everywhere on R. (Further generalizations of these results see in [3]).

The author [4] studied relations between the classes BVp and classes of continuous functions with a restriction on the fractality of their graphs.

Definition 1. Let f: R ^ R be a bounded 2n-periodic function. By the modulus of fractality of the function f, we call the function v(f, e) which, for all e > 0, gives the minimal number of closed squares with sides of length e parallel to the coordinate axes that cover the graph of the function f on [—n,n].

Definition 2. Let ^: (0, ^ R be a nonincreasing continuous function such that

lim u(e) = We define the functional class

£^0

F^ := {f € C2n : v(f,e) = O(^(e))}. In the case ^(e) = l/ea, where 1 ^ a ^ 2, we will write Fa instead of F1/£".

1This work was supported by the Russian Science Foundation (project no. 14-11-00702).

The following statements were proved in [4]:

BV = BV! = F\ [4, Theorem 1]; (2)

BVP C F2-l/p, p> 1 [4, Theorem 2].

The latter is unimprovable; i.e., BVv+£ ^ F2-1/p for all e > 0.

In the present paper, we study the pointwise behavior of the Fourier series of continuous functions from F^.

Theorem 1. Let ^: (0, — R be a nonincreasing continuous function, let epi(e) be a

nonincreasing function, and let

lim e^(e) = (3)

Then there exists a continuous function F^ whose Fourier series does not converge everywhere. Proof of Theorem 1. We will require that

t-"1 < /x(vrt) < 2t"i, t- € (0,1]. (4)

By (3), the former inequality holds on an interval (0,5) and, changing the function ^ on the interval 1), we will obtain the same class FThe latter inequality can only reduce the class FThus, if we find a required function in the narrower class, it will belong to the wider class immediately.

To obtain a function f € FM with divergent Fourier series, we modify Lebesgue's example from [1, Ch. 1, Sect. 46]. We start with defining an increasing sequence of natural numbers {ak} as follows. Let a0 = 1. Suppose that the first k elements a0,a1,..., ak-1 have been already defined. From inequalities (4), it follows that

ak-1 \ak-1

and, for b ^ (6ak-1)2,

b2 (n

ak-1 \b,

Then, by continuity, there exists the smallest number a such that

a2 „ in

= 3 ¡j, ak-1 \a

As ak, we take the largest integer such that ak ^ a and the fraction ak/ak-1 is integer. It is not hard to understand that ak belongs to [a — ak-1, a], and, in view of the inequalities

ak >£^1 = 3//^!-! >2, (5)

afc-1 afc-1 \aj a

we conclude that ak > ak-1.

The definition of ak implies the inequality

^ 3/J,(tts), e €

e2ak-i

1

ak '

(6)

The definition of ak, inequalities (5), and condition (3) imply that

ak

ak-1

^ k ^ +<x. (7)

Consider the half-open intervals

h=\ —

n n

ak ak-l

k e N.

Let {ki}°=0, k0 = 1, be an increasing sequence, on which, in what follows, two additional conditions will be imposed. Let

Ck

1

ln ak/ak-

k e {ki}°=oi

0, k / {ki}=o-

Finally, we define the function f on the interval [—

f (x) = Ck sin akx, x e Ik, f (0) = 0, f (-x) = f (x).

We extend the function f to R periodically. The resulting function is continuous on each Ik and, since ak/ak-\ is integer, is continuous and vanishes at the points ±n/ak. Thus, the function f is continuous on [—n,n].

Since f has only a finite number of maxima and minima on [5, n], 5 > 0, it has bounded variation on this interval (and on [—n, -5] as well). Thus, its Fourier series converges at every x e [—n, n]\{0}.

Consider now the sequence of partial sums of the Fourier series of f at the point x = 0. As is known [1, Ch. 1, Sect. 32, formula (32.5)], for the function f, we have

Sk(x, /) = - f f(x + tf-^dt + o(l); n J-n t

hence, for x = 0,

Sk(0J) = - r f(t)S-^dt+ 0(1). n J-n t

The function f is even; therefore,

Sk(0J) = - ^ f(tf-^dt +o(l). n J o t

Let us show that, after an appropriate choice of {ki}

sin aki t

rn <

Ji = / №-

Jo

t

dt — i —

Then Sak. (0, f) — as i — i.e., the Fourier series of f diverges at x = 0. To estimate Ji, we divide it into three terms:

r/aki „ n sin ak.t fn/aki-i sin ak.t, fn

■h= \ f(t)—-~2—dt + / f(t)-r^dt +

t

>n/ak

We have

Hence,

t

sin aki t

'n/ak.-1

„, .sinak■ t , i /' , .

f(t)—-^-dt = Ji+Ji +Jt . (8)

t

< aki.

n

lJiK n majc l/№fci— = KCki+1 = 0(1).

(9)

i

o

Suppose that kl}... ,ki-l have been already defined. Then the function f (t)/t is defined, bounded, and continuous on (n/aki-l,n]. Extending this function by zero to [—and assuming that ki are large enough (this is the first of two conditions on k), we can make the Fourier coefficient aki of the obtained function small enough; more precisely,

I ''' .

iJi I =

f (t)

- sin a,kit at

<

1

(10)

It remains to estimate J . We have

Ji

rn/ak.

In/ak.

ck. sinakt

rh ln dki

sin akit t

,11

cn/aki-i i - cos2akit

dt

'n/aki

cki fn/aki-1 cos2akit

2

aki-1

2

'n/ak.

t

dt.

According to the second mean value theorem, taking into account that the function 1/t is positive and monotone, we find that

r/aki-i cos 2ak.t 'n/ak. t

dt

<

n

/n/aki

cos 2aki t dt

<

akj 2

it 2 ak.

1 n

Thus,

j" cki i aki . x

Ji =^fln-—+ o(l).

2 aki — 1

Combining (8), (9), (10), and (11), and taking into account (7), we conclude that

(11)

■h = f In -^L. + 0(1) = i ./in + 0(1) +oo. 2 aki — 1 2 V aki — 1

Let us now estimate the modulus of fractality v(f,e). Denote by v(f,e)[a,b] the minimal number of squares with sides of length e parallel to the coordinate axes that cover the graph of the function f on [a, b].

If k1,...,ki-1 have been already defined, then the function f is defined on the interval [n/aki-1,n] and has bounded variation; hence, by (2),

v (f,e)

[n/aki_i

-Tl 0\£

Condition (3) allows us to take ki such that, for ne € (0, n/aki],

v (f, ne)

[n/aki-i

< ß(ne)-

(12)

This is the second condition on ki.

Let 0 < e ^ 1. Then there exists i € N such that e € [1/aki+1, 1/aki]. Let us prove the inequality v(f,ne) ^ C^(ne) with some constant C. It follows from what is proved above that the required inequality holds for the covering of the graph on [n/aki-1, n]. The inequality also holds for the intervals [n/aki+1-1, n/aki] and [n/aki-1, n/aki-1 ] where f is identically zero; hence,

v (f, ne) r , , i + v (f, ne)r , , i

[n/aki+1-1,n/aki\ [n/aki-1,n/aki-1\

n

i-1 J e

n

1

t

n

Covering the whole rectangle \_0,n/aki+1 -1] x [—cki ,cki] and using (6), we can obtain the estimate

v(f, ne)

[o,n/aki

<

i+1-

n

aki+1-ine ne

<

8

24

2 < —

aki+1-ine2 n

(14)

here and in what follows, \x] stands for the rounding of x upward.

It remains to cover the graph on the interval [n/aki,n/aki-i] where f (x) = cki sin akix. We can divide this interval into N = 2aki/aki-1 — 2 intervals of monotonicity of f: [n/aki + n(n — l)/2aki ,n/aki + nn/2aki~\, n = 1,...,N. Let us show that, to cover the graph of f on each of these intervals, we need at most 8/ne squares. Using the definition of the length of a curve, we can show that the length of the graph of f on these intervals is at most n/2aki + 2cki. Squares with sides of length ne can cover the graph of a monotone function of length at least ne. Hence,

u(f TTF) . IV 7T . _ \ 11 . 8

( , ) [n/aki +n(n-1)/2aki ,n/aki +nn/2aki ]

<

n \ 1

-— + 2 cki — 2aki V ne

<

From (6) and the monotonicity of e^(e), we obtain

v (f, ne) r ,

Ln/aki ,n/aki

<

4aki

<

(n \ ( n \ n

— 12«— —alité) aki aki aki

ne

12

neak.

n2e^(ne)

< -^e). (15) n

Finally, by (12), (13), (14), and (15), we obtain the following estimate for the modulus of fractality of f:

V(f,ne) ^ 2v(f,neW] < 2(V(f,ne)[c)n/aki+1-1] + V(f,ne^[n/ak+1-1,n/akt] +V (f, ne) [n/ah ,n/akH-i] + V (f, ne) [n/a^/a^ ] + V (f ^ [n/a*_i, n] )

/24 n 12 \

^2 —At(vrt) + - + —ii(ire) + At(vrt) = OUi(ire)), \ n e n J

i.e., f € F».

The theorem is proved.

1

neaki-i

1

REFERENCES

1. Bari N.K. Trigonometric series. Moscow: Fizmatgiz, 1961. 936 p. [in Russian].

2. Salem R. Essais sur les series trigonometriques. Actual. Sci. et Industr. Vol. 862, 1940.

3. Waterman D. On converges of Fourier series of functions of generalized bounded variation // Studia Mathematica, 1972. Vol. 44. P. 107-117.

4. Gridnev M. L. About classes of functions with a restriction on the fractality of their graphs // CEUR-WS Proceedings, 2017. Vol.1894: Proceedings of the 48th Intern. Youth School-Conf.: Modern Problems in Mathematics and its Applications, Ekaterinburg, February 5-11, 2017. P. 167-173. http://ceur-ws.org/Vol-1894/appr5.pdf [in Russian].