Contributions of the theory of asymptotic distribution modulo I

Contributions to the theory of

asymptotic distribution modulo i

1387

720

Bll A Delft

CONTRIBUTIONS TO THE

THEORY OF ASYMPTOTIC

DISTRIBUTION MODULO I

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP-PEN AAN DE TECHNISCHE HOGESCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAG-NIFICUS, IR. H. J. DE WIJS, HOOGLERAAR IN DE AFDELING DER MIJNBOUWKUNDE, VOOR EEN COMMISSIE U I T DE SENAAT TE VERDEDIGEN OP WOENSDAG 7 JULI 1965

DES NAMIDDAGS TE 4 UUR

DOOR

PETRUS JOHANNES HOLEWIJN

geboren te U t r e c h t

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. D R . L . KUIPERS

Aan mijn ouders aan Bartje aan Peter

CONTENTS

Introduction 9 CHAPTER I Uniform distribution of functions 11

1.1 Definitions and criteria 11 1.2 Relation between B-u.d. and C-u.d 15

1.3 Not uniformly-distributed functions 16 1.4 A continuous analogue of a theorem from the discrete

theory 16 1.5 The continuous analogue of a generalization of VAN DER

CORPUT'S theorem 21

CHAPTER II Uniform distribution of sequences of numbers 27

2.1 Definition and criteria 27 2.2 A relation between uniformly distributed functions and

uniformly distributed sequences 28 2.3 Not uniformly distributed functions giving rise to not

uniformly distributed sequences 29 2.4 An analogue for the discrete case of a theorem from the

continuous theory 31

CHAPTER III Another proof of W E Y L ' S criterion 35

CHAPTER IV A metrical theorem 43

References 48

I N T R O D U C T I O N

In Chapter I of this thesis we start from the definition of the uniform distribu-tion of a real funcdistribu-tion of a real variable as given by E. HLAWKA in his paper ,,Über C-Gleichverteilung" (1.1.1). According to this definition we call a function f{t) A-uniformly distributed modulo 1 [A-uA.) if for every interval

T

[a,/3) C [0,1] lim /' a{t,T)6{{f[t)}\ a,P)dt = (i-a ÏÏ 0 is the characteristic function of the interval [a,ft) and if {ƒ(/)} denotes the value oïf{t) modulo 1.

T

The weight/unction a[t,T^ has the property that \ a[t,T)dt = 1 for each 7" > 0.

(i

T

In the special case that a[t,T) = b{t)lu^^^ji we speak of B-uniJorm distribution

0

modulo 1 {B-uA.) and if, moreover, b{t) = 1 (then a{t,T) = IjT, T> 0),

about C-uniform distribution modulo 1 (C-u.d.).

In general we will restrict ourselves to the cases j5-u.d. and C-u.d.

Contrary to the method of E. HLAWKA we will not make use of the notion of

discrepancy (except for a special case considered in Chapter I I I ) . We will use

an analogue of a criterion introduced by H. W E Y L which is of major impor-tance in the theory of uniform distribution.

Having proved this analogue (1.1.3) we show by a simple transformation that the cases 5-u.d. and C-u.d. are in fact equivalent (1.2.1). Hence we will be able to state some theorems for the case B-\xA. which were already proved for the case C-u.d. (1.2.1 and 1.3.1).

We proceed by transforming a theorem due to M. Tsuji which formulates a property concerning weighted means of the residues (mod 1) of the elements of certain sequences of numbers. The continuous analogue of this discrete theorem is proved in 1.4.1.

By the stated equivalence of B-uA. and C-u.d. wc then also obtain some properties for the case C-u.d. (1.4.1, theorem 8).

In a similar way we generalize a theorem given by L. KUIPERS, which is the continuous analogue of a well-known theorem of J . G. VAN DER CORPUT (1.5.1), for the case 5-u.d.. Again transformation yields a property for C-u.d. (1.5.1, theorem 10).

With every function ƒ (/) (< > 0) a sequence ƒ (<) {t = 1,2,. . .) can be associated. In Chapter II we define the uniform distribution of such a sequence. The elements of this sequence are the values of the function ƒ([<]), where [t] =

= t— {/} is the largest integer < t. We call the sequence ƒ (n) An-uniformly

distributed modulo 1 [n = 1,2,. . .) (A^-u.d.) if the function ƒ([/]) is B-uA. where the numbers A» depend on the function b{t). W E Y L ' S criterion can be deduced directly from the continuous case (2.1).

As a first application of this criterion we investigate some relations between the B-uA. of a function ƒ (<) and the A,j-u.d. of the sequence ƒ(«) {n = 1,2,...) (2.2). Here EULER'S summation formula plays an important role.

Moreover, we deduce for the discrete case an analogue of the already mentioned theorem from the continuous theory which establishes the equiv-alence of the cases B-uA. and C-u.d. (2.4.1).

T o this end the sequence ƒ« (considered as function of n, n = 1,2,. . .) must be extended to a function ƒ (<) of the continuous variable t > 0, which still has to satisfy certain conditions.

Also in the proof of this theorem EULER'S summation formula is a starting point.

The results obtained are illustrated by simple examples.

In Chapter I I I we prove W E Y L ' S criterion for the ^-u.d. for a second time in this thesis. In this second proof we use properties from the theory of probability concerning distribution functions and characteristic functions. In the discrete theory some investigations in this direction were made by W . J . L E V E Q U E .

It turns out that necessity of W E Y L ' S criterion can be proved by the well-known continuity theorem from probability theory (3.1).

The continuity theorem, however, is not able to prove sufficiency of the criterion. In order to prove sufficiency we need an estimation of the so called

discrepancy which is defined by DT = sup \FT{X)—X\, where FT[X) = = \a{t,T)d{{f{t)]-%x)dt (3.2).

d

In Chapter IV we prove a metrical theorem for a set of functions7i(/) [t > 0) depending on a parameter x (0 < A; < 1). As a starting point we use the well-known monotone convergence theorem.

In the applications we restrict ourselves to the case that the set of functions is of the ïorm fx{t) = x.f{t), 0 < x < 1. We prove that under certain con-ditions onf{t) these functions are C-u.d. for almost all x from [0,1] (4.4 and 4.5). We will always refer, if necessary, to the litterature consulted which can be found at the end of this thesis.

C H A P T E R I

U N I F O R M D I S T R I B U T I O N O F F U N C T I O N S

1.1. Definitions a n d criteria

1.1.1. Let the real-valued function a{t,T), defined for / > 0 and 7" > 0, be nonnegative and integrable with respect to t in each interval 0 < / < T, while

T

I a{t,T)dt = 1 for 7" > 0; then we define:

Ó

D e f i n i t i o n . (E. H L A W K A , [2]). The real-valued function ƒ(<), defined for i > 0 and integrable on each interval 0 < / < 7" is called A-uniformly distributed

modulo 1 [A-uA.) if for each interval [a,fi') C [0,1] T

lim fa{t,T)Oi{f{t)};aSdt = fi-a,

where d{u;a,ji) denotes the characteristic function of the interval a s^ u < p and {fit)} means the value off{t) modulo 1. (The considered integral has a meaning as Lebesgue-integral).

Special cases. b[t)

1Ïa{t,T) = where b{t) is nonnegative and integrable in each interval H[T) T

0 < < < 7"and where fi{T) = \ b{t)dt for 7" > 0, we speak of B-uniform distri-()

bution modulo 1 [B-uA.), and if, moreover, b{t) = 1 (then fi{T) = 7") we speak 0Ï C-uniform distribution modulo 1 (C-u.d.).

1.1.2. By C[0,1] we denote the space of all continuous complex-valued functions!»(a) defined on the interval 0 < a < 1 with the property !X'(0)=w(l). We define a norm in C[0,1] by \\w\\ = max !;»(«)

|-We now prove os;"<i T H E O R E M 1. The function ƒ (0, integrable for i > 0, is ^-u.d. if and only if for each function w belonging to C[0,1]

T 1

lim i a[t,T)w{{f{t)])dt = f w{u)du.

P r o o f Without restriction we assume that w{u) is real. We first show that the condition is necessary.

For this purpose we divide the interval [0,1] into s equal parts; the sub-interval {a„,fi„) be denoted by r„.

We define

M ( T „ ) = the least upperbound o[ w{u) in T„,

N ( T J = the greatest lowerbound of !X'(M) in r„. Now for every t > 0

SN(r„)0({/(0}; a^A) < w{{fm <

«.,/?„).

a a

So, multiplying by a{t,T) > 0 and integrating with respect to t from 0 to T' and taking the limit as 7 ^ ^ oo, we obtain

T

2N(T„)(/^„-aJ < lim la{t,T)w{{fit)})dt<

a T—>• CO Q T

< iï^ I'a{t,T)w{{fit)})dt<i:M{r„)ift^-a„).

T —>- CO n a

If now we refine the division of the interval [0,1] we see

T 1

lim /' a[t,T)w{{f{t)})dt = /' w{u)du

r - > OP (j ,j

if w belongs to C[0,1] as was to be proved.

T h e sufficiency of the condition follows directly from the fact that the characteristic function of the interval [«,/?) C [0,1] can be approximated from the upper and the lower side by functions w from C[0,1] such that their integrals approximate fi — a.

1.1.3. T H E O R E M 2 ( W E Y L ' S criterion).

T h e function ƒ (<), integrable for < > 0, is A-\iA. if and only if 7'

lim /' a{t,T)e""'"'yt = 0, k = 1,2,3,. . . (1)

For the proof of theorem 2 we make use of the following lemma (see for in-stance [9]).

L E M M A 1. Let a family of complex linear functionals 0T [T > 0), defined on some normed space X, have the properties

(i) lim 0T{X) exists for all x belonging to some subset Q of X, which is

7 - - > o o

dense in X;

(ii) there exists a constant c > 0 such that for all 7" > 0 [|<PT|| < c (where

II'JT'II is the smallest number M for which | ^ T ( ^ ) | < M-||x|j for all A; from Z ) ; then lim 07'(;c) exists for all A: from X.

P r o o f of l e m m a 1.

E

Choose X from X a n d f > 0 ; then a point x' in Q exists with ||.v —.ir'[[ < — . Since lim <prp{x')

exists there is a n u m b e r 7" > 0 such that for T^^T^ > T r-^-co \1>T,{^')-t>T.V)\ < 3

a n d thus, since \<1>T{X) \ < \\^T\\-\\x\\ for x in AT

\t>T,{x)-1>T.,{x) I = \<PT^{X-X') + 0T^{X')-0T^{X')+<1>T,{X'-X) 1 <

< \0TSX-X') 1+ [ 0 T , M -fT.ix') \+ \0T,(x'~x) \ <

< \\0T,\\-\\x-x'\\+\0T,ix')-'PT,{x')\+\\0T,\\-\\x-^x\\<cj^ + ^ + c^^=e, which proves t h e lemma.

P r o o f of theorem 2.

Necessity of the condition follows directly from theorem 1 in choosing w{u) =

= e^""'" (this function belongs to C[0,1] for h = 1,2,. . . ) .

In order to prove sufficiency we introduce for 7" > 0 a family of linear functionals, defined on C[0,1], by

T

0r{w) = \ a{t,T)w{{f[t)])dt, ()

where w belongs to C[0,1].

Since \0T{W)\ < \\w\\ we have \\0T\\ < 1 for 7" > 0 (it even holds \\0T\\ = 1, since for w{u) = 1 we get \0T{W)\ = 1 =

||!^1|)-If we consider the subspace £ C C[0,1], consisting of all finite complex linear combinations of functions

cf,,„{u) =e""""',m = 0, ± l , + 2 , . . .

defined on the interval [0,1], some function w^ from £ may be written as

w^{u) = a,e"""'' + .. . + «j.^'"-V, j , ^ji \{ k ^ l ,

and we obtain

V^ ^ „••»> lO, i f j f c ^ O , / t = 1,...,6 lim (I>T[W,) = ) aic- Wm j a[t,T)e'""^f^"dt = \ ' -^ ^ ' ' '-^^ 7 " - ^ » ^ ^ r ^ c o (•) I aic, lijk = 0

because of (1).

Moreover we see for w^ from £

;• , w V I ' ^ - w | o , i f i . 7 ^ o , A - = i , . . . , / ,

/ Wi^[u)du = y aic j e 'I' du =

Ü f-\ (•) I «ft, iO* = 0 Thus lïw^ belongs to £ lim <PT{W^ exists and

However, since £ is dense in C[0,1] (this follows from WEIERSTRASS'S second approximation theorem, see for instance [1]) we deduce from lemma 1 that lim 0T{W) exists for w from C[0,1]) so that (2) holds i[ w is in C [0,1], since

1

I lim 0T{W)— j w{u)du\ = \ lim 07'(w —«;^) + ( lim 0T{yog) —

T — > OT (J T —>• 03 T — > <» I I

— / w^[u)du)+ j {ws,{u)-w{u))du\ < 7 , + ^ = e, if Ikf —!»|| < ^ •

( i d ^ ^ *

Hence, the proof of theorem 2 is supplied by theorem 1.

1.1.4. Applications

1. The function ƒ ( 0 = log t is B-uA. if b[t) = l/< (for < > 1; we define

b{t) = 0 if 0 < < < 1).

Proof. From theorem 2 it follows {/i{T) = log T, T > 1):

I 7 ^"""""'^' = ^ ^ - T I T T ^ ' ^ 0 if r ^ oo (/, = 1,2,.. .).

log T j' t 2mh log T

2. The function ƒ(<) = log / is not 5-u.d. ii'b{t) = t", a> — l(i > 0).

T""" + 1

Proof. Wc have, if we take h = 1, since ft{T) = c t + l «4-1 /• „ ., , a-\-\ „ ., „

r " ' , / « + l + 2 7 r z which does not tend to zero as 7" tends to infinity.

1.1.5. We define

D e f i n i t i o n . A nonnegative function b{t) defined for i > 0 is said to satisfy property {A) if

(i) for some /o > 0 b{t) = 0 if 0 < / < io (ii) h{t) is continuous for t > <o

(iii) //(<) = / b{x)dx ^ oo if / - ^ oo.

'Ó

We observe that if the function b[t) satisfies property {A) then //(<) is dif-ferentiablefor t > to, the inverse M{u){ = t) of/;(/)( = M) exists (/((i) is monotone) and is differentiable for u > /((^). In the sequel we will write g{u) for f [M{u)).

Further we will write, if necessary, b{t) {t > to) if we want to indicate that

b{t) = 0 for 0 < < < to.

1.2. R e l a t i o n bet^veen B-u.d. a n d C-u.d.

1.2.1. We prove

T H E O R E M 3. Let b[t) satisfy property [A); let the function ƒ(/) be inte-grable on each interval 0 < i < T; then f[t) is 5-u.d. if and only if the function g(u) is C-u.d..

Proof. The proof follows directly from W E Y L ' S criterion and the identity

- 1 - j biDe'^-'-^^'^dt = - ^ 'fr'-'-'-V^, {h = 1,2,...).

We obtain a direct consequence of theorem 3 if we consider the following theorem due to L. KUIPERS (the theorem follows from the proof of theorem 5 Chapter IV in [3]).

T H E O R E M 4. Let the function ƒ (0 be differentiable for < > ^n > 0; l e t / ( 0 be monotonie increasing with ƒ(^) -^ oo if i-> oo; let ƒ'(<) be monotone for

t > to and let tf'{t) -^ oo'iït-^ oo; then the function ƒ (<) is C-u.d.

Now, \ï b{t) satisfies property [A) then g[u) is differentiable and monotonie

/'(O

increasing to infinity as M ^- oo ifƒ(0 is, and we have, iï b[t) 7^ 0, g'[u) = -Tyr-/«(O

(since M' (u) • 11' [t) = M'[u)-h[t) = 1), so that ug'{u) = ƒ'(<), and

applica-b{t)

tion of theorem 4 to the function g{u) gives by theorem 3

T H E O R E M 5. Let b{t) satisfy property {A), let the function ƒ (<) be differen-tiable for t > to > 0 and let ƒ (<) be monotonie increasing to infinity as i ^^ 00,

f'{t)

let b{t) =?^ 0 and let —— be monotone for t > to;

uU) *(')

then, if f'{t) -> 00 as / - > 00, the function ƒ (/) is B-u.d.

h{t)

1.2.2. Applications

3. The f u n c t i o n / ( 0 = a{logt)" is if^-u-d. ( < > 1) if a > 1, /3 > ^ 1 , a > 0; 4. The function ƒ (<) = a{logt)" is r '-u.d. {t > 1) if a > 0, « > 0;

5. The f u n c t i o n / ( 0 = (log t)" is (log O^-u.d. {t > 1) if/9 > 0 and a - / ? > 1; 6. The function7(0 = /" is ^''-u.d. (/ > 1) if a > 0, ,3 > - 1 .

P r o o f s . We only remark that it is simple to verify that the conditions of theorem 5 are satisfied.

1.3. N o t u n i f o r m l y - d i s t r i b u t e d f u n c t i o n s

1.3.1. In 1.2 we considered uniformly distributed functions for which

ug'{u) I = 'v-— •ƒ' (/) tends to infinity as « -;• oo. We now consider

not-uniform-ly distributed functions with bounded ug'[u). L. KUIPERS and B. M E U L E N -BELD ([5]) proved that differentiable functions ƒ (?) for which tf'[t) is bounded are not C-u.d., so that application of theorem 3 yields

T H E O R E M 6. If the positive function b{t) satisfies property [A), i f / ( 0 is differentiable and if a fixed number Uo ^ Q exists such that for ;/ > «o

\ug'{u)\ < K, K a positive constant; then f {t) is not -6-u.d.. Remark.

Since -, = tf'{t) we see that the condition of theorem 6 is satisfied in the special case that both tf'(t) and are bounded. Further we observe that, if é(i!) is differentiable for t > 0, the boundedness of is garanteed if

b'{t) > 0 which can be seen from ^ ^ tb(t)— / xb'{x)dx

0 < ^ = — - ^ ^ — — - < ^ = 1

tb[t) tb[t) tb{t)

We summarize the above mentioned considerations in the following corollary: C O R O L L A R Y 1. If the positive function b{t) satisfies property {A), if b{t) is differentiable with b'{t) > 0, i f / ( 0 is differentiable and if tf'{t) is bounded, then ƒ(/) is not 5-u.d.

1.3.2. Applications

1. T h e function ƒ (<) = sin t is not «'-u.d.

P r o o f . The assertion follows from theorem 6 since |M^'(M)| = |cos t\ < 1. 8. T h e function/(O = a(log t)" is not è(log /)'*-u.d. ( « > 1) if a > 0, é > 0. 0 < « < 1 , / J > 0 . ^ doeO'^-' P r o o f . We apply corollary 1. Here b'{t) = bft ^-^^ > Ofor/ > 1; also, evidently, tf'{t) is bounded.

1.4. A c o n t i n u o u s a n a l o g u e of a t h e o r e m f r o m the d i s c r e t e theory 1.4.1. We prove a continuous analogue of a theorem due to M. Tsuji ([8]). 16

T H E O R E M 7. Let the function a{t) be decreasing; let b*{t) = a{t). b{t) and let b{t) and b*{t) satisfy property [A) (with to = 0); let the function ƒ(/) be con-tinuous for t > 0; then, iff{t) is B-u.d., f (t) is also 5*-u.d..

The proof of theorem 7 will be preceded by the lemma

L E M M A 2. Let To and T be two given integers with T> T'o > 0; let the function b{t) be continuous for 0 < < < T; let the function ƒ (<) be continuous for 0 < / < T; let /i{To) > 1 and let (p{t) = è(0 • e^™''^'";

then for £ > 0 an integer m > 0 exists for which it holds, uniformly in those integers n satisfying Tom < K < Tm

fi{nlm) ó' <p{t)dt n — 1

-L

2^H^,

P r o o f . Let e > 0 be given. Since the function ^>{() is uniformly continuous on the interval 0 < f < 7^a number (5i(e, To, T) > 0 exists such that

\(p{t2) -(p{tl) I < ^ if l'2-^li < ^1 ,

and if ^1,^2 belong to [0,7^]. ,

If we now choose an integer mi such that — < ^i we have for all integers n satisfying Tomi < K < 7mi mi

/V'(o

dt-^y,~) ^ y I

mi i—\ \mi -i-J ;

'Pit)~v\ — \]dt

<

< y I 9'(o-9'

(3)

For the same reason as above a number b^ exists such that, if m-z is an integer with — < f52, we have for all n satisfying 71)^2 < « < 7^2

mi

/*

V

<

(4)

Now let ö = min((3i,Ó2) then, if ??2 is an integer with — < d, both (3) and (4)

m

are satisfied if we replace mi and m-i by m respectively. T h e n we finally have for Tom < ;z < Tm

I fi^njm) nlin q'{t)dt-

z

/ i l

' (|;,„._Lv,(i))^(Lv,(r))( '

' Vb''

This proves the lemma

= b{-\ and since /< — 1 > 1. ml \m/

P r o o f of theorem 7. Let O < e < 1. Then we shall prove that a positive integer Ti (e) exists such that for 7" > Ti [T integer)

T

"é*(Oe^™''^"V<| < e (^ = a fixed integer, ^ 0), 1

/.*(r)'ö'

this being sufficient to prove that ƒ(/) is B*-u.d.. For this purpose we choose 0 < e < 1 a n d determine: (i) the positive integer 7o(fi) such that

/ ' * ( 7 ^ o ) > l , (5) which is possible since b*{t) satisfies property {A), and such that for

T > To {Tinteger)

—"—I lb(t)e'''''J"'dt\ < — e ,

this being possible since f{t) is B-u.d.;

(ii) the integer 7"i(£) > 7o such that for T > Ti (7" integer) / * * ( r o ) < ^ £ . / * * ( r ) .

(6)

(7)

We now choose an arbitrary (but fixed) integer 7" > Ti. The interval [0,7"] will be divided into subintervals with length 1/m {m integer) so that the end-points of these intervals form the sequence njm, n = 0,1,2,.. . .,Tm.

We choose the integer m = m[e,To,T) so large that for all integers n satis-fying Tom < K < Tm

m L-i \m m L-J \m' u[n m] A 16

i.=o v = 0 r-x I J 0

a n d v = n— 1 1 sr^ . . IV m

z

N o w from (6) a n d (8) it follows for Tm < n < Tm t h a t

i:'ör-

v = 0

1 1 1

< — E H £ = - e ,

16 16 8 '

w h i l e , of course, also for I <, n < Tom

(10) ' I D ; i 2 ) F u r t h e r i t follows from ( 9 ) , (7) a n d (5) T,m - 1 1 VI . . / ) ' \ 1 „ , 1 / 6* - < - £ + , a * ( 7 ^ „ ) <^eil+/,*{T)) < 8if,*{T)+,^*{T)) m i^ \m' 4 4 4 m o r e o v e r w e see from (9) OT / j \ m ' 4 = 2^/'*(7'); (13) (14) = 0 W e m a y n o w w r i t e , if a„ = a - , b,. = -b\—\ a n d a-n = ; <p\-\m/ m \ml Z J \m v = 0 Tm-\ , . I Tm-\ 1 m L— \m/ \m l—i \m •2mhf{vlm) m 1

|aoO'0 + ü!l('Tl~fJo) + - • •+«7'm-l(o-T?n-l—<7Tm-2)

= - |ö-o(ao—ai)+ffi(ai — ^ 2 ) + - • •+CTTm-iaTm-i| <

m

1

< — (|cro|(ao — a i ) + - . •+|ffTom-2|(aTom-2 —flTom-l)) +

ffZ

< [bo{ao—ai) + {bo + bi){ai — a2)+. . . + (éo + . • .+èTom-2)(«rom-2 —«Tom-i)] + + 06- [(*() + • • • + ^T„m-l)(flTom-l —aT(,m)+. . . + (éo + - • • + iTm-l)fl7'OT-l], (15)

8

according to (12) and (11) respectively.

Now we may also write, observing that the mixed terms in (15) vanish,

^Z'*Ö'-'"""|<Z«A + ^ 2 « A <

v = 0

< 2 ^ - / ' * ( ^ + 8 ^ - ( / ' * ( ^ ) +\^) < ^ ^ 7 ' * ( 7 ^ ) , (16) where the last two inequalities follow from (13), (14) and (15) respectively.

From (16) together with (10) we finally have 1 3

/ b*[t)e''''''''^'^dt\ <-e + ^F.-/i*{T) <R-fi*{T),

0 4

which proves the theorem.

Special case.

If we write ^*(/) =f{M*{t)), where M*{t) is the inverse of/?*(<), then appli-cation of theorem 7 with b{t) ^^ 1 together with theorem 3 gives

T H E O R E M 8. Let the function b*{t) be decreasing and let b*{t) satisfy property {A). Then, if the function ƒ(<), continuous for < > 0, is C-u.d., the function g*{t) is also C-u.d.

1.4.2. Applications. We apply theorem 8.

9. If the continuous function ƒ(<) is C-u.d. then alsoƒ(«*) is C-u.d.

P r o o f . We choose b{t) = !/<(/ > 1). Then M{t) = e^ and ^(0 =f[M{t) =

= f{e').

10. If the continuous function ƒ(<) is C-u.d. then also/(^") is C-u.d. (a > 1). P r o o f . Here b[t) = - /<'-""", so i,[t) = t"" and t = M[u) = u".

a

Remark. L. KUIPERS proved ([3]) that for n > \ [n integer) the function

n

F' sin 2ntis C-u.d. If we choose a = n and a = respectively then we directly n—1

deduce that also the functions t"'sin27it'' and <"'"""" sin 27rf''""" are C-u.d. (re > 1, ?2 an integer).

11. If b{t) satisfies property {A), if b{t) is decreasing and if b{t) ^> c > 0 if 1 1

t ~> oo [c constant), then M'{u) = , -^ - if u ^ oo. Hence, under these b{t) c

conditions on b{t): if the continuous function ƒ(<) is C-u.d. t h e n / ( A / ( / ) ) is C-u.d. if M'[t) -^ fi? > 0 for M ^> oo [d constant). (In [3] L. KUIPERS proved more generally that if M'[t) -^ d > 0 for / ^ oo then f {M{t)) is C-u.d. if

fit) is).

1.5. T h e c o n t i n u o u s analogue of a generalization of v a n der Corput's

t h e o r e m

1.5.1. We will prove a continuous analogue of a generalization of VAN DER CORPUT'S fundamental theorem from the „discrete" theory as is given by M. TSUJI ([8]).

Before formulating this theorem we will give the definition of some class of functions b{t) and afterwards we will prove two lemmas necessary for the proof of the theorem.

D e f i n i t i o n . A nonnegative function b{t) defined for / > 0 satisfies prop-erty (B) if

(i) b{t) satisfies property {A) (with to = 0) (ii) b{t) ^ 0

(iii) b{t) is decreasing for / > 0

(iv) the function is decreasing in t for t > 0 and for k = 1,2,. . .

b{t-\-k)

We now prove

L E M M A 3. If the function h{t) satisfies property (B) then

bit)

lim T7^r, = l f o r ^ = 1,2,... r->co b{t-\-k)

b{t) *„' b{t+i)

Proof. S i n c e - - — = TT 7- -.——, the proof is delivered when the

prop-b{t+k) /lob{t+i+l)' ^ ^ ^

erty is proved for k = 1. This last proof runs as follows: assume that

b{n) . . . bin)

lim = a > 1 (n a positive integer), then > a, so that

„_>o„ b{n+l) bin+l)

1 1 1 1

é ( « + l ) < - bin), and thus b{n+l) < - b{n) < — è ( n - l ) < . . . < — ^i(l).

a a a^ a"

" " 1 " 1

Hence ƒ é ( i ' ) < é ( l ) - N — < è ( l ) - N ~, which contradicts the fact

r = l r=l y=\

<= " n i l

(We remark that in the case lim bit) = a > 0 the validity of the

prop-( - J > OJ

erty is obvious; we also draw attention to the fact that the condition //(i) -^ oo if ^-^ oo used in the given proof is essential, for if we take A(/) = e^^ then — = e^, this being a decreasing function in t which does not tend to 1 if

bit^k)

t-> oo. Indeed here //(/) = 1—«^', which does not tend to infinity if < ^- oo).

We also have

L E M M A 4. If the function è(i) satisfies property (5) then lim — — — — = 1 f o r / t = 1,2,. . . ^ ^ " 1''^^'^ P r o o f . As in lemma 3 we may take /; = 1. (TA-W

Then we deduce the property from the identity

/t(r) + (//(r+i)-,»(r)) A/.(r+i) . ^'^^^^_^,,

= 1 A , where A / ( ( y + l ) =

I'iT) /liT) ' ' ^ -^ '

= /iiT-\-l)—/iiT); and from the fact that lim ——— = 0 because

r I 1 r - > a , / ' ( J )

A/f(7"-|-l) = I bit)dt is bounded since bit) is decreasing.

We now reach the main purpose of this paragraph. We prove

T H E O R E M 9. If the function bit) satisfies property (5) and if the con-tinuous function ƒ(<) has the property that the function ƒ ( / + /)—ƒ(<) is B-u.d. for / = 1,2,. . .; then ƒ (0 is S-u.d.

P r o o f . For 7" > 0 we define a function uit) for all real t by I 0 for < < 0 and i! > T

V ' I ^2n.i,m forO <t< T Now the proof runs in different steps.

(i) Let q he an arbitrary positive integer, then we have according to the definition of «(/) for Q = 0,1,. . ., r/—1

T]q I T\-,l-l-Q T I bia — Q)uia — Q)da = / bia)uia)da = ƒ bia)uia)da,

() - e () and so

T+q-\ 7 ^ ' T

I ^ bia-'Q)uia—Q)da = q j bia)uia)da. ill)

•I r~o ° (ii) From SGHWARZ'S inequality we deduce

/ \^bi<^-Q)ui'^-Q) ' o so that r i r / - l ? - ' da Thq-\ , ' - 1 < /•é(c7)rfcT- /• - ^ y bia~Q)uia~e) I y bia — Q)uia~Q)dc (>=0 < / . ( r + < ? - i ) . / r + ? ~ ' ] I ' ' ^

«' H-)l„^

(iii) From (17) and (18) and from the definition of «(^ it follows

da, da. (18) / bia)uia)da

T+q-< I

7^

^'^''""^"'^'^

, y • ^(T -F

ï(ff)'

2 J 2 J

/"^•yy-^^+2. r^-yy-^^-f+

_{é(ff) z ^ ^}

(iv) We calculate

7 - + ? - i . ï - i

/ = / TT-^-y ibio-Q)uia-Q)-bia-^Q)üia-q))

, ba] z—i da

and estimate this integral.

For p = 0, 1,. . ., q—l we have from the definition of «(i)

7 - + ? - ! 1 T+q-\-a 1 T

, 1 i + . - . - . ' I ^ 32(^) ' ATT b^ia~Q)uia~Q)uia-Q)da = , - , , b'^ia)uia)uia)da = T ^ T ^ N ^'^'

'•'-^ -l b{a + Q) 0- é(<T+e) 0 bia)

from which we see

/ • - / •

1 1

,' \bia) bia+l)

+

...-1

bia + q~r •b^ia)da.

It follows from lemma 3 that for arbitrary e > 0 a number aoie,q) exists such that for CT > ao

so t h a t , if 7" > CTo

a„ T a„

[< [ + q{l+e).fbia)da < f + qil+e)f,iT).

1 d a„ (')

1 "."

Hence, if we choose Ti > ffo such that j < e for 7" > Ti, we have for

T > Ti / ' ( - ' ) Ö

J - / • < . + 9 ( l + £ ) . (20)

(v) We calculate

/ = / jTf.- 2_,l^ibi'^^e)<'^-Q)-bi(y-v)aia-v))da r> p

and also estimate this integral.

If we first introduce a new integration-variable t for fixed v by setting

a—V = t and then for fixed Q we introduce a new summation-variable / by

putting a—Q = t^l (so that / = V—Q is a positive integer) we obtain, because of the definition of uit)

T+q-i J

/ j----bia~Q)uia—Q) •bia — v)üia—v)da = Ö «"W Tl-q-l->' , = /' . , , , M , bit+l)bit)uit + l)üit)dt = -i bit + l+g)

= f uJfA. ,bit+l)bit)e"'"-in'''^~mdt,

/ Z. Z. ,/ é(.+/-K,)

Now, since is decreasing and since the function ƒ (<+/)—ƒ (0 i^

bit+l+g)

7?-u.d. it follows from theorem 7 that for a number £ > 0 a number 7"'(£) exists such that for T > T'

T-l

bit + l) T-l bit+l) bil)

I Kt) J ? ^ - . ,2«7,{/(M./)-/(„}^^ <e. ibit) "'^ ' ^ dt < £ - 7 7 7 ^ . -lAT). I ^^ bit+l + g) J "•' bit + l+g) bil+g)'^ '

Thus we may also say that for e > 0 a number T2is,q) exists such that for T> T2

IJiiT) < e-^(?), (21)

where Aiq) is some constant depending on q. Finally we deduce from (19), (20) and (21) that

T

^t[T+q-l)./iiT) bia)uia)da <E + qi\+E)+2EAiq) if T > max(7"i,71j).

From this and from lemma 4 it follows for each fixed positive integer q T

-r- 1

lim —-—-r->a> fi^iT)

bit)uit)dt

1 lim

7-_>oo A((7^)ó

\bit)e^'"''l'^'^dt = 0.

This proves theorem 9.

We see, if bit) satisfies property ( 5 ) :

if the f u n c t i o n / ( A f ( 0 + / ) — ƒ ( M ( 0 ) ( / ( 0 continuous) is C-u.d. (/ = 1,2,. . .) then by theorem 3 fit+l)—fit) is B-u.d. (/ = 1,2,. . .) so that, by theorem 9,

fit) is jB-u.d., and we obtain

T H E O R E M 10. Let the function bit) satisfy property ( 5 ) . If the continuous functionƒ(<) has the property t h a t / ( M ( < ) + / ) —/(Af(/)) is C-u.d. f o r / = 1,2,..., then the function/(Af (<)) is C-u.d.

1.5.2. Applications

12. If bit) satisfies property (5) and if P'-'\M{t)) y UjiMit))! («J real) is a polynomial in Mit) from degree riur 7^ 0) (r > 1) and if at least one coefficient aj ^ 0 ( j > 0) then the function /''''(M(i)) is C-u.d.

P r o o f For r = l 7""(M(/!)) = ao + aiA7(/) (with «i 7^ 0). Since M'iu) is monotone because hit) is so (property (5)) and since uM'iu)

bit)!

= -> 00 if M -^ 00 we see from theorem 4 that P*"(M(i)) is C-u.d. Now \ bit)]

we assume that the assertion is true for r = s and we give the proof for r = s+\ with the help of theorem 10.

s + l

P^'+^\Mit)+l)-P^'+^\Mit)) = yaj{iMit)+l)i-iMit))l} = a polynomial in Mit) from degree s with the leading coefficient /• J- «,+,, 7^ 0, which is C-.u.d. for/ = 1,2,. . . Hence from theorem 10 the polynomial P ' " ''(A7(i!)) is C-u.d.

Remark. We draw attention to the fact that we could also have proved

example 12 for Mit) ^^t and then could have applied theorem 8 (observe that the conditions on bit) in theorem 10 are more stringent than in theorem 8.) We shall now give an example where application of theorem 8 is not possible. 13. If 'F(M) is a continuous function defined on — 1 < M < 1 then the function te'+Wisin27ie*) is C-u.d.

Proof, outfit) =tlogt+ f'(sin27T«) and Mit) = «« (è(0 = Ijt, t > 1). Then / ( M ( 0 ) = te^+Wisin 2nei) and / ( M ( i ) + / ) - / ( M ( / ) ) = logfl + -^V* +

-l-/log(«* + /). ^ "

We make use of the following lemma. L E M M A 5 (L. KUIPERS [3]).

I f / ( ; ) is C-u.d. and if the function F ( 0 satisfies Hm if it)—Fit)) = 0 then

Fit) is C-u.d. ' ^ "

Here we have ( / ( A f ( < ) - h / ) - / ( M ( / ) ) ) - / ( l + 0 -> O i f / ^ 00 ( / = 1 , 2 , . . . ) so that we may conclude from lemma 5 t h a t / ( M ( < ) + / ) —ƒ(A^(0) is C-u.d. for / = 1,2,... and hence by theorem 10:/(A/(i)) = fe« + f (sin 2ne^) is C-u.d.

CHAPTER II

U N I F O R M D I S T R I B U T I O N O F S E Q U E N C E S O F N U M B E R S

2.1. Definition a n d criteria

In 1.1.1 we gave the definition of a 5-uniformly distributed function ƒ (/). If in the expression

1 "

— - - / bit)di{fit)}; a,fi)dt, /'(A"),/

N a positive integer, we replace ƒ (i) by ƒ([<]), where [t] = t—{t} = n, we obtain

n = N

I bit)Oi{fin)}; a,ft)dt = ^ ^ ^«0({/(«)}; a,ft), if

/l« =//(«)—//(re—1) (re = 1,2,...) so that//.(A) = ) /l„ 1. « = A'

n = l

This last mentioned expression suggests the following definition of uniform distribution of sequences of numbers:

D e f i n i t i o n . Let the function bit) he defined and nonnegative for t > 0 and let bit) he integrable on each interval 0 < / < re (re integer, > 0 ) ; let //(re) =

n

= / bit)dt in = 1,2,...) and let X„ = /^,(re)—//(re —1) (re = 1,2,...); let the func-d

tion ƒ(/) be defined and integrable on each interval 0 < i < re (re integer, > 0 ) ; then the sequence/(re) (re = 1,2,...) is called ^„-uniformly distributed modulo 1

iXn-n.d.) in = 1,2,...) if the function/([<]) is B-u.d..

More explicitely: the sequence ƒ (re) (re = 1,2,...) is Xn-u.d. if for each inter-val [«,/?) C [0,1]

n = N

Ar_^„ //(TV) L-i

n— 1

In further replacing/(/) by ƒ([/]) in 1.1 we directly obtain

T H E O R E M 11. T h e sequence/(re) (re = 1,2,...) is A„-u.d. if and only if for each function w from C[0,1]

lim -YTf^y ^nwiifin)]) = \ wiu)du;

T H E O R E M 12 ( W E Y L ' S criterion).

The sequence/(re) (re = 1,2,...) is Xn-n.d. if a n d only if

n - .V lim

. v ^ » /((A)

y Xnc'^-'f'"^ = 0, A = 1,2,...

Special case.

If//(re) s^ re and thus A^ = 1 we obtain the ordinary case of uniform distribution modulo 1 of a sequence of numbers. In our notation we write in this case 1-u.d. 2.2. A relation b e t w e e n u n i f o r m l y d i s t r i b u t e d functions a n d

uni-f o r m l y d i s t r i b u t e d s e q u e n c e s

2.2.1. First we give a generalization of EULER'S summation formula.

L E M M A 6. Let the f u n c t i o n / ( / ) be differentiable for / > 0; l e t / ' ( / ) be integrable on the interval [0,7"], 7" a positive integer; let bit) be nonnegative and continuous for < > 0 and let ƒ*(<) = /<(/)-/<([<]); then

n — / y

y-y A,/™''""' = 2mh \Pit)fit)e"''''''^'^dt + \ bit)e"'"'f^'^dt. ih = 1,2,...)

Proof. For an integer 7" > 0 we have

il<.i[t])de''"""' = V^,(n)(r'-^*/-(«H,_^-2.,7,/(„,) _ _ V 2„,^-V('"+^,( T)/-"•"ƒ<^'.

Further

l=T T

j/i.it)de"'"'^"^ = - \bit)e"''''f^''dt+iiiT)e'""'i''^'^\

Subtraction of these two equalities gives the required result. We now have

T H E O R E M 13. Let/(<) have a derivative for t > 0, for each positive in-teger r integrable in [0,7"]; l e t / ( 0 be B-u.d. and let

1 ?^

/ Pi^)f'it)dt-^ Oif 7"-^ oo; then the sequence/(re) (re = 1,2,...) is

l'iT)ó

An-U.d.

Proof. We only remark that the assertion follows directly if wc apply lemma 6. 2.2.2. Applications

14. From example 3 (paragraph 1.2.2) we know that the function a(log t)" is

t"-u.d. it> 1) (a > 0, a > 1, ^ > - 1 ) .

Here In = //(re)—//(re—1) = (re'* ^ ' —(re—1)'' ' ) . We want to show that also the sequence «(log re)" is (re'*' ' — (re—l)'*+')-u.d. (re = 1,2,...).

We have to estimate / Pit)f'it)dt. It holds Pit) < /((<)—//(<—1) < | é ( / ) i f / 3 > 0 '"

< by the mean-value-theorem of the integral cal-I bit-l) i f - 1 < / 3 < 0

cuius. Hence for £ > 0, if we choose a number 7o(£) such that for

I T 1 ':» t > To: fit) < £, wc see - - — / Pit)f'it)dt <-— j +

E ] / / ( r ) - / * ( i ) i f / ? > o . /,{T-\) , .^

H —- • , so that, since here — > 1 if

/./(T) I / / ( r - l ) i f - l < ^ < 0 niT) T

T-^ oo, lim ƒ Pit)f'it)dt < £ which completes the proof".

In the same way one proves (compare with the examples 4, 5 and 6 li-om 1.2.2) 1

15. The sequence «(log re)" is log 1 -| -u.d. (re = 2,3,...) if a > 0, a > 0; „ \ re—1/

16. The sequence (log re)" is /" (log /)''a'^u.d. (re = 2,3,...) if fi > 0, a~/J > 1;

n ' - l

17. The sequence re" is ( r e ' ' ^ ' - ( r e - l ) " ^')-u.d. (re = 1,2,...) iff)<a< 1,

^ > - l . ^+^

2.3. N o t u n i f o r m l y d i s t r i b u t e d f u n c t i o n s giving rise to not

uni-f o r m l y d i s t r i b u t e d s e q u e n c e s

2.3.1. In the following we will write A//(0 = /((O—//(/—1). We first prove

L E M M A 7. If the integrable f u n c t i o n / ( O is not B-u.d.; if J-^ ^ 0 if

1 T /'it) t - > oo; then the expression / bit)e^'""^'^dt = y(7") does not tend to zero

/ ' ( ^ ) d

if T runs through the positive integers.

P r o o f . We observe that if———> 0,< -> oo then also > 0,t^- oo. ^ ^ ^/^it+l) . r „ / ' ( < + ! ) ''^^^ For from — ——^ 0,i -> oo it follows ——; ^ 1,/ —> oo and consequently

flit +1) nit)

//(<+i)-//(0

Since the function ipiT) does not tend to zero if 7" (continuously) tends to infinity the set of numbers y>iT) has at least one point of accumulation I 7^ 0 and thus a sequence 7"i,7"2,... exists with T/c^ 00 if A: ^^ 00, for which

fiTic) ^ 1 if k-^ 00. Now it can be seen that also the sequence ifilTk]) ik = 1,2,...) tends to | if A; -» 00. For WiTk)~wi[Tk])\

f +

which tends to zero if k -^ 00.

^g/'(7".)-//([7".]) ^

Now we have

T H E O R E M 14. Let the function fit) have a derivative for each positive integer 7" being integrable in the interval [0,7"]; let the function bit) satisfy

A//(0

property (^4) and let

flit) 0 if < ^- 00; let f u r t h e r / ( / ) not be B-u.d. and

let ug'iu) be bounded; then the sequence/(re) is not /l,ru.d. (re = 1,2,...).

bit)

P r o o f . From \ug'iu)\ < K (constant) we see that | / ' ( 0 < ^ so that

flit)

jPit)f'it)dt<Kjbit)^dt.

» d I'y'-)

A/t(0

Now, if we assume that for t > 7"()(£): < e, we obtain

flit)

T T„

\ Pit)f'it)dt< K \ + K.E-ifiiT)-^fiiTo)), d ()

from which it follows

lim - - - Pit)f'it)dt = 0,

r - > » / ' ( ^ ) , j

and the proof of theorem 14 is completed by the lemmas 6 (for A = 1) and 7. 2.3.2. Applications

As in examples 14 and 16 one shows 30

18. For a > 0 , 0 < a < 1, / ? > —1 the sequence «(log re)" is not (re'* + ' - ( r e - l ) ' * + ' ) - u . d . (re= 1,2,...);

/ ? + l

19. For a > 0 , 0 < a < l , / ? > 0 the sequence «(log re)" is not

n

j ilog t)''dt-u.d. in = 2,3,...).

n - l

2.4. A n a n a l o g u e for the d i s c r e t e c a s e o f a t h e o r e m f r o m the

c o n t i n u o u s theory

2.4.1. We shall prove an analogue of theorem 3 for the case where we are dealing with sequences of numbers.

T H E O R E M 15. Let the f u n c t i o n / ( / ) be difTerentiable for t > 0 and let

fit) he integrable in the interval [0,A] (A an arbitrary positive integer); let

the function bit) satisfy property iA); let further for / > 0 (i) | M ^ ' ( " ) | < Ku'' iK and v constants),

(ii) A//(<) < Lu" (7 and g constans), where p < 1, j ' < 1 and g+v < 1;

then the sequence/(re) is An-u.d. if and only if the sequence ^(re) is 1-u.d. (re = 1,2,...).

We remark that, as usual, u = //(O, i = A/(M) (M(re) is the inverse of//(<)),

giu) =fiMiu)) and A//(0 = f,[t)-fiit-\).

P r o o f of theorem 15. If 7" > 1 then the following identity in 7" holds (see lemma 6)

y An^'"*^'"' = j'bit)e-""-^^yt+2mk />(/)/'(Or'"'''^*''^/. (A = 1,2,...) (22) If we put flit) = re we obtain from (22), s i n c e / ' ( < ) = g'iu) -bit) (so that

fit)dt=g'iu)du)

"=IJJ /'(tn) Mm)

y 2,/"'''""' = /'«2-'««rfre + 27zfA f {M-//([M(M)])}^'(re)r'-''^f''Vre. (23)

If in (22) we take bit) = 1 and replace [7"] by [/<(7')] then application of (22) to the function giu) gives

[''(I")] [,,(7-)] [//(T)l

By subtracting (23) and (24) we obtain

„ = [7-] lum] [„(T)] MT)]

\ ;i„g2m7,/w _ \g2.ii,,w _ _ I'/''''•«'''^du + 2mh j {[u]-f,i[Miu)])}-g'iu)e-''""'"^du

hi «=i "ffn) d

-2mh f{u-fii[Miu)])}-g'iu)e"'""''"du = h+h-h. (25)

In order to obtain an estimation for the right hand member of (25) we observe that

ir iT^^ nT^)l ^ 1 ^I'iT) i^^lT]) < [fiT)] ' ^ ^ ' ^ ^ ^ ^ " ^ ' ^ ^ " ^ ^ ^ ' " l 1 i f / . ( [ r ] ) > [ / , ( r ) ] ' so that always

| [ / / ( r ) ] - / / ( [ r ] ) i < A//(r) + i,

and it follows from condition (ii) and since g < 1 that ^ 0 if 7" (continuously) -> oo. Further we see

|[re]-//([Ar(«)])| = lre-//([M(re)]) + [re]-«| < P ( 0 + 1 < A//(0 + l < L-u'+\ so that, from (i) and (ii) (and if we suppose [//(7")] > 1)

l - i i < C +\ L.ti^-'.\ug'iu)\du + \\g'iu)\du< Znri (0.1) f i

< C +

(It holds C < / \g'iu)\du = I \f'it)\dt; this integral is assumed to exist).

(0,1) li ' (j

Hence, since v < 1 and since g+v < 1

^ 0 if 7" (continuously) -> oo.

L/nT")]

In the same way one sees, since M —//,([M(re)]) = Pit) < A//(<) (and if we suppose //([7"]) > 1)

' - ^ < lK-L-u<'+"^^du, 2nh I

from which we have 1/21

> 0 if 7" (continuously) -> 00.

mf]

Hence we obtain from (25) that

,« = [7"] b,(T)]

^ Xnc^""-'''"^ - y e^"''"'"^) ^ 0 if r(continuously) -> 00.

[/'(^)]Vé^ ^

(A = 1 , 2 , . . . ) (26) Now we first suppose that the sequence ^(re) is 1-u.d. (re — 1,2,...); then (A integer) n = N Hm - y e2™v«(„) ^ Q ^^ _ J ^2,. ; ^ c o A Z J '^ • - „^1 or also

- — = - y e^"''"''"' -^ 0 if ^(continuously) ~> co.

1 1 = I

Then it follows from (26)

- — — y A^e'"*''"' ^ 0 if r(continuously) -> 00. (27)

win] ^

From condition (ii) we see

/ / ( [ r ] ) > / / ( T - l ) > //(7") ( l - Z - ( / / ( 7 " ) ) ^ ' - ' ) ( w i t h e < 1), so that (if we choose 7"so large that 1 L- (/t(7"))-^' > 0)

< L y 3,2.i7,/(„) < ^ L . V

//([r])| Z J " r i - z . ( / / ( r ) ) ' - ' /.(7")|Z.'

[/'(T")] If-^

and we deduce from (28) and (27) that (A''integer)

ji = A'

.V

Hm - ^ ^ y A„r™''^'"' = 0 (A = 1,2,..

fiiN) -\

Now, conversely, we assume that the sequence/(re) is /l„-u.d. (re = 1,2,...); then iN integer)

ii = A'

lim ^ y ^«^""'''"'= 0 (A =1,2,...),

or also " = [7-] "^^ /l„«^'"*''<"' ^ 0 if r(continuously) -> oo.

MIT]) A.

HIT])

U'iT)]

y A„«2™-''/'"'

[//(r)] + i | Z .

Now from (26) it follows that (A''integer)

.I = J V

lim

n = I

y ^2.i7«(„) ^ g ^^ _ J ^2,...)

n = I

which finally proves the theorem.

In connection with the conditions (i) and (ii) in theorem 15 we call attention to the following example.

log re / 1 \

As we know the sequence is log 1 H -u.d. (« > 1) (re = 2,3,...) log« \ re — 1' , ,

. log t

(see example 15 paragraph 2.2.2 with « = 1); here/(<) = ;; and u = /((<) = log a

= log t it + \) so that 5 = 0; further v = 1 since giu) =

Thus here we have the case g + v = \. °

n . 1 .

Indeed, the sequence ^(re) = (re = 1,2,...) is not 1-u.d. if is , log a log a

rational. ° " 2.4.2. Application

20. We know (example 16 with /? = 1) that the sequence (log re)" is (re log ^ ^ - F l o g ( r e - l ) - l j - u . d . (a > 2) (re = 2,3,...).

t

H e r e / / ( 0 = i ( l o g i — 1 ) and A/i(0 = Hog + log(i!—1) —1 < L - ( / / ( / ) ) ' ' Further ug'iu) = ' ^ / ' ( O = « ( l - i ^ ) - ( l o g 0 " ^ ' < ^-(/'(O)"'* say (so we have g = v = 1/4).

Hence, if we denote the inverse function of f i it) = /(log t—l) by *f (re), the sequence ^(re) = (log('7'(re)))" is 1-u.d., a > 2, (re = 2,3,...).

C H A P T E R III

A N O T H E R P R O O F O F WEYL'S C R I T E R I O N

3.1. Let the function «(/,7") be nonnegative, let «(/, 7") be integrable with

T

respect to t in each interval 0 < / < 7"(7"> 0) and let ƒ ait,T)dt = 1 for

d

7 " > 0; let further ö(M;a,/9) be the characteristic function of the interval

0<a<u<p< 1 and let the function fit) be integrable in 0 < / < 7";

then we define for 0 < x < 1

T

Frix) = l'ait,T)0i{fit)};O,x)dt. (29)

We shall prove that the function Frix) as defined above is right-continuous in X for each 0 < A; < 1.

P r o o f . If A is some positive number and if x > 0, x + h < 1 then we have to show that for each (fixed) 7" > 0

lim iFTix + h)-FTix)) = 0.

/ i - ^ O

T o this end we observe that

T

iFTix+h)-FTix)) = jait,T)6i{fit)};x,x+h)dt = j'ait,T)dt,

d . A-//,

if Mn denotes the set of numbers t from [0,7"] for which x < {fit)} < x+ljh

iMh is measurable).

However, since for each (fixed) t:x < {fit)} < x+h cannot be satisfied for A small enough it follows that lim Mu = 0 .

/ i - > 0

Now, if we define for ^ > 0 (and fixed 7" > 0) ( « ( / , 7 " ) i f « ( / , r ) < y t

akit,T) =

\k ifait,T) > k

then for given e > 0 and each fixed k a number ó(£,^) > 0 exists such that for A < Ó (/< = measure)

l'akit,T)dt<k-fiiMn) <'h;

Ml,

T

I iait,T)-akit,T))dt < [ iait,T)-akit,T))dt < 'jo,

Ai,, 6 so that for each fixed k > Kis), if A < f5(e,A;)

I ait,T)dt < £.

Al,,

This completes the proof.

Thus, summarizing, Frix) has the properties (i) Frix) is non-decreasing in x

(ii) FriO) = O a n d F T ( l ) = 1

(iii) Frix) is right-continuous in x for 0 < x < 1.

Because of these properties we call FT(X) a distribution function. Obviously/(/) is ^-u.d. if and only if F T -> F ( 7 " ^ oo), where

Fix) = X (0 < X < 1). (30)

Belonging to the distribution function FT the characteristic function '/-r (in the sense of probability theory) is for each real u defined by

1

Xriu) = l'e""dFTix). (31) d

O n e easily calculates for the characteristic function x of the distribution function F with F(x) = x (0 < x < 1)

( e'"—\

} . if re 7^ 0

Ziu) = I e'"'dx = ' lu . (32)

" [ 1 if re = 0 W e proceed in proving the following theorem.

T H E O R E M 16. IfXf is the characteristic function belonging to the distribu-tion funcdistribu-tion FT then

r

Xriu) = j ait,T)e""<''"'}dt. (33)

(I

P r o o f . We prove (33) for the real part, the proof of the imaginary part being similar.

For this purpose we divide the interval [0,1] into 5 equal parts. A sub-interval with endpoints a and ji will be denoted by a. We define

M„ia,P) = upper-bound of COS(MX) if X is in a; iV„(a,(3) = lower bound of cos(rex) if x is in a.

Obviously it holds

r r S N^ia,p) I'ait,T)di{fit)}; a,fi)dt < j'ait,T) cos iu{fit)})dt <

" d d T

< 2 M„(a,^) f «(/,r)0({/(/)}; /.,/?)<//,

2 N^ia,P)iFTiP)-FTia)) < /'«(/,T) cos (re{/(0})6?/ <

o (')

< S M „ ( a , / J ) ( 7 ' T ( ^ ) - F T ( « ) ) .

CT

Now, if s ->- oo, the left-hand side and the right-hand side of this inequality 1

tend to / cosiux)dpTix) so that d

T I

/ «(/,7") cos iu{fit)})dt = j cos (rex)(/FT(x),

d o" which proves the theorem.

In order to prove necessity of W E Y L ' S criterion we will also make use of the well-known continuity theorem ([7]), where we replace the discrete para-meter re by a continuously varying parapara-meter 7" (this is permissable since vergence in a continuous varying parameter 7"-^ oo is equivalent to the con-vergence in every sequence 7"^-> oo, re = 1,2,..., and since a distribution-function is uniquely determined by the corresponding characteristic distribution-function) and we have

T H E O R E M 17. Let FT be a family of distribution functions (7" > 0), let IT he the family of corresponding characteristic functions. If F T ^ F( 7" ^ oo) then XT -^ ^ ( 7 " ^ oo), where X is the characteristic function corresponding to the distribution function F.

Conversely, if ^ T ^ X(7"-> oo) and if X is continuous at re = 0, then FT converges to some distribution function F ( F T -^ F, T ^^ oo) where X is the characteristic function of F.

For the proof of theorem 17 we refer to [7].

Now from theorem 16, from the first assertion of theorem 17 and from (30) and (32) we deduce: i f / ( 0 is ^ - u . d . then

lim j ait,T)e'-'{''f'ndt = xiu) e'"-l

—.— if re 7^ O tu

1 if re = O

However, since %(2JTA) = 0(A = 1,2,...), we may conclude (necessity of W E Y L ' S criterion): i f / ( O is ^-u.d. then

T

l i m j'ait,T)e""'-^^'^dt = 0 (A = 1 , 2 , . . . ) .

3.2. Before proving sufficiency of W E Y L ' S criterion we remark that for this purpose the converse from theorem 17 cannot be applied, for, if one knows from W E Y L ' S criterion that

<e'"-\

lim j'ait,T)e"'if"^}dt = xi»)

if re 7^ 0

lU

1 if re = 0

holds for re = 27ih ih = 1,2,...), this equality is not proved to hold for all real a.

Therefore we shall have to prove sufficiency fo W E Y L ' S criterion via another way.

T o this end we first prove

T H E O R E M 18. If lim Frix) = x (0 < x < 1) then this convergence is uniform in x. ^^"^

P r o o f . Choose 0 < £ < 1, r = [3IE] + \ and ro(£) such that for T > To and for i = \,...,r

i\ I

F T \ - \ - _< 1

(34)

Since r > 3/e we have 1/r < £/3 < e so that l/r^ < £2 < g and with (34) we also have FT 1 < - , < £ . y2 (35) \rl r

Now suppose 0 < X < 1 and put R = [rx] (thus R integer, > 0), then

rx—1 < R <rx and R < r-l, (36) so that

/ R\ I R I 1

FT ( - < pTix) < FT ( ^ ^ (37)

From (34) it follows

-l/r2 <FTfn - J < l/»-' (38)

and hence from (37), (38) and (36)

FTix) < ^ ^ + i < iR+\) ( + M < (rx+1) (^ +

-r -r^ \-r -r^j \-r -r^'

= x+ljr+xjr+ljr^ < x+ljr+llr+ljr = x-(-3/r < x + £. In the same way it follows from (36), (37) and (38), if R > \

FTix) > ^ ^ ( 7 ) > 7 - ^2 > ^ (7 - . y > t - l^ ^''-'^ =

= X—1/r—x/r+l/r^ > x — Sjr > x —£,

while, in the case R = 0, we have from (36) x < 1/r from which one sees X —3/r < 0 and x —e < 0 so that, in this case, we trivially have FT(X) >

> 0 > X —£.

Thus, if 0 < X < 1 and T > Fo(£), we proved

x - e <FTix) < x + E, (39) this relation, however, being also satisfied if x = 1 since F T ( 1 ) = 1.

This proves the theorem. We now define

D e f i n i t i o n . The expression

DT = sup | F T ( X ) — x |

0 E ; * S ; 1

is called the discrepancy of the function/(/). Hence we conclude from theorem 18:

T H E O R E M 19. The function/(/) is ^-u.d. if and only if Hm DT = 0. r ^ . c o

T h e way we follow in order to prove sufficiency of W E Y L ' S criterion runs via an estimation formula for the discrepancy DT. We prove

T H E O R E M 20. The discrepancy DT of the function/(/) satisfies

T

/ , » I/•«(/, r).^-"^<'V/|2\v.

\ A = l

P r o o f . Consider the function/(i) for 0 < / < T. T h e function FT(X) defined for 0 < X < 1 is periodically extended (period 1) to the function F T * (x). T h e n

the Fourier series of F T * ( X ) converges since FT(x) (0 < x < 1) is of bounded total fluctuation and at the continuity points of F T * (x) holds

FT*ix) = y Cne'""'\

/ i = — oo where

Cn = \ e-''^"-FTix)dx ih = 0, ± 1, ± 2 , . . . ) .

We calculate the Fourier coefficients Cu by integration by parts in (31). This gives Xriu) = e'"-iu j «""FT(X)//X, so that C\ \-Xri-2nh) . -2mh if A ^ 0 \FTix)dx ifA = 0

Consequently, at the continuity points of F T * (x)

Since

also the series

1 « - • »

FT*ix) = \FTix)dx + y

= - I , ^-+^ = ^ - V 2 (0 < X < 1), /i=-tx) • h^-\ ( A * 0 ) vn XTi-2nh)e-'2.mh.\ /l= —CO {/, :• 0 ) 2mh

is convergent for 0 < x < 1 and at the continuity points of Fr(x) we have

V

/ i = - c o 2mh

Grix) = Frix) —x; g^ = j GTix)dx,

then, at the continuity points of FT(X)

A = <.

GTix)-gj, = 2_^

,2m/ix

(/i = 0)

so that, by PARSEVAL'S equality

I /i = oo I

,- Y~i X-pi — 27rA)

j iGTix)—g^)^dx = / J

since ^Ti—u) = ^T(re). Hence we obtain by (33)

( G T ( X ) - ^ ^ ) 2 Ö ' X =

1 " _{/ T ( 2 7 I A ) | 2} 1 V ^ •

'«(«,r)«^--"''^"V/|2

(40)

Now, in order to estimate ƒ (GT(X) —^j,)V.r, we observe from the definition

Ó

of DT that at least one of the regions lying between the x-axis and the graph of Grix) contains a right isosceles triangle of side DT with base, say, the interval

Xo < X < XO + DT (see figure 1, where we assumed, without restriction, that GT(X) > 0 if XO < X < XO + DT).

As one sees, the graph of GTix) lies in a region bounded by the sides (of length 1 and 2DT respectively) of a rectangle and the hypothenusas of two right isosceles triangles of side DT and one deduces from the figure

/'(GT(x)-^y)2(/x > fx^dx + \ xHx, (41)

d Ó d

the two surfaces on the right hand side of (41) being indicated in the figure. Thus

1

f iGTix)-gTy'dx > g^^Dr-g^D^T+^lsD^T = / ( ^ T ) -d

Now, since/(^j,) > / ( I / 2 7 ) T ) = ^IIZD^T we have

\iGTix)-g^)^dx>^li2D^T. (42)

d

Finally, combining (42) and (40) we obtain the desired result: r

/"«(<, r)e^'"''-^"'rfi|2y/3

D

\7r2 Z_j A2 /

h=\ I

From this inequality sufficiency of W E Y L ' S criterion directly follows if one observes that summation and taking the limit T-^ oo may be changed since the series is uniformly convergent which follows from WEIERSTRASS'S test.

Remark. We remark that, since | G T ( X ) | < DT and | ^ j , | < DT we also have

1 iiGTix)-g^)'-dx < 4 7 ) 2 T d SO t h a t T

47)2^ > — y -^

which implies, since the last series consists of positive terms, that also r

lim I ait,T)e"'"'^''-yt = 0 (A = 1,2,...) this again proving necessity of W E Y L ' S criterion (compare with 3.1). 42

CHAPTER IV

A M E T R I C A L T H E O R E M

4.1. In this chapter we shall restrict ourselves to the case of C-u.d. O u r starting point is a consequence of the so-called monotone convergence theorem

(see for instance [7]). In this theorem we replace the discrete parameter re by a continuously varying parameter F ( F > Fo > 0).

The continuous analogue of the above mentioned consequence is

T H E O R E M 21. Let g^ix) for F > Fo > 0 be a nondecreasing family of nonnegative integrable functions of x, defined, say, on the interval 0 < x < 1

1

and let Hm j g^ix)dx exist; then the family of functions g^ix) converges for r ^ c o (j

almost all x from [0,1] to a function ^(x) which is integrable on [0,1].

P r o o f . For the proof of theorem 21 if F runs through the integers we refer to [7]. Then we know that gr^iix) -^ gix) ( F - > oo) if x is not contained in Ü

[Ü some fixed subset of [0,1] of measure zero, ^(x) some integrable function).

However, since 0 < ^(x)—^y(x) < (^x)—^^^-.(x), -^ 0 for T -^ oo, if x does not belong to Q we then also have gTi'^)~K{^) ^ 0 ( F - ^ oo) if x is not con-tained in Q.

4.2. Let/x(re) be a continuous family of functions of x and re for 0 < x < 1, a > 0. Then we define for a non zero integer A and for F > Fo > 0

1 '• T

p^[x) = - I .2-*^^("Vre and ^^(x) = / \p,ix)\'dt.

d t„ Now we apply theorem 21 to the increasing family of integrable functions

g,pix) and we have: if

1 T

Hm /" (/' Ip^ix) \^dt)dx < oo (43) r - > a , Q f^

then also

I' il\Ptix)\^dt)dx <oo. (44)

From (44) it follows, however, that

CO

lor all X from [0,1], possibly except for values of x belonging to some subset of [0,1], i3(A) say, of measure zero, from which we conclude, taking into account that/>^(x) is a continuous function of t for each x from [0,1], that

Hm p^ix) lim .imhUu du = 0

for those x from [0,1] not belonging to I?(A).

If we observe that a countable union of sets i3(A) of measure zero (A= 1,2,...) still has measure zero and if we change the order of integration in (43) (which is permissable since/'^(x) is a continuous function of x and t) we obtain T H E O R E M 22. If the condnuous family of functions fiu) (0 < x < 1; re > 0) has the property that for A 7^ 0 and for some Fo > 0

7 I

for A 7^ 0 and almost all x from [0,1]; i.e. the family of functions/t:(re) is C-u.d. for almost all x from [0,1].

4.3. It is not difficult to find a simple family of functions to which theorem 22 cannot be applied. Take for instance/^(re) = x-re.

It is easy to see that this family of functions is C-u.d. for all x from (0,1]. We have in this case for x from (0,1]

1 1 ,o_ 7 / "du = _{t 2nihx} •1)

from which we calculate that 1 '. - / e''"''"'du t , sin^jihxt inhxt)^ and • 1 • \ - j e'-'"'"Vre d ^ (i

and we see that (we take Fo = 1)

-r, }

II

It is, however, possible to generalize theorem 22 so that the application possibilities are enlarged.

We shall prove

T H E O R E M 23. If positive numbers ci and C2 exist such that the continuous family of functions/x(re) satisfies

/

1 '• _imhfJu)

du dx Cl

(logo

for t > to> 1, where the constants Ci and c-z may depend on A, then the family

fxiu) is C-u.d. for almost all x from [0,1].

Proof. Put t„ = / '', where q = (so 0 < « < 1) and g > I. Then

•ImhfJu, du

eo

dx Cl Cl

and we have for v > I

Q = v , 1

'• {l

( l o g g " - e'+^'

which is bounded if v -> oo.

From this we conclude, if we write

gM = j I /V^^<"Vre

dg,

that

lim j g,.ix)dx < oo by an obvious change of order of integration in (45).

Now application of theorem 21 to the family ^„(x) gives that

f I

for all X from [0,1], possibly except for values of x belonging to some subset i3(A) of [0,1] of measure zero, from which it finally follows that