ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V II (1974) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I : PRACE MATEMATYCZNE X V II (1974)
E. T
a b e r s k i(Poznan)
On general Diriclilet’s integrals
1 . I n t r o d u c t i o n . Let E be the class of all real functions f(t) Lebesgue- integrable over any finite interval, such that
T + c —T
(1) 2 lim-if ^oo 1 J \f(t)\dt = 0 = T-+co 1 _rf/_c lim-i- Г \f(t)\dt
for every fixed c > 0. E. g. the condition
<->•±00 lim
№
t = о implies ( 1 ), whenever these integrals exist.
Given a function f e E and a positive number l, we set
for X e ( — oo, oo), n = 0 , 1 , 2 , ...
Clearly, the last sums can be represented in the form
( 2 )
where
1
7 J f (u) Dln(u — x) d u , - i
k= 1
sin( 2 R -fl) izt 21
2 sin 711 21
In Section 4 we shall prove that, for some /’s of class E, the above general Hirichlet’s integrals ( 2 ) tend to f(x ) as l -> oo, n ->■ oo and Ijn -> 0 .
13 — Roczniki PTM — Prace M atem atyczne XVII.
500 E. T ab erski
Start with two obvious identities
l + x l —x
{ J f { x - t ) D ln(t)dt + J f{x + t)Dln{t)dtY
—l + x —l —x
l + x l —x
1 = iy { J DnW dt+ /
—l + x —l —x
which yield
l —x
K (æ ;/ ) -/ ( « ) = y y f {f (x + t ) —2f(æ )+ f(x -t)} l> ln(t)dt+
—l + x
l + x —l + x
+ т у f ^ œ~ t^~^œ^ I)l^ dt + ^ ï / { / ( ж + * ) - / И } ^ ( * ) ^
l —x —l —x
= I lW + U lM + V '+ c c )
for же<0, V). Writing <px(t) = /(ж-И) — 2f (x )- sr f (x — t), we have
1
l ~ xl lni®) = y j <Px{t)&n(t)M
0
l l
= y f ^W-DnW^-y =^(®)+^И-
0 Z— x
Therefore
«!,(*; / )-/ (* ) = J 'n W + v 'M + v ’M + w ü x ) .
If 0 and \f(x)\^.K (а ,Ъ ,К mean constants), then
\Uln(®)\
2 x —l
---1--- {J \т\т+2ыЛ.
4Zsin— (l — b) ~l By ( 1 ),
2
X — llimy f \f(t)\dt= 0
Z->oo v J
uniformly in же <a, b). Consequently,
lim Uln(x) = 0
Z—>co
lim У^(ж) = 0 = lim Wln{x).
Z—>■ oo Z—xx>
and, analogously,
Thus (3)
General DirichleVs integrals 501
S ln{ o r,f)-f(x ) = J ln{x) + o{ 1) as Z-> oo,
uniformly in xe (a , b ) (0 < a < b < oo), n = 0 , 1 , 2 , , whenever /e ÆJ remains bounded in <a, fe>.
It is easy to see that, in the last assertion, the assumption 0 < a < b
< oo can be replaced by — oo < a < b < oo.
In the sequel, we shall denote by M (u), N (u) [resp. M (u ),N (tt)]
and M k(u), Nk(u) [M k(u ), 1Ул(г«)] (к = 0, 1, 2) the suitable pairs of non
negative convex functions complementary in the sense of Young ([ 2 ], p. 16-20). For the inverse functions the symbols M~l (v), N~x(v) etc.
will be used.
2 . A u x i l i a r y r e s u l t s c o n n e c t e d w i t h g e n e r a l i z e d v a r i a t i o n s . As in [4], Section 1 , let FM(/; a, b) and V*M(f, a, b) be the first and the second M -variation of the real function f in a finite interval (a , 6 ). Write
a, °°) - lim V m (/; b), У*м(Л - oo,b) = lim V*M{f] a, b).
b -* oo a —>—oo
Eetain the definition of the strong majorant, given in [4], Section 3.
By an argument similar to that of [4], Theorem 3.1, we obtain L emma 1 . Suppose that V*M{f\H, oo) < oo — oo, H) < oo]
for some fixed H, and that N (u) has the strong majorant N (и) in an interval
<0, rf). Then
lim F^(/; h, оо) = 0 [ lim F^(/ï - h) = 0 ].
h —>oo Л -> ~ oo
Consider now the functions
Z Z
Ф'п(г) = j f D l„(t)dt, Wln( z ) = j f t & M d t (*> 0).
0 0
L emma 2. Let
(i) M 0(u) = uq (q > 1 ), or
(ii) M 0(u) = и (log(2/^))_y (y > 1 )
for ue ( 0 , |>. Then, under the restriction l > 0 and n — 0 , 1 , 2 , ..., Ум0(Ф1п> ^
where A is a constant depending resp. on g and y, only.
P ro o f in case (ii). Write
502 R. T ab er ski
If <a, £> c <0, cq>,
(S fS ax
( 4 ) j y f Dln{t)dt = y j \FlJt)\dt J dt < 1 .
.( 6 )
In case <ct, / 8 > c <am, an+1> (w = 1, 2, n),
1 2
J Dln{t)dt na n <■ miz
2 1 и'п-Я (2“ + 1 ) ^ Clearly, (4) and (5) imply
a/c+l
(6) J J \D‘M \ à t<
ft + 1 for A; = 0 , 1 , 2 , ..., n.
Take an arbitrary partition
( i ) 0 = < c tx <c t2 tr <C i = 1 j and pnt AF{%) = F (tp+1) — F(t„).
1° If ak < ti+ l< . .. < tj < а л+1 (0 < 7v < n, j > i + 1), then, by ( 6 ),
i- 1 j - 1
у ж 0 (иФ^а>]) = у to i jiog 2 { - у
и < о ) Г ) . 3 - 1
{log 2 (& + l)}} i V лф \ (fc + l){log2(fc-f-l)}>
2 ° In case а* < % < а й+1 < а л+2 < ... < ат 4+i (1 < A' < — 1,
1 < r < r), inequality (5) leads to
{ l o g Z k n y ^
(& + l){log2(fc + l)}>’
3° Finally, if a 0 < tv < ax < a 2 < ... < am < /„+1 (0 < r < r, 1 < m <
< n + l), estimates (4), (5) give
Ж (\АФ1 (t )|) < i < ___é___
°d « (,)!)< {iog|}y ^ 3 (log|У Hence
У ЛГ„(ИФ(,(*,) 1 ) < 2 у
v -=0 /с =0 (* + 1) {log2 (* + 1)>” 3{log(f)} 3\■> у >
General Diriclilet's integrals 503
i. e. (see [4], Section 1) V *
3 I
q« i 0 , l) < 1 +
rn=l
2 m (log2m)y
4 3{iog(|)F’
and the desired result follows (cf. [ 6 ], Section 7).
Since
uq < Lu for и e ( 0 ,f>,
with a certain constant L (depending only on q and y), the thesis in case (i) is an immediate consequence of the preceding assertion.
L emma 3. Given any q > 1, let M 0(u) = uq for и > 0. Suppose that 0 < l < C(n + l ) 1-1/® (n = 0 ,1 , 2, ...), with a certain constant C. Then
0, I K l + 2!+ 1 P .
P roo f. Let am (m = 0 , 1 , ..., n + 1) be as before, and let
( 8 )
А(ж) = ж/sin ж ( 0 < ж < 7 г / 2 ).
In case <(a, /?> <= < 0 , ax>, we have
2n-\-4 r ai l 1
IT / w ' ^ dt / tdt
21 о J 2?г + 1
If (a , /?> cz ( a m, an+1} (m = 1 , 2, ..., n), the mean-value theorem yields
P
(9) l tDln(t)dt 1 / 71 ft \ f . 2 Z
< — Я —- sin( 2 w + l) —- dt < ---
\ 2 1 / J v ' 2Z (2 W + 1)тс
Prom ( 8 ) and (9) follows that
"A; 4-1
( 10 ) f 7 Z
j t\Dln( t ) ld t ^ —- - - - for Jc = 0 , 1 , 2 , n . ak
Choose partition (7), retain the symbol ЛF (tv). By (8)-(10), in cases 1°— 3° considered in the last proof, we get successively
V jr,(i jy i< o i) < j J / w ln(K)\< ( T n F ’
v=i v=$i ' '
( * 21
2 w + l ( 2 п + 1 )тт) H \ 2 ю + 21 1 \ q
504 R. T a b ersk i
Consequently,
(n + l f - 1 ’ 2q+lF
and the thesis is now evident.
Similarly, the following result can easily be obtained.
L em m a 4. Let с, y be two positive constants, y > 1, and let
0 < Z < cmin{[log(% + 3)]y, {2n + l)} (n = 0 ,1 , 2 , ...).
Then, taking
we have
^ 0 ( П ; 0 , ! ) < l + {2 + (log 2 r ’’}c.
For the sake of completeness, we shall yet give an important estimate proved in [3], Theorem 2.3 (cf. [5], (10.9)).
L em m a 5. I f F^f l ( F ; a, b) < oo, V*M2{0; a, b) < oo and
3 . A n a l o g u e s o f t h e R i e m a n n - L e b e s g u e t h e o r e m . Throughout this section the real-valued functions f(t) are Lebesgue-integrable over all finite intervals and subject to further restrictions specified in particular statements.
T heorem 1 . Suppose that
OO
then
ь
J { F (t)-F (a )} d G (t)j< oVMl(F-, a, b)V*„2(G-, a, b).
a
Then, for each real a, b (a < b) and ô > 0,
■uniformly in xe <a , by, provided Z-> oo ,n -> oo.
General Dirichlefs integrals 505 P ro o f. By the assumption, for any positive s there is a A > b such that
/ J f i î M i * <
uniformly in же (a , b>. Hence
1
T
г
j f ( x ± t ) B ln(t)dt
A
< 2 ’ when же <a, ft) and Z > А (n = 0 ,1 , 2 , ...).
Now, it is enough to prove that the functions
A
Qt,nix ) = j j f ( x ± t ) B ln(t)dt d
are numerically less than e/ 2 , uniformly in же 6 ), if Ijn are sufficiently small.
1 ° Suppose first that f(t) is continuous in the interval <a — <5, b + A}.
Write [i = max|/(Z)| in <5, &-f zl>. Choose the partition ô = t0 < tx < t2 < ... < ts = A
for which
Osc f{ x ± t)< SÔ
Т а (h = 0 , 1 , 2 , . . . , s - 1 ).
By the mean-value theorem,
Qt,nix) = ---- j f { x ± t ) s m ( 2n + l ) ^j- dt (l > ô, ô < £ < A).
2 Z sin^— 6 21 Consequently,
j f (x ± t) sin (2 n - f 1 ) 7U t dt
Without loss of generality, let us consider the integral
£ TCt
#«(*) = J /(a? + Z)sin( 2 u + l)-^-<ZZ,
506 R. T ab ersk i
only. If tr < £ < tr+1 (0 < Г < S — 1),
r — î *к +1 ç
Ц п(я) = У A-=° i k Z. j sin[2n + 1 ) ~ dt + J/(a>-H)sin( 2 w + l ) - ^
T — l O f c + l
dt
k = о t
f {f(x + t ) - f ( x + tk)}sin(2n + l ) ~ d t + s
+ J {f(oc + t ) - f ( t r)}sin(2n + l ) — dt +
tr
y ^ f ( x + tk) J sin( 2 u + l ) ~ d t + f { t r) J mn{2n + l ) ~ d t .
*k + 1
Hence ti
ed 4:1
î. e.
j.v sÔ
\Itn(x )\ ^ ~7~7 ~ à) [л{г. + 1 ) — < — ?
4zl ( 2 w + 1 ) tc 2
1 Фмг ,(ж)1 < eM as Ijn are small enough.
2 ° In general case when f(t) is Lehesgue-integrable in (a — ô, Ъ + A ), лее can find a function g{t) continuous in this interval snch that
b + A
f \f(t)-g{t)\dt <
Erddently, a — ô
eô
~2
Qt,n(x) = ~ j { f{ x ± t)-g (x ± t)} D ln{t)dt + j j g ( x ± t ) D ln(t)dt
Ô Ô
and, by the mean-value theorem, the first term does not exceed in absolute
value л
--- f l/(#±*)-0O»±*)|d< < j ,
2 Zsin—- s
21 ^
uniformly in xe {a , by, whenever l ^ Ô (n = 0 , 1, 2, ...). By lo,
A
J f g (x ± t)D ln(t)dt < — (a < x < b) 4:
if l Jn is small; whence
( 1 1 ) I Q f » ( * ) l < e / 2
for these l, n, x, and the desired result follows.
Bern ark. Since
i
lim — f Dln(t)dt = 0 (ô > 0 ),
(j/nb 0 l j
General DiricMeVs integrals 507 the hypothesis of Theorem 1 can be replaced by
— I
/
f ( t ) - c 1 00
dt-\- [ f(t) <?2
t J l t dt < oo,
where c1} c2 signify two real constants.
T heorem 2. Let f(t)/t be of bounded variation over the intervals ( — oo ,
— H ), (II, o o), with a certain positive И. Then the thesis of Theorem 1 remains valid.
P ro o f. In view of the well-known Jordan theorem m
t = /i (*)-/*(<),
where f k(t) (h = 1 , 2 ) are non-negative and non-increasing [non-decreasing]
in (Л , oo) [ ( - oo, — Я>].
Taking a A ^ max(H — a, H + b, 1) and putting i
Щ п(я) = j § f{x± t)B n{t)dt, л i
l£ j (x ± t)fk(x ± t)D ln(t)dt
A
{a < x ^ b , l > A, h = 1 , 2), we have
4n(3>) = I t ~ I i- By the mean-value theorem
( t O C J — —sm ^ + i)— tk f * 1 Tit A 2 sin-^-
21 i k •— 7it
1 r 21 . 4 7rf ,
± - J --- ~ s m ( 2n + l ) - m J sin —
21
{ X r . -Kt
= f k{ x ± A )l---— j sm( 2 n + l) — d t±
} tc ZI
2 bsin---- J 21 _I . sk
1 21 r . ut )
± ---— sin (2w + 1 ) Vf dtf
77 . 7X§k J, 21 \
Sin --- —
£. 21
(Л < I, < l, A < Й < A < £ < £k, J c = l , 2),
508 R. T aberski Therefore
™ ^ ±A)- ( 2 ^ V T +1}
< - ^ ^ у ^ т ах{Л(Я ),/л(-Я)}{1пах(|а|, |ô|) + l} . Consequently, for any given s > 0,
\Bfn{x)\ < «/2 (&<#<&), whenever Ijn is small enough.
Clearly, the estimate (11) also holds. Thus, the proof is completed.
T heorem 3. Suppose that f(t) is of bounded the second Ж-variation over the intervals ( — сю, —H}, (Н , сю) (H > 0 ), where
(i) M (u) = u p( p > 1 ), or
(ii) M (u) = exp( — l/ u a) (0 < a < |)
for sufficiently small и > 0 . Then the assertion of Theorem 1 is also true.
P ro o f in case (ii). Retain the symbols Ф1 п{%), Qt,n(x ) and Щ Л Х) used above, with l > A > тах(£Г — a, H -f 6 ).
Consider M 0(u) as in Lemma 2 (ii), and put
M (u) = exp( — 1 / m «) (0 < a < a < \) for small и > 0. Then
, 2 \7 — / 1 \—x/«
M 01{v)^i v\log—\ , M 1{ v ) = n og—I , , i \ - i/a _ / i \—i/â N (и) < и [log — I , N (и) ^ \u I log
'W/ \ 'll i
for positive u, v small enough (see [ 2 ], p. 25). Therefore, N (u) is the strong majorant of N (u) and
oo
N î ' ( l ) N - ' { l ) + È M o l { l ) S ‘ ( l )
Clearly,
<y < oo if 1 + y < 1 1 a .
i
Rtn(®) = J {f{æ± z)-f{x± A )}d< Pln{z) +
A
l
+ / ( ж ± Л ) - у f l ) U t ) d t = J f + J f .
A
General Dirichlet’s integrals 509 In view of Lemmas 6 and 2,
|J±| < oAV^(F±-, A, l ) (I > A, n = 0 ,1 , 2, ...), where
F±{z) = f (x ± z ) ( a ^ x ^ b ) . Let e be a positive number. Then, by Lemma 1,
r v ( F i ; A , l ) K r ^ ( f ; a + A, со) <
and
A ^ X V ^ i f ; - o o ^ - A X ^ L - ,
provided A is large. Consequently, u f i <
8
4 for x e (a , by, l > A, n — 0, 1, 2 , ...
Since х-\-А е(Д , oo), x — Ae{ — oo there is a constant K{A) such that \f(x±A)\ < K (A ) when же<«, ft). Applying the mean-value theorem, we obtain
2 1
(2^ + 1 )^ ^ < 4 ’
uniformly in xe <a , &), if l/n is small enough.
Finally, inequality (11) remains valid. Hence the theorem.
In case (i) we argue analogously.
T heorem 4. Let f(t)jt be of bounded the second M -variation over the intervals ( — oo, —Н у ,(Н , oo) ( H > 0 ), with M (u) considered above.
Then, in case (i) [(ii)] of Theorem 3, the thesis of Theorem 1 holds under the conditions of Lemma 3 [Lemma 4] concerning l, n, whenever 1/p + l I g
> l [ y < (1 — a ) / а ].
The proof is similar to that of Theorems 2, 3.
4 . C r i t e r i a o f t h e D i n i a n d Y o u n g t y p e . Eetaining the previous notation, we restrict now to functions fe E satisfying the conditions given in any one of Theorems 1-3. The symbol f { x ± 0) will means limf{x-±t) as t - * 0 +.
T heorem 5. Suppose that, for any xe {a, by (a < b),
f
\<Рх(1)\ dt < oo,
t
510 E, T ab ersk i
with some r — r(x) > 0. Then
(1 2 ) lim S ln(x-,f) = f(æ ),
(Z/n )-*0
provided l -> oo,n-> oo and a < x < 6 . 7// is continuous at every xe {a, 6 ) and if
r— lim M ) +
T
