Series I: COMMENTATIONES MATHEMATICAE XXIX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXIX (1990)
St a n is l a w Si u d u t (Krakôw)
Some remarks on theorems of Romanovski and Faddeev type
Abstract. In this paper an extensive class of 2ir-periodic kernels К is found for which it is impossible to widen the set Zc in Theorem 1 of Romanovski type. Similar results for Theorem 3 of Faddeev type are given. The nonperiodic case is briefly discussed, too.
1. Introduction. We shall denote by G a real Banach space with the norm
|| ||. Let E a R and £0 be an accumulation point of E. We denote by K{t, £) a function defined for all te R and £ e E, 2n-periodic, even, nonnegative with respect to t for every fixed ÇeE. We assume that this function satisfies the conditions
(1) K (t, £) is nonincreasing in t on <0, я> for every £ e £ ;
(2) \im ] K{t,Ç)dt = \ (£e£);
«-«о -я
(3) lim K(ô, Ç) = 0 for every <5 in (0, я ) (ÇeE).
We define the sets J?2k(G), S?lKi Z c by the identities Jjf2n(G) = {/I /: R-+G is 2n-periodic
and Lebesgue-Bochner.integrable on < — я, я)},
(3') Z c = Z c (K) = { (x ,Q e R x E : |x - x 0| • K(0, 0 ^ C},
where C > 0, x 0e R and the Lebesgue-Bochner integral is described in [2], Chapter IV.f1) (*)
(*) The name Bochner is omitted in [2].
For the singular integrals
U(x, £ )= U{x, ( J ) = f K ( t - x ,( ) f ( t ) d t ,
— n the following theorem holds.
Theorem 1. I f f e l ? 2n(G), C > 0, x 0eR and
(4) then
l i m ^ h -> O h
x 0 + h
xo
f(t)d t = / ( x 0),
(5) U {x, Ç) tends to f ( x 0) as (x, £) -* (x0, £0), (x, £)eZc.
This result in the special case G = R was obtained by R. Taberski with the aid of some generalizations of Natanson’s lemma (see [4], p. 174-176). Since the suitable generalizations of this lemma are shown in [3], the proof of Theorem 1 is similar to Taberski’s proof, and therefore it is omitted here.
Let us define for r > 0
X r = {(x, £) £ R2: ( x - x0)2 + ( £ - £ 0)2 ^ r2}.
Let A be a nonempty subset of R x E and let (x0, £0) be an accumulation point of A.
If the relation
(6) 3r > 0 3C > 0 A n Ж r c Zc holds, then, under the assumptions of Theorem 1, we have (7) U (x, £)-> /(x0) as (x, £)->(x0, f 0), (x, Ç)eA.
If relation (6) does not hold, i.e. if
(6') V r > 0 V C > 0 ( A n J f r) \ Z c Ф 0 ,
then (7) may be false. We shall show this in Section 3. In other words, we shall prove that for an extensive class of the kernels К it is impossible to widen the sets Zc in Theorem 1 in essential way.
2. Auxiliary results. Let us introduce the following useful definitions:
Defin itio n 1. Let A be the set described in Introduction. We call the set A jutting with respect to the family {Zc}c>0 if relation (6') holds.
De fin itio n 2. Suppose that K{ . R x E x -*R and K 2: R x E 2 ->R, where Ej c= R, E2 cz R and £0 is an accumulation point of Et and E2. If c E2 or E2 cz E± then, by definition, K x >- K 2 if
(8) V M > 0 3 C > 0 3 r > 0 Z c i K f j n J f r ^ Z n i K J n j r , , where Z C( K X), Z M( K 2) are given by (3').
If K x > K 2 and K 2 > K x, then we write K x ~ K 2.
Definition 3. The function K: R x E -* R will be called a maximal kernel, if K (x, Ç) is nonnegative and measurable in t for every fixed ÇeE and
l«-«o|
(9) liminf j K{t, Ç)dt = В > 0.
«-«о О
We shall need the following lemmas:
Lemma 1. Let K x, K 2 be the functions as in Definition 2. I f the set A is jutting with respect to {ZM(X1)}M>0 and K x > K 2, then A is jutting with respect to {ZC(K2)}C>0. I f in addition K x ~ K 2, then A is jutting with respect to {ZM(KX)}M>0 iff A is jutting with respect to {ZC(K2)}C>0.
The proof is easy and we omit it. The next lemma is obvious.
Lemma 2. The relation ~ is an equivalence relation.
Lemma 3. Let {£„}, ( y j be sequences of nonnegative real numbers. Suppose that:
(i) limn_>00 yn = 0 and yn > 0 for all neN;
(ii) i j y n < /?„ for each n e N , where {/?„} is such that £ ®= i f n < oo.
Then
Km \ £ = 0.
i - 0 + n {n:y„üh}
P ro o f. If we define n0 = n0(h) = min {n: yn ^ h}, then y„o ^ ft and therefore we get
1 ft l
{n-.yn^h}
- Y Ç Y — - y Y — - y ^
ft > " v > v y v > v Упо
,V п ^ п о S t l o n ^ n o S n J n o n & n o S n
I
Abecause (i) and (ii) hold. Hence, by (ii), we obtain
t
Z
Zn<Z A.">° as ft
-+0+,
^ {myn^h} n>no
because n0 (ft)->oo as ft-+ 0 +. Thus, the proof is completed.
3. Impossibility of essential extension of the sets Zc. Throughout this section we assume that R э х0 is fixed, E c (0, 1), £0 = 0 and R x E zd A is a nonempty set which has the accumulation point (x0, £0).
For ÇeE, k e Z and t e ( — n , n ) we define
P(t, £ )= [ l ?
u , t e ( - n , 7 c > \ ( — £ ) ,
P(t + 2kn, ® = P(t, a
Thus we have a 27i-periodic and maximal kernel.
Theorem 2. Assume that the function К : R x E ^ R is nonnegative, measurable and 2n-periodic with respect to the first variable for every fixed value of the second one.
If, in addition, К satisfies condition (9), К > P and the set A is jutting with respect to {ZC(K)}C>0, then there exists a function fe £ P 2n{G) satisfying the condition
for which (7) does not hold.
P ro o f. It is sufficient to prove the theorem in the case G = R. Indeed, if there is a function / in <£ 2n for which the theorem is true(2), then the theorem holds for the function/• e e 2n(G), where eeG, ||e|| = 1. For that reason we assume G = R in the sequel.
We now present the idea of the proof:
(A) We shall construct sequences of real numbers, {x„}, {£„}, such that 1° \x„ — x0| > 0, <^„ > 0, (x„, Çn)e A for n e N and the sequence {|x„ —x0|} is decreasing;
3° the intervals Pn = (x„, *„ + £„) are disjoint, i.e. if m ,n e N and т ф п then P „ n P m = 0 and x 0фPn for eacfy neN ;
4° the point x 0 is the Lebesgue point of the characteristic function %P of the set P — | J n=tPn, i.e. condition (4') holds for %P instead of f.
(B) We shall define / as the 27t-periodic extension of the restriction ip to the interval < — n, л).
(C) We shall calculate U(xn, £„,/) and
We shall see that it is impossible that lim^^^ U (x„, = f ( x 0). Thus the proof will be finished.
We first realize (A).
Construction. Since К > P, by Lemma 1 the set A is jutting with respect to {ZC(P)}C>0, where
h \
0
(10)
Let us define Cn = 2”, n = 1 ,2 ,...
(2) We write | | instead of || || in (4').
(ax) Since A is jutting with respect to {ZC(P)}C>0, there exists (xl5 ^1) е А п Ж 1 such that (x1? Ç1)$ Z Cl(P), whence we have
(hi) |X i- x0| < l , ^ ^ i l x j - X o l .
(a2) If we take in (a^ x, £, C with subscript 2 instead of 1 and we take
^ \x i- x 0\i2 instead of Ж x, then we get
(b2) Х2 ф Х0, |x2 XQ| < 2 |Xj X0| < 2>
£2 < |x2 —x0|/23, P x n P 2 = 0 (see 3°).
(a„) If we take in (at) x, £, C with subscript n instead of 1 and we take Ж |*„-1-хо|/2 instead of Ж lt then, by (b„_i), (10) and the definition of Ж г, we obtain
(b„) * „ # x 0, |x . - x0| < f e - - V - ^ < ... < | X i - x 0| ^ 1
>fl-l >n- 1 » i . < ÿ + î I*. - *ol < 2 ^ 4 ■ I*. - 1 - *ol < • • • < 2 ^ ’
The consideration above implies that the sequence (|x„—x0|) is decreasing, lim„^ J x „ - x 0| = 0; the sequence { Q is positive and lim ,,^ £„ = £0 = 0.
Moreover, the intervals Pn (n e N ) are disjoint. Hence we have (A), 1°, 2°, 3° and consequently, for h > 0 we get
x o + h
(11) Xp(t)àt = ~ J 1 dt
xo P n ( x o , x o + h}
1 °°
= - y
h k% Pk<^(Xo,Xo + hy
I « .•
n {„:\xn - х о \ * Щ
Since (b j implies
«■ < 1
| x „ - * o l '
2
" + 1 ’using Lemma 3 for yn = |x„ — x0|, f}„ = 1/2"+ 1 we obtain from (11) xo + h
lim y h~>0+ "
XP(t)dt = 0.
Similarly
x o + h
lim
Л-0-h XP (t) dt = 0;
hence we have 4° since Xp (x0) = 0.
Now we define / as in (B) and observe that
V(x„, £„,/) = £ l f ( t ) K ( t - x „ , Q d t
к = 1 P k P „
= J l - K ( t - x „ i j d t = / K(r, { ,)* . Hence, for sufficiently large n we have
U(xn, £„) ^ ilim in ffK (t, Ç)dt = i B > 0, S-*o+ о
because К is a maximal kernel. On the other hand,
lim
h-*0 ‘ f ( x 0 + t)dt = 0 = f ( x 0);
therefore it is impossible that lim,,.,^ U (xn, £n, f ) = f { x 0). Thus relation (7) does not hold, and the proof is finished. ■
4. Some corollaries. Let
2n (1 — r)2 + 4rsin2| t
be the Abel-Poisson kernel with parameter re(0 , 1) ([1], pp. 46, 53). Setting s /(t, Ç) = p1- i {t), we get
«
s /{ t, Qdt = 1 2 n
1 - ( 1
dt >1 - £
£2 + 4 (l —£)sin2\ t " 2 n J { 2 + ( l - { ) t dt
= f - 7 L = a r c t g x/ T 3 |
which tends to i as £ -* 0 +. Thus s i is the maximal kernel (as £ -> 0 +, £ e(0, 1),
£o = 0).
Since P(0, £) < s i (0, £) < 4P(0, £) (^g(0, 1)), we have s i > P; moreover, s i ~ P.
For £e(0, 1) the triangle kernel T(t, f) is defined by
T(t,
l o ,
T(t + 2kn, f) = T(t, 0 ,
t e ( - n , 7 c > \ ( - f , £);
t e ( —n, тс), k e Z .
It is easy verify that T is a maximal kernel and T ~ P. Moreover, P is a maximal kernel, too.
Note that the kernels s /, T, P satisfy all the assumptions listed in Introduction (for K).
Therefore by the above considerations, Definition 2, Theorem 1 and Theorem 2, we get
Co r o l l a r y 1. Suppose that f e ^ 2n(G), C > 0, x 0eR and К is one of the kernels s é , T, P. I f (4) holds, then
(12) U (x, £ ,/) -* f(x 0) as (x, £)->(x0, 0), |x - x 0| ^ C£.
I f the set A cz R x(0, 1) is jutting with respect to |((x, x (0, 1):
|л: — x0| < C£}}c>o> then there exists a function g eJ? 2n(G) whose Lebesgue point is x0, and for which the relation
U{x, Ç, g)^>g{x0) as (x, Ç) -> (x0, 0), (x, Ç)eA, does not hold.
As an application of the generalized lemmas of Natanson type from [3]
one obtains the following counterpart of Theorem 3.2 ([4]):
Th e o r e m 3. Let K (t, Ç), K*(t, Ç) be the functions defined for all te R and ÇeE, 2n-periodic, even, bounded and measurable with respect to t for every fixed ÇeE.
Assume that fe S P 2(G), C > 0 and
(i)
I
K (t, £)|< K*(l, 0
for each (t, Q e R x E ;(ii) l i m ^ s u p ^ ^ J K (t, £)| = 0 for all <5e(0,7t> (feE);
(iii) there is В > 0 such that §n- nK*(t, Ç)dt < В on E;
(iv) condition (2) for К and condition (1) for K* hold.
Then relation (4') implies
(50 U (x, U ) -> /(*0) (x, - (x0, £0), (x, Ç)eZ*c , where
Z$ = {(x,Ç )eR xE : |x - x 0| K*(0, Q < C}.
Let the modified Fejér, Fejér-Korovkin, Jackson kernels be defined by
= ([1], pp. 43, 51, (2.2.24)),
J f ( t, <3 = f K Jt) ([1], p. 79, (1.6.2)),
([1], PP- 60, 61, 9),
12 — Roczniki HIM — Prace Matematyczne XXIX
respectively, where Ç = 1 /п, n eN . Write E = {1 /n: n e N }, £0 = 0. There exist majorants X * , X * and J * of X , X and J , respectively, such that the assumptions of Theorem 3 are satisfied and X ~ J* *, X ~ X * , # ~ (see, e.g., [1], p. 51, problem (6) for F).
One can prove that X >- P, X > P, # >- P and they are maximal kernels (with E — {1/n: neN}).
Furthermore, they are nonnegative and equivalent to P. Thus we obtain (similarly to Corollary 1)
Corollary 2. Assume that f e X 2n (G), C > 0, x 0eR and К is one of the kernels X , X , # . I f (4') holds, then the relation (12) is satisfied.
Moreover, the part of Corollary 1 below (12) is again true.
The de la Vallée Poussin kernel is defined for n e N by (cf. [1], p. 112)
#„(*) = (nl)2 (2л)! 2cos:
2n
Set
v ( t , £) = l
2k where £ = - , n e N . n
The kernel V(t, Ç) satisfies all the assumptions listed in Introduction, but it is not maximal. Indeed,
l /и 1 In
V(t, ®dt = - n
2 л2n Sn(t)dt = 1 (n!)2 2
2n (2n)\ cos 2ntdt 1 (n!)222n 1 _
^ — • — ---0
2k (2n)\ 2 as n 00
by Stirling’s formula. Moreover, V is not equivalent to P (again by this formula).
5. Nonperiodic case. In this section we shall denote by K (t, if) a function defined on R x E , even, nonnegative and satisfying the conditions
(a) K (t, £) is nonincreasing in t on <0, oo) for every ÇeE;
(P) lim f K (t,Ç )dt = 1 (£e£);
“ Qo 00
(y) lim J K (t, £)dt = 0 for all ô > 0 (ÇeE).
Ô
Let X (G ) be the set of all functions f : R -^ G which are Lebesgue-Bochner integrable on R.
As a counterpart to Theorem 1 one has
Th e o r e m 4. J//eJS?(G), C > 0, x 0eR and (4) holds, К is as above, then (5')
J
K (t — x, Ç )f(t)dt-> f(x0) as {x, Ç)-*{x0, Ç0), (x, Ç)eZc.— 00
Similarly to Theorem 2 one proves
Th e o r e m 5. Suppose that К : R x E - * R is nonnegative, measurable with respect to the first variable for every fixed value of the second one. I f К > P and (9) holds and A is the set described in Theorem 2 then there exists a function feJ£(G ) satisfying (4') and for which (7) does not hold (with
U(x, Ç) = $°200K (t-x ,Ç )f(t)d t).
Finally, we shall present 'the complementary
Le m m a 4. Suppose that f 0e SB(Я), / 0 ^ 0, and f 0{0) > 0, Jo /o( 0 ^ > 0- I f for £e(0, 1),-
A:(t, « = then К ~ P and К is a maximal kernel.
P ro o f. We have
Z C{K)= ^(x, Ç)eRxE: |x —xo|--* /o(0) < C \, then К ~ P, because / 0 (0) > 0. Moreover,
К (t, <*) dt = | i / 0 dt = J/o (s) ds > 0
0 0 0
and the proof is complete. ■
Let us define the modified Gauss-Weierstrass and Cauchy-Poisson kernels by
Их>а = ^Техр(~(Н)2 )’ {e((u)’
C(x,•£) = -•É 1 1 1 1
я x2 + r n £ (x te (0 , 1), + 1
respectively (compare [1], pp. 125, 126). According to Theorem 4, Theorem 5 and Lemma 4, we can obtain the nonperiodic version of Corollary 1 for the kernels W, C.
The nonperiodic version of Theorem 3 and Corollary 2 may be stated, too.
As an example, one can consider the singular integral of Fejér on the real line (see [1], p. 122) (or the singular integral of Jackson-de la Vallée Poussin, [1], p. 130, 14), with parameter £ = \ /q instead of q.
Acknowledgement. I am very much indebted to Professor Roman Taberski for his valuable suggestions and remarks.
References
[1] P. L. B u tz e r , R. J. N e s s e l, Fourier Analysis and Approximation, Vol. I, Birkhâuser Verlag, Basel-Stuttgart 1971.
[2 ] L. S c h w a r tz , Cours d ’analyse, Paris 1981.
[3] S. S iu d u t , Generalizations o f Natanson’s lemma, Comment. Math. 29 (1990), 277-286.
[4] R. T a b e r s k i, Singular integrals depending on two parameters, Roczniki PTM, Séria I: Prace Mat. 7 (1962), 173-179.
