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Lp-inverse theorems for beta operators
OF APPROXIMATION
THEORY
L p-Inverse
66, 279-287 (1991)
Theorems
for Beta Operators*
DING-XUAN
ZHOU
Department qf Mathematics, Zhejiang L%iversity and Advanced Institute of Mathematics. Hangzhou, Zhejiang, People’s Republic of‘ China, 310027 Communicared hy Vilmos Torik Received October 30, 1989; revised June 13, 1990
In this paper we give the direct and inverse theorems for beta operators. Some other approximation properties of these operators are also given. 6? 1991 Academic Press, Inc.
1. IN?‘KODUCTION
Beta transform was introduced for studying conditions for the regularity of sequence-to-sequence transformations. Beta operators are given by
wherefcLj’[O,
co) (1
co), and
n-1 l&(x, 2+=x” u B(a, n) (1 + XU)2n9 B(n, n) = ((n - 1)!)2/(2n - l)!.
(1.2) (1.3)
Some papers [19,20] have considered the approximation properties of beta operators. In this paper we give the direct and inverse theorem for these operators in Lp[O, CC) (1 < p < co ). For consideration of the connection between the rate of approximation and the smoothness of the functions we use the modulus of smoothness given by Z. Ditzian and V. Totik [9], which in our result is given by
* Supported by the National
Science Foundation
of China.
279 0021-9045/91 83.W Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form rererved
280
DING-XUAN
wherefELj’[O,
ZHOU
co), q(x) = x, and
A;f(x)=f(x+h)-2f(x)+f(x-h) Aif
for XE [h, co);
= 0, otherwise.
This modified modulus of smoothness has the known properties of moduli of smoothness which were found by V. Totik [ 151. It is a valuable tool for dealing with the rate of approximation, inverse theorems, and imbedding theorems. We use the technique of interpolation spaces and characterization of K-functional which has been used for inverse theorems by a number of mathematicians [4, 5, 8,9, 141. Now let us give some notations. For 1 < p < GO,let D = { g: g E Lp [ 0, 00), g’ absolutely continuous locally, cp’g” E LPIO, co) > (1.5 be the weighted Sobolev space, and define
%T) = 92g”;
(1.6
Il~k)llp= II(P2g”llLpp,4. ForfELPIO, Kp,2(f,
co), let
LIP=;$
{Iv-- gllp + t Il~k)ll,~
(t>O)
(1.7)
be the so-called K-functional. Z. Ditzian and V. Totik proved (see [9, Chaps. l-31) the interpolation theorem, that is, the equivalence of this K-functional to the above modulus of smoothness. Now we can give our direct and inverse theorem for beta operators as follows. 1. For j-6 LP[O, co), 1
THEOREM
statements are equivalent:
(1) IIPnU) -fll, = wm; (2)
&2(f,
t)p = O(t%
(3) IlA;,fllww = O(W; (5) w2,(f, f)p= o(tZa). For p = 00 we restrict our discussion to continuous functions.
(1.8) (1.9) (1.10) (1.11)
Lp-INVERSE THEOREMS FOR BETA OPERATORS
281
Remark 1. It is easy to prove by the methods in [IO, 12, 131 that our theorem is still valid for a = 1. Remark 2. The L” case of our theorem can be derived from a result of V. Tot& [ 161 by a minor change. The results in Section 2 show that beta operators have some interesting properties. 2. LEMMAS First we give some formulae and estimates. For 1 < p < 00, define 4 :=PAP--1. LEMMA
2.1. Beta operators have the following properties: fi,(ln t, x) = In x;
(2‘1)
/?,(t”, x) = B(n + m, n-m) x”/B(n, n),
(2.2)
for FEZ, m
AAt, xl = nx/(n - 1); /?,(t’, x)=n(n+ Ml/t, ProoJ
1) x’/((n-
l)(n-2));
(2.4)
xl =nl((n - 1)x).
(2.5)
For m < n, we have
= B(n + m, n -m) x”/B(n, n). In the same way we have /3,(ln t, x) -In x = /IJln(t/x), x) f-1 02 1 =---------In I 0 B(n,n) (1 + t)2n = 0. LEMMA
2.2. For 1~ p < CO,f e D, we have
ll4o!‘llpG ll(P2f”llP.
tdt
282
DING-XUAN
ZHOU
ProoJ: First let us prove that lim, ~ o. f’(x) = 0 for f~ D. Forp=l, x, y>O, we have If'(x)-f'(y)1
= lj: u2f"(u) up2 dui Q Ilcp2fnlll/min{x2,
y'}.
For p= co, we have lfyx)-f’(y)1
For l
d II~2f”ll.&Wx~
~4
we have
If’(x) -f’(v)l
G (/r: 18f”WIP A I)‘“( lj; !.-Q dui)liq
Thus we know that hm,,,, a, If’(x) -f’( y)l = 0 and lim,, m f’(x) exists, but the condition f E Lp[O, co) implies lim, _ mf’(x) = 0. Now the proof of our lemma can be derived easily from the Hardy inequality for 1 < p < co, g > 0, and j > 0 given by
The casep = cc is simpler and our proof is complete. LEMMA
2.3. For 1
llDnllpG n/b - 1). ProoJ:
(2.8)
For 1~ p < co, f~ Lp[O, m), we have
IIMf)ll;
d jamjamUx, f) If(W
dt dx
= joa If(W)lp dt jomb,(x, t) dx
= n llfll g/b - 1). The proofs in the casesp = 1 and cc are trivial. The following property of beta operators is very interesting.
LEMMA 2.4. Suppose r E N, f +l) qff @)EL*[O, co). We have
is absolutely continuous locally, and
PuooJ: Note that
We can differentiate below the integration sign on [c, + co) for any c > 0 since the integration formula X
--r-l
03 s
V
n-3
x (x/v)~‘~‘(x/v) 0 B(n,n)( l+ u)*~ v*
dv
is uniformly convergent on [c, + co), hence our proof is trivial. With all the above preparations we can now prove our theorem. 3. PROOFOF THEINVERSE THEOREM
From the following lemmas we know that (1) and (2) are equivalent by a result of A. Grundmann [lo]. The equivalence of (2) and (3) has been proved in [14] and our proof of Theorem 1 is complete. LEMMA 3.1. For 1 < p < GO,f E D, we have the estimate (3.1)
llB,~f)-fll,~~,ll~(f)ll,l~~ where A, is a constant depending only on p.
Note that
ProoJ:
f(t)-f(x)=j’(l/u-l/t)(u:f’(u))‘d~+(l/~-l/~)x’f’(x). Y
(3.2)
We have
IMA XI -f(x)l
d lxfY(x) B,(llx - l/t, XII + ,8,, [’ (l/u - l/t)(~~f’(~))’ du , x (I x
I
1
d Ixf’(xMn - 1) +LL(lWirl
j~~l,.r’;.,,‘ldu~,s).
(3.3)
284
DING-XUAN
ZHOU
Thus for p = co, we can estimate easily as follows,
IIMf) -fll ccG Ilcpf’llm/(n - 1)
For 1 < p < co, we use the maximal function M((q*f’)‘)(~)
llMf)-flip<
and obtain
I140f’llpl(n- 1) + M((cp2f’)‘)(x)P, t2-2;;+x24)~~ ( II P < II4of’ll,l(n - 1) + 2 ll~((~*f’)‘)ll,l(n - 1) G llcp2f”llpl(~- 1) + 24 IIcp2f”IIpl(~- 1) GA, llW)ll,l~.
The proof for the casep = 1 is somewhat different. By Taylor’s formula f(t)-f(x)=(t-x)f’(x)+j’(t--u)f.“(u)du:=I+.I,
(3.4)
x
we can estimate by Fubibi’s theorem as follows,
llB?zKXIIIL’[O,m)= IIxf’(x)II J(n - 1) G 2 llW)ll h;
(3.5)
b,(x, t) jl’l (l/t - u) f”(u) du dt dx
IIMJ, XhqO,m)=
x
b,(x,
<
t)
j”’ (l/t - u) If”(u)1 du dt x
+ jm b,(x, t) j* (u - l/t) If”(u)] du dt dx l/x
l/f
>
= jam dx jm If”(u)1 du j;‘” b,(x, t)( l/t - u) dt x + jm dx jX If”(u)1 du jm b,(x, t)(u0 0 llu
l/t) dt
LP-INVERSETHEOREMSFOR BETA OPERATORS
ZZ
joa If”(u)l
285
du (1; .i,:i” b,(x, t)( l/t - U) dt dx
+ jm j” b,(x, t)(u0 l/u
l/t) dt dx
>
=lam lf”(u)l du (j;(nxlb -1)-~1 dx +s nt-*/(n=s O”(u- l/t)
1) dt
l/u
omu* lf”(u)l W(n - 1) G 2 IIWN l/n.
Thus our proof of Lemma 3.1 has been obtained. LEMMA
3.2. For 1 <~PKJ,
f ED, n32,
we have
(3.6) By Lemma 2.4, we have for cp*f" E L”[O, m)
ProoJ:
llcp*(P,f)“ll, 6 IlP,((~*f”)ll, Gn llcp2f”llpl(~- 1) 6 2 IIWN,. The following result is the so-called Bernstein-type inequality. LEMMA
3.3. For 1
IlW,f)ll, Proof.
G 32 llfll,.
(3.7)
By simple calculations we have
-2nu”x”(l
+xu)-‘“-~)
du,
63.e)
286
DING-XUAN
ZHOU
X2(&f)“(X) =jom g$ (n(n - 1)zFIXn( 1+XU)-2n -4n2u”l”+‘,(l +XU)2n+1 =sm-nun-lxy(n+ fU/u) 0Bhn)
l)(xu-
l)‘-2)(1
+xu)-2”-QU. (3.9)
Thus for 1~ p < 03, we obtain
f0O”Ix2uLf)“lpQ!x
cx If(llu)l nUnplXn
0
n+l (1 + XU)2n
B(n, n)
- 4(n+l)ux-2 (1 +x?#n+2 n(n + 1) b,(x, U) -
<
B(n+l,n+l)nb Bh n)
n+ 1(x, u)
PI4
x(4n+4-2/(u)) X om If(llu)l s
)) du
p ( n(n + 1) b,(x,
u) - n'b,+
l(x,
u)
x(4n+4-:)/(4n+Z)du)dx < (3n)“‘” omIf(llu)lWn s
+ 1) B,(W, u)Iu
- (4n + 4) n2PH+l(llf, u)l(u(4n + 2)) + 2n2B,+ 1(4 u)l(u2(4n + 2))) du < (3n)qm
0
If(l/u)lP/u2 du;
hence the Bernstein-type inequality
IIwJ-)ll,
G 32 U-II,
holds. The casesp = 1 and CCare trivial and the proof is complete.
L “-ISVERSE
THEOREMS FOR BL:TA OPERATORS
287
ACKNOWLEDGMENTS The author is very grateful to Professor Zhu-Rui Guo for his kind help and encouragemeni. Thanks are also due to Professor V. Totik for his comments.
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