- Email: [email protected]
Saturation of positive operators
JOURNAL
OF APPROXIMATION
5, 413-424
THEORY
Saturation
(1472)
of Positive Operators*
G. G. LORENTZ AND L. L. Department
of Mathematics,
Uninersity
SCHUMAKER
of Texas, Austin, Texas 78712
ReceivedOctober 20, 1971 DEDICATEDTOPROFW~RJ.L.
WALSH ONTHE OC~ASIONOF H~~~~THBIRT~A~
1. HISTORICAL INTRODUCTION The theory of saturation of nontrigonometric operators originated with the results about Bernstein polynomials B,(f, x) by DeLeeuw f6], by one of the present authors [13], and by Bajanski and Bojanid f2]. In particular, d [x(1 - ~)/2n] IV&0 < x < 1: Lorentz [13] proved that ] B,(f, x) -f(x)\ if and only if the functionf has a derivative that belongs to the class Lip, 1. Bajanski and Bojanid proved that if B,(L x) -f(x) = o(n-‘) at each point 0 9 x < 1, thenfis linear. The method of the paper [13] has been further developed by DeEuca 171on one hand, and by Ikeno, Suzuki and Watanabe j&21-23] on the other, They showed that the proof of [13] can be formulated in a more general form, which allows one to apply it with successto many concrete problems. The method of Bajanski and Bojanid found further development in the papers of Amel’kovic [l] and Miihlbach [15]. They have noticed that asymptotic relations (of the type appearing in the theorem of Voronovskaja, see [12, p. 221) are of great help in this problem. Recently, the method of semigroups of operators and of infinitesimal generators has been applied to the saturation problems by Schnabl [lg], Micchelli [14] and Karlin and Ziegler [ll]. (Butzer was the first to discover the usefulness of this method in trigonometric saturation problems; see for example, Butzer and Berens [3]). In the present paper we shall further develop the second method. Since there does not exist a comparative study of the merits of the three methods described above, let us stress its advantages: it leads to simple proofs (Section 4), allows local assumptions to be treated (compare o(1) in * This work has been, in part, supported by the grants AFOSR 69-1512and AFOSR 69-1826.
413 0 1972by AcademicPress,Inc.
414
LORENTZ AND SCHUMAKER
Theorem 4.3, which is not uniform), and it can be applied to a wide variety of known special approximation operators (sections 5, 6). Basic for us will be the existence of an appropriate asymptotic formula and the positivity of the approximating operator. An important tool will be the theory of Cebysev systems, developed by Karlin and Studden [lo]. As a byproduct of our results, we shall see that a sequence of positive linear operators cannot satisfy an asymptotic relation of the form (4.1) with k > 2. 2. DIFFERENTIAL OPERATORS, CEBY~EV SYSTEMS AND CONVEXITY Let w0,..., wk be continuous, strictly positive functions on (a, b), such that j = o,..., k. We define the operators
Wj E ck-',
mY~) = Km(~>>f(x>l’, and
j = O,..., k - 1,
(2.1)
Df(x) = [l/w.&)] b-1 *-- DofW,
(2.2)
‘!f(x) =[l/&+&)]Dk-“3 -** D&x).
(2.3)
With the operator D we associate the functions (where c is a fixed point, atctb) uow = wow, (2.4)
The functions u0,..., r&--l. form a fundamental set of solutions of the differential equation Du = 0; the function ug satisfies Du, G 1. According to Karlin and Studden [lo, Chap. 1I] the functions (2.4) form an extended complete CebyBev system. This means that they belong to Ck(a, b) and that each set u,,,..., zlj, 0 ,( j 2( k, is a CebySev system on each closed subinterval of (a, b). In particular, the determinants
whenever a < x0 < *** < xi < b, 0 4 j < k.
SATURATION OF POSITIVE OPERATORS
415
With the system(2.4) we sometimesassociatethe reducedsystem vO,.*., cg+ j defined by
vjw = b~~+Iw/%(41’~
j = Q,..., k - 1.
(2.6)
The functions tli belong to C”-l, and are given by formulas similar to (2.4) involving the functions w1 ,. .., IQ ; they too form an extended complete CebySevsystem. Moreover [lo, p. 3831,
A function rj of the form 4 = CFli a+j will be called &ear. A functionf on (u, b) is convex with respect to u,, ,..., ukPI if k > 2, and if
> o
u
2
xO ,.**,
Xk-l
a .< x0 < .~. < xp < b.
(2.5)
, xk
For i = 2, 3,..., k, the function ui is convex with respect to u. ,I.., uiel , An important theorem of Karlin and Studden [lo] characterizes convex functions jI (A simple proof of this theorem, due to Mtihlbach [17], will appear in this Journal.) The relevant properties are: f E Ck-B(~,b) and f has right and left derivatives f~“‘(~), OF-” for II < x < b, which are, respectively, right and left continuous. Moreover, Ll,f(x)
= [l/W&l(X)] Dipe DI,-, -.. D,f(x)
(2.9)
is right continuous and increasing on (a, b). For any functionf(x) on (a, b) and any set of points a
‘.. < x-1
< b,
there exists a unique interpolating function q5= ‘j$’ ajllj , which coincides withffor x = xi ,j = 0,..., k - 1. The following two lemmas will be used in Section 4. LEMMA 2.1. A continuous fumaion f is convex on (a, b) with respect to uo : Ul if and only if for every a -=cx, < x2 < b the linear fu~~etio~ 4 = aOuo+ a,u, interpolating f at x1, x:, satisfies 4(x)
416
LORENTZAND SCHUMAKER
The simple proofs are left to the reader. Compare also [9, Theorem 2.2, p. 2821. 3. LIPSCHITZ CONDITIONS We shall say that a function h, continuous on (a, b), satisfies the Lipschitz condition or belongs to the class Lip,W+l with respect to u,, ,..., rlk , with M > 0, if h has a right continuous derivative Iz$-l’ and a left continuous k’t-l’ and if Ah(x,) - Ah(x,) 3 --M jz’ wk(t) dt, “0
R < x,, < .x1< b,
(3.1)
where for each of the values of Ah one can substitute A, h and AL Iz. Similarly, h belongs to the class Lip,-1, if Ah(x,) - Ah(x,,) < A4 j” wk(t) dt. 20
(3.2)
Finally, h belongs to the class Lip,1 with respect to the-system U, ,..., uk if h(k-l) is absolutely continuous, and if almost everywhere 1Dh(x)l < M.
(3.3)
LEMMA 3.1. A contirmousfunction f 012(a, b) belongs to the class Lip,+1 with respect to uO,..., uk if and only if Mu, + f is convex with respect to 2lf-J )...) U&l .
Proof. If Mu, + f is convex, the required smoothness off follows from the theorem of Karlin and Studden mentioned in Section 2. The condition that Muk + f is convex is equivalent to the validity of the inequality (3.4) for all possible points a < x0 < ... < xR < b. To each of the two determinants U in this inequality, we apply the formula (2.7); for the first of them we replace uk by f and u-~ by D,,f. Then, using the positivity of w,, and the fact that the intervals (x0, .x1)....,(x k- 1 , x,) are disjoint, we see that (3.4)
SATURATION OF POSITIVE OPERATORS
LJjf
holds for all selections of the points a < x0 < *‘. < xiC < b, if and only if
holds for all selections of a -=Cto < ... < ttiM1< 6. We can repeat this process, obtaining reduced systems [see(2.6)] of higher order. After k - 1 steps we reduce (3.4) to
(We use here the fact that a continuous function is the integral of its right derivative if the latter exists everywhere and is integrable). l3ut this is equivalent to Af(x,) - Af(x,) 3 --M iL’ w,(t) dt. d 20 THEOREM3.2. The following are equivalerzt for f E C(a, b):
.f E Lip,+1 n Lip,--1
with respect to u. ,..., z.J;,.
Of is absolutely continuous on (a, b) ad j Ai fELip,
- Af(x,)l d M i’,“’ w&) c/t. 0 1 with respect to u. ,..~, ulz ~
(3.6) ‘(3.7)
(3.8)
PPOO~.The inequality (3.7) is equivalent to (3.6) by definition. If in this inequality we make x0 and x1 approach, from left and right, some point x of (a, b), we seethat d J(X) = A,f(x) at tbis point. Thus (3.7) is equivalent to the absolute continuity off’“+ and to 1@f(x)1 < M ae. 4. THE SATURATION CLASSES We assume that L, , n = 1, 2,..., are linear positive operators that map C[a, b] into itself. In addition, we assume that L, satisfy the following asymptotic fomula:
f;$JL[Ldf(x) - fW1 = PC4WC4
a
(4.1)
where X, > 0 converges to $ co with n, p(x) > 0 on [a, b], and p(x) > 0 on
418
LORENTZ
AND SCHUMAKER
(a, b), and D is a differential operator of type (2.2). We assume that (4.1) holds whenever f belongs to the class Ck (where k is the degree of D) in a neighborhood of the point x. In the following lemma we write r,(j) = &cf) -J: LEMMA4.1. If
Lr,Cf, J-4b --M&l
+ o(l),
(4.2)
a
then the fimction Muk f f is convex with respect to u, , u, . Proof. Assume that Mun: + f is not convex. Then, since the set of all convex functions is closed in the space of continuous functions, the function g = f + (M + E) uk is not convex for some E > 0. According to Lemma 2.1, this implies the following. There exist points a < x, < y < x2 < b such that for the function q5 interpolating g at x, , x2 , +(JJ) < g(v). We consider the function
[g(x) - twl/h(~>.
(4.3)
It is continuous on I = [x1, x3], vanishes at the endpoints x1, x2, and has value >0 at ~7.Let m be its maximum on I; suppose it is achieved at z with x1 < z < x, . The linear function dl(x) = d(x) + mwO(x) coincides with g at z and satisfies g(x) < +&) for x E I. We can extend g from I to a continuous function G on [a, b], for which
Then, applying (4.1) to the function g - G and to r& (for which Du, = l), we obtain hrd.fi
Z)
=
&r&f-l-
=
br,(G,
(M 4
-
i(111
~1 uk, +
4
z> p(z)
-
W +
+
d
hnrn(uk,
Z)
o(l).
Since L, is positive, it follows by (4.4) and (4.1) that
Lrdf 4 = LL(G 4 - G(z)1- CM+ 4 P(Z)+ 4) < MUA , 4 - A(z)I - W + 4 P(Z)+ 40 < -CM + 4 P(Z)+ 4). But this contradicts (4.2). This completes the proof of Lemma 4.1.
SATURATION OF POSITIVE OPERATORS
414
then the function Muk -f is convex. As a corollary we have THEOREM 4.2. A sequence of positive linear operators L, cannot satiffv an asumptoticformula of the form (4.1) with k i 2.
Proof. hssume the contrary. Then (4.1) implies that (4.2) is satisfied for M = Cl and f = -ul . But then -uz is convex with respect to zfO9 ug : a contradiction. From now on we shall assume k = 2 in (4.1). The following theorem identifies the saturation class of the operators L, . THEOREM 4.3.
A function f E C[a, b] sati$es
(where o(l) converges to zero for n -+ CO,but not necessarily unz~orz~l~~~@” and only if f E Lip, 1 with respect to u1 , u1 ) uZ .
The theorem follows from the results of Section 3, Lemma 4.3 and the following lemma. LEMMA4.4. If Mu, + f is convex with respect to ug , uI , then (4.2) is satisjed. Proof. Let a < x,, -C b be fixed, By Lemma 2.2, there exists a linear function q3with the properties
&cJ = (#4& g(x) 2 9(X)>
~(4x3 adx
where g = Mu, + f. Then
Q-n(g, x0>= LL(g,
4 - b&d1 Z MLCA 4 - #bJl = 4Qs
by (4.1). Hence &r&T x0) 3 --h,Mr,(u, as required.
, x0) t- o(l) = --1Mp(x,j + O(1)T
420
LORENTZ AND SCHUMAKER
Remark 1. From the case M = 0 of Theorem 4.3 we obtain: If ;\,r,(f x) = o(l) for a < x < b, thenfis linear. This is a result of BajanskiBojanid type. Remark 2. Sometimes it is possible to prove a slightly more satisfying statement, namely, that forf6 C [a, b] one has
Aa I La
4 -fW
< MPG4
a
n = 1, 2,...,
(4.7)
if and only iffE Liphl 1. This is true if L, reproduces the functions U, , U, , and (4.7) holds forf = u2 . Only the sufficiency of the condition is to be shown; the proof is the same as that of Lemma 4.4. 5. APPROXIMATION OF FUNCTIONS OF Two VARIABLES
Let G, C R2 be an open domain with compact closure G. We consider a sequenceof positive linear operators L, that map C(G) into itself and satisfy an asymptotic formula
ti &[-&(.fi x, J’) - f(-u,Y)] = P(KJ’)Df 6, J%
(~2 v) E Go 3
(5.1)
where h, --f + co, p(x, u) > 0, and D is an elliptic operator Df = 4x, .Y>(W~X~>+ 2&, v>(W~x 3~1+ 4x, YXW~~~Y~) (5.2) with continuously differentiable functions a, b, c in G that satisfy b2 - ac < 0 in G,. We shall need the following definitions and facts about operators (5.2) (see, for example, [5]). There exists a solution U E C2(G,) of the equation DU = 1. A solution 4 of 04 = 0 will be called harmonic. For each disc y C Go, and for arbitrary continuous boundary values on the boundary of y, the Dirichlet problem for y is solvable with a harmonic r$ (since the assumptions on D assure it is uniformly elliptic on the disc y). A function u E C2(G,) n C(G) will be called subharmonic, if for each y C G, one has v < 4 on the disc bounded by y, where $ is the solution of the Dir-i&let problem with boundary values v on y. The subharmonic functions are characterized by the inequality Dv(x, y) > 0 on G,, . After these preparations, we can solve the saturation problem, at least for smooth functions f: THEOREM 5.1.
The class of all functions f E C2(G,,) n C(G) that satisfy
A, I =Lzcf; x, VI -f(x,
VI < WG,
v) + o(l),
(x, ~4 E Go
(5.3)
d"] ~-A
SATURATION OF POSITIVE OPERATORS
is identical with the class of all functions for which both MU + f and MU - f are subharmonic.
The proof of the necessity of the conditions is the same as that of Lemma 4.1. If they are satisfied, then D(MU j-f) > 0. This implies that / Df / < AlO = M. Then (5.3) follows from (5.1).
6. APPLICATION
AND EXAMPLES
For sequencesof positive linear operators operating on functions of one variable, the saturation theorem of Section 4 permits the derivation of many known results as well as a wide variety of new ones. Many positive linear operators satisfy an asymptotic relation of the form $i+i Xn[Ln(J
x) -,f(x)]
= a(x)f’(xj
+ /3(xjf”(x)~
a -=I x < 6,
(6.9)
where LX(X)> 0, /I(x) > 0 in (a, b). Then if p(x) is any function, positive on (a, b), the right hand side of (6.1) can be written as P(X)Do with
where c is some fixed point in (a, b). As a first example, we consider the operators of Cheney and Sharma [4]9
where t < 0 is a parameter, and L, is the Laguerre polynomial of degree V. For t = 0, P, reduce to the operators of Meyer-K&rig and Zeller. For the operators (6.3), Watanabe and Suzuki [23] established relation (6.1) with X, = 2(rz + l), CL(X)= -2tx, and P(X) = x(1 - x)“. To illustrate the technique we shall give two different saturation results for P, . Choosing p(x) = 1, the saturation class of the Gheney-Sharma operators P,, with respect to the relation 2tn f l)l P,(f,
X) -f(x)i
d M + 41)
consists precisely of all f with absolutely continuous f’, for which j -2txf’(x)
+ x(1 - X)2s”(x); < M,
a.e.
(6.4)
422
LORENTZ
AND
SCHUMAKER
On the other hand, if we choose p(x) = x(1 - x)* e-Q(“),where
then w2 = 1 and l/w, = e*(“). We conclude that the saturation class of P, relative to the relation 2(n + I)] P,(f, x) --f(x)/
< x(1 - x)” e-+)M
+ o(l)
(6.5)
consists of all functions f such thatf’e” is absolutely continuous and belongs to the classical Lipschitz class Lip, 1. In the special case t = 0, the second result above yields a theorem of Watanabe and Suzuki [22] for the Meyer-Konig and Zeller operators. The saturation results for the operators of Szasz,and of Baskakov, proved in [8] and [21] by other methods, also follow from Theorem 4.2. The Bernstein polynomials &(.f, x) satisfy relation (6.1) with A, = ~1, 01= 0 and /I(x) = x(1 - x)/2. The choice p(x) = &x(1 - X) leads to the theorem of Lorentz [13] mentioned in the introduction. Other choices of p lead to new saturation theorems. For example, taking P(X) = 1 and Df(x) = +x(1 - X)f “(x), we obtain
n I UL 4 -f(x>l
G MT
O
(6.6)
if and only iff’ is absolutely continuous and If”(x)\ < 2M/[x(l -x)]
a.e.
Still another example of a sequenceof positive linear operators satisfying an asymptotic relation of the form (6.1) with (y.# 0 can be found in Mtihlbach [ 161. For functions f(x, JJ) of two variables, there exist natural definitions of their Bernstein polynomials Bn(,f; x, JJ)for the case of the square, 0 < x, y < 1, and of the triangle; x > 0, JJ> 0, x + y < 1. Compare [12, p. 51, formulas (12) and (13), respectively]. Stancu [20, 191 has obtained for the two casesthe relation
where @Yx, v) =
X(I
-
2
X)
ay
ax2+
~(1 - ~4 azf 2
ay2
(6.8)
423
SATURATION OF POSITIVE OPERATORS
for the square and
Df(x , y) = xc1 -X) 2
Tf
=+xY&+
YU -Y> a?-
2
-@
(6.3)
for the triangle. The differential operators (6.8) and (6.9) are elliptic, and results of our Section 5 apply. (We note that in [19] Stancu assumes only the existence and continuity of the derivatives appearing in (6.91, which is not enough, instead of fg I?).
REFERENCES 1. V. G. AMEL'KOVIC, A theorem converse to a theorem of Voronovskaja type- Teuu. Funk&i, FcmkGonal. Anal. i Priloien. 2 (1969, 67-74. 2. B. BAJANSKI AND R. BOJAN&, A note on approximation by Bernstein polynom.ials. Bull. Amer. Math. Sot. 70 (1964), 675-677. 3. P. L. BUTZER AND H. BERENS, Semigroups of Operators and Approximation, Springer-
Verlag, Berlin, 1967. 4. E. W. CHE~~~EY AND A. SHARMA,Bernstein power series, Canad. J. Math. 16 (19&Q, 241-253. 5. R. CO~RANT AND D. HILBERT, “Methods of Mathematical Physics,” Vol. II, Interscience, New York, 1962. 6. K. DE LEEUW, On the degree of approximation by Bernstein polynomials, J. d’Ana!. 7 (1959), 89-104. 7. L. J. DELUCA, Algebraic approximation and saturation classes, Ph.D. dissertation, Syracuse University, June 1966, unpublished. 8. K. IKENO AND Y. SUZUKI, Some remarks on saturation problem in the local approximation, Tdhoku Math. J. 20 (1968), 214233. 9. S. J. KARL~N, “Total Positivity,” Vol. 1, Stanford University Press, Stanford, CA, 1968. 10. S, J. KARLIN AND W. J. STUDDEN, “Tchebycheff Systems,” Interscience: New York, 1966. 11. S. KARLKN AND Z. ZIEGLER, Iteration of positive approximation operators, J. Appraximation Theory 3 (1970), 310-339.
12. 6. G. LORENTZ, “Bernstein Polynomials,” University of Toronto Press, Toronto, Canada, 1953. 13. G. 6. LORENTZ, Inequalities and saturation classes of Bernstein polynomialb, [n “On Approximation Theory,” pp. 200-207, Proc. Conference Oberwolfach, 1963, Birkhauser Verlag, Base1 and Stuttgart, 1964. 14. C. A. MICCHELLI, The saturation classes and iterates of Bernstein operators, to appear in .I. Approximation Theory. 15. G. M~ACH, Operatoren vom Bernsteinschen Typ, J. Approximation Theoq? 3 (1970), 274-292. 16. G. M&LBACH, Uber das Approximationsverhalten gewisser positiver linearer Operatoren, Dissertation, Hanover, W. Germany, 1969, 93 pp. 17. G. M~JHLBACH, A recurrence formula for generalized divided differences and some applications, to appear in J. Approximation Theory. 64Q/5/4-6
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LORENTZ
AND SCHUMAKER
18. R. SCHNABL, Zum globalen Saturationsproblem der Folge der Bernsteinoperatoren, Acta Math. Szeged, to appear. 19. D. D. STANCLJ, Some polynomials of two variables of type Bernstein and some of their applications, Dokl. Akad. Nauk SSSR 134 (1960), 48-52. 20. D. D. STANCLJ,Evaluation of the remainder term in approximation formulas by Bernstein polynomials, Math. Conzg. 17 (1963), 270-278. 21. Y. SUZUKI, Saturation of local approximation by linear positive operators of Bernstein type, T6floku Math. J. 19 (1967), 429-453. 22. Y. SUZUKI AND S. WATANABE, Some remarks on saturation problem in the local approximation. T6koko Math. .I. 21 (1969), 65-83. 23. S. WATANABE AND Y. SUZUKI, Approximation of functions by generalized MeyerKiinig and Zeller operators. BUN. Yamagnta Uniu. (Nat. Science) 7 (1969), 123-128.
OF APPROXIMATION
5, 413-424
THEORY
Saturation
(1472)
of Positive Operators*
G. G. LORENTZ AND L. L. Department
of Mathematics,
Uninersity
SCHUMAKER
of Texas, Austin, Texas 78712
ReceivedOctober 20, 1971 DEDICATEDTOPROFW~RJ.L.
WALSH ONTHE OC~ASIONOF H~~~~THBIRT~A~
1. HISTORICAL INTRODUCTION The theory of saturation of nontrigonometric operators originated with the results about Bernstein polynomials B,(f, x) by DeLeeuw f6], by one of the present authors [13], and by Bajanski and Bojanid f2]. In particular, d [x(1 - ~)/2n] IV&0 < x < 1: Lorentz [13] proved that ] B,(f, x) -f(x)\ if and only if the functionf has a derivative that belongs to the class Lip, 1. Bajanski and Bojanid proved that if B,(L x) -f(x) = o(n-‘) at each point 0 9 x < 1, thenfis linear. The method of the paper [13] has been further developed by DeEuca 171on one hand, and by Ikeno, Suzuki and Watanabe j&21-23] on the other, They showed that the proof of [13] can be formulated in a more general form, which allows one to apply it with successto many concrete problems. The method of Bajanski and Bojanid found further development in the papers of Amel’kovic [l] and Miihlbach [15]. They have noticed that asymptotic relations (of the type appearing in the theorem of Voronovskaja, see [12, p. 221) are of great help in this problem. Recently, the method of semigroups of operators and of infinitesimal generators has been applied to the saturation problems by Schnabl [lg], Micchelli [14] and Karlin and Ziegler [ll]. (Butzer was the first to discover the usefulness of this method in trigonometric saturation problems; see for example, Butzer and Berens [3]). In the present paper we shall further develop the second method. Since there does not exist a comparative study of the merits of the three methods described above, let us stress its advantages: it leads to simple proofs (Section 4), allows local assumptions to be treated (compare o(1) in * This work has been, in part, supported by the grants AFOSR 69-1512and AFOSR 69-1826.
413 0 1972by AcademicPress,Inc.
414
LORENTZ AND SCHUMAKER
Theorem 4.3, which is not uniform), and it can be applied to a wide variety of known special approximation operators (sections 5, 6). Basic for us will be the existence of an appropriate asymptotic formula and the positivity of the approximating operator. An important tool will be the theory of Cebysev systems, developed by Karlin and Studden [lo]. As a byproduct of our results, we shall see that a sequence of positive linear operators cannot satisfy an asymptotic relation of the form (4.1) with k > 2. 2. DIFFERENTIAL OPERATORS, CEBY~EV SYSTEMS AND CONVEXITY Let w0,..., wk be continuous, strictly positive functions on (a, b), such that j = o,..., k. We define the operators
Wj E ck-',
mY~) = Km(~>>f(x>l’, and
j = O,..., k - 1,
(2.1)
Df(x) = [l/w.&)] b-1 *-- DofW,
(2.2)
‘!f(x) =[l/&+&)]Dk-“3 -** D&x).
(2.3)
With the operator D we associate the functions (where c is a fixed point, atctb) uow = wow, (2.4)
The functions u0,..., r&--l. form a fundamental set of solutions of the differential equation Du = 0; the function ug satisfies Du, G 1. According to Karlin and Studden [lo, Chap. 1I] the functions (2.4) form an extended complete CebyBev system. This means that they belong to Ck(a, b) and that each set u,,,..., zlj, 0 ,( j 2( k, is a CebySev system on each closed subinterval of (a, b). In particular, the determinants
whenever a < x0 < *** < xi < b, 0 4 j < k.
SATURATION OF POSITIVE OPERATORS
415
With the system(2.4) we sometimesassociatethe reducedsystem vO,.*., cg+ j defined by
vjw = b~~+Iw/%(41’~
j = Q,..., k - 1.
(2.6)
The functions tli belong to C”-l, and are given by formulas similar to (2.4) involving the functions w1 ,. .., IQ ; they too form an extended complete CebySevsystem. Moreover [lo, p. 3831,
A function rj of the form 4 = CFli a+j will be called &ear. A functionf on (u, b) is convex with respect to u,, ,..., ukPI if k > 2, and if
> o
u
2
xO ,.**,
Xk-l
a .< x0 < .~. < xp < b.
(2.5)
, xk
For i = 2, 3,..., k, the function ui is convex with respect to u. ,I.., uiel , An important theorem of Karlin and Studden [lo] characterizes convex functions jI (A simple proof of this theorem, due to Mtihlbach [17], will appear in this Journal.) The relevant properties are: f E Ck-B(~,b) and f has right and left derivatives f~“‘(~), OF-” for II < x < b, which are, respectively, right and left continuous. Moreover, Ll,f(x)
= [l/W&l(X)] Dipe DI,-, -.. D,f(x)
(2.9)
is right continuous and increasing on (a, b). For any functionf(x) on (a, b) and any set of points a
‘.. < x-1
< b,
there exists a unique interpolating function q5= ‘j$’ ajllj , which coincides withffor x = xi ,j = 0,..., k - 1. The following two lemmas will be used in Section 4. LEMMA 2.1. A continuous fumaion f is convex on (a, b) with respect to uo : Ul if and only if for every a -=cx, < x2 < b the linear fu~~etio~ 4 = aOuo+ a,u, interpolating f at x1, x:, satisfies 4(x)
416
LORENTZAND SCHUMAKER
The simple proofs are left to the reader. Compare also [9, Theorem 2.2, p. 2821. 3. LIPSCHITZ CONDITIONS We shall say that a function h, continuous on (a, b), satisfies the Lipschitz condition or belongs to the class Lip,W+l with respect to u,, ,..., rlk , with M > 0, if h has a right continuous derivative Iz$-l’ and a left continuous k’t-l’ and if Ah(x,) - Ah(x,) 3 --M jz’ wk(t) dt, “0
R < x,, < .x1< b,
(3.1)
where for each of the values of Ah one can substitute A, h and AL Iz. Similarly, h belongs to the class Lip,-1, if Ah(x,) - Ah(x,,) < A4 j” wk(t) dt. 20
(3.2)
Finally, h belongs to the class Lip,1 with respect to the-system U, ,..., uk if h(k-l) is absolutely continuous, and if almost everywhere 1Dh(x)l < M.
(3.3)
LEMMA 3.1. A contirmousfunction f 012(a, b) belongs to the class Lip,+1 with respect to uO,..., uk if and only if Mu, + f is convex with respect to 2lf-J )...) U&l .
Proof. If Mu, + f is convex, the required smoothness off follows from the theorem of Karlin and Studden mentioned in Section 2. The condition that Muk + f is convex is equivalent to the validity of the inequality (3.4) for all possible points a < x0 < ... < xR < b. To each of the two determinants U in this inequality, we apply the formula (2.7); for the first of them we replace uk by f and u-~ by D,,f. Then, using the positivity of w,, and the fact that the intervals (x0, .x1)....,(x k- 1 , x,) are disjoint, we see that (3.4)
SATURATION OF POSITIVE OPERATORS
LJjf
holds for all selections of the points a < x0 < *‘. < xiC < b, if and only if
holds for all selections of a -=Cto < ... < ttiM1< 6. We can repeat this process, obtaining reduced systems [see(2.6)] of higher order. After k - 1 steps we reduce (3.4) to
(We use here the fact that a continuous function is the integral of its right derivative if the latter exists everywhere and is integrable). l3ut this is equivalent to Af(x,) - Af(x,) 3 --M iL’ w,(t) dt. d 20 THEOREM3.2. The following are equivalerzt for f E C(a, b):
.f E Lip,+1 n Lip,--1
with respect to u. ,..., z.J;,.
Of is absolutely continuous on (a, b) ad j Ai fELip,
- Af(x,)l d M i’,“’ w&) c/t. 0 1 with respect to u. ,..~, ulz ~
(3.6) ‘(3.7)
(3.8)
PPOO~.The inequality (3.7) is equivalent to (3.6) by definition. If in this inequality we make x0 and x1 approach, from left and right, some point x of (a, b), we seethat d J(X) = A,f(x) at tbis point. Thus (3.7) is equivalent to the absolute continuity off’“+ and to 1@f(x)1 < M ae. 4. THE SATURATION CLASSES We assume that L, , n = 1, 2,..., are linear positive operators that map C[a, b] into itself. In addition, we assume that L, satisfy the following asymptotic fomula:
f;$JL[Ldf(x) - fW1 = PC4WC4
a
(4.1)
where X, > 0 converges to $ co with n, p(x) > 0 on [a, b], and p(x) > 0 on
418
LORENTZ
AND SCHUMAKER
(a, b), and D is a differential operator of type (2.2). We assume that (4.1) holds whenever f belongs to the class Ck (where k is the degree of D) in a neighborhood of the point x. In the following lemma we write r,(j) = &cf) -J: LEMMA4.1. If
Lr,Cf, J-4b --M&l
+ o(l),
(4.2)
a
then the fimction Muk f f is convex with respect to u, , u, . Proof. Assume that Mun: + f is not convex. Then, since the set of all convex functions is closed in the space of continuous functions, the function g = f + (M + E) uk is not convex for some E > 0. According to Lemma 2.1, this implies the following. There exist points a < x, < y < x2 < b such that for the function q5 interpolating g at x, , x2 , +(JJ) < g(v). We consider the function
[g(x) - twl/h(~>.
(4.3)
It is continuous on I = [x1, x3], vanishes at the endpoints x1, x2, and has value >0 at ~7.Let m be its maximum on I; suppose it is achieved at z with x1 < z < x, . The linear function dl(x) = d(x) + mwO(x) coincides with g at z and satisfies g(x) < +&) for x E I. We can extend g from I to a continuous function G on [a, b], for which
Then, applying (4.1) to the function g - G and to r& (for which Du, = l), we obtain hrd.fi
Z)
=
&r&f-l-
=
br,(G,
(M 4
-
i(111
~1 uk, +
4
z> p(z)
-
W +
+
d
hnrn(uk,
Z)
o(l).
Since L, is positive, it follows by (4.4) and (4.1) that
Lrdf 4 = LL(G 4 - G(z)1- CM+ 4 P(Z)+ 4) < MUA , 4 - A(z)I - W + 4 P(Z)+ 40 < -CM + 4 P(Z)+ 4). But this contradicts (4.2). This completes the proof of Lemma 4.1.
SATURATION OF POSITIVE OPERATORS
414
then the function Muk -f is convex. As a corollary we have THEOREM 4.2. A sequence of positive linear operators L, cannot satiffv an asumptoticformula of the form (4.1) with k i 2.
Proof. hssume the contrary. Then (4.1) implies that (4.2) is satisfied for M = Cl and f = -ul . But then -uz is convex with respect to zfO9 ug : a contradiction. From now on we shall assume k = 2 in (4.1). The following theorem identifies the saturation class of the operators L, . THEOREM 4.3.
A function f E C[a, b] sati$es
(where o(l) converges to zero for n -+ CO,but not necessarily unz~orz~l~~~@” and only if f E Lip, 1 with respect to u1 , u1 ) uZ .
The theorem follows from the results of Section 3, Lemma 4.3 and the following lemma. LEMMA4.4. If Mu, + f is convex with respect to ug , uI , then (4.2) is satisjed. Proof. Let a < x,, -C b be fixed, By Lemma 2.2, there exists a linear function q3with the properties
&cJ = (#4& g(x) 2 9(X)>
~(4x3 adx
where g = Mu, + f. Then
Q-n(g, x0>= LL(g,
4 - b&d1 Z MLCA 4 - #bJl = 4Qs
by (4.1). Hence &r&T x0) 3 --h,Mr,(u, as required.
, x0) t- o(l) = --1Mp(x,j + O(1)T
420
LORENTZ AND SCHUMAKER
Remark 1. From the case M = 0 of Theorem 4.3 we obtain: If ;\,r,(f x) = o(l) for a < x < b, thenfis linear. This is a result of BajanskiBojanid type. Remark 2. Sometimes it is possible to prove a slightly more satisfying statement, namely, that forf6 C [a, b] one has
Aa I La
4 -fW
< MPG4
a
n = 1, 2,...,
(4.7)
if and only iffE Liphl 1. This is true if L, reproduces the functions U, , U, , and (4.7) holds forf = u2 . Only the sufficiency of the condition is to be shown; the proof is the same as that of Lemma 4.4. 5. APPROXIMATION OF FUNCTIONS OF Two VARIABLES
Let G, C R2 be an open domain with compact closure G. We consider a sequenceof positive linear operators L, that map C(G) into itself and satisfy an asymptotic formula
ti &[-&(.fi x, J’) - f(-u,Y)] = P(KJ’)Df 6, J%
(~2 v) E Go 3
(5.1)
where h, --f + co, p(x, u) > 0, and D is an elliptic operator Df = 4x, .Y>(W~X~>+ 2&, v>(W~x 3~1+ 4x, YXW~~~Y~) (5.2) with continuously differentiable functions a, b, c in G that satisfy b2 - ac < 0 in G,. We shall need the following definitions and facts about operators (5.2) (see, for example, [5]). There exists a solution U E C2(G,) of the equation DU = 1. A solution 4 of 04 = 0 will be called harmonic. For each disc y C Go, and for arbitrary continuous boundary values on the boundary of y, the Dirichlet problem for y is solvable with a harmonic r$ (since the assumptions on D assure it is uniformly elliptic on the disc y). A function u E C2(G,) n C(G) will be called subharmonic, if for each y C G, one has v < 4 on the disc bounded by y, where $ is the solution of the Dir-i&let problem with boundary values v on y. The subharmonic functions are characterized by the inequality Dv(x, y) > 0 on G,, . After these preparations, we can solve the saturation problem, at least for smooth functions f: THEOREM 5.1.
The class of all functions f E C2(G,,) n C(G) that satisfy
A, I =Lzcf; x, VI -f(x,
VI < WG,
v) + o(l),
(x, ~4 E Go
(5.3)
d"] ~-A
SATURATION OF POSITIVE OPERATORS
is identical with the class of all functions for which both MU + f and MU - f are subharmonic.
The proof of the necessity of the conditions is the same as that of Lemma 4.1. If they are satisfied, then D(MU j-f) > 0. This implies that / Df / < AlO = M. Then (5.3) follows from (5.1).
6. APPLICATION
AND EXAMPLES
For sequencesof positive linear operators operating on functions of one variable, the saturation theorem of Section 4 permits the derivation of many known results as well as a wide variety of new ones. Many positive linear operators satisfy an asymptotic relation of the form $i+i Xn[Ln(J
x) -,f(x)]
= a(x)f’(xj
+ /3(xjf”(x)~
a -=I x < 6,
(6.9)
where LX(X)> 0, /I(x) > 0 in (a, b). Then if p(x) is any function, positive on (a, b), the right hand side of (6.1) can be written as P(X)Do with
where c is some fixed point in (a, b). As a first example, we consider the operators of Cheney and Sharma [4]9
where t < 0 is a parameter, and L, is the Laguerre polynomial of degree V. For t = 0, P, reduce to the operators of Meyer-K&rig and Zeller. For the operators (6.3), Watanabe and Suzuki [23] established relation (6.1) with X, = 2(rz + l), CL(X)= -2tx, and P(X) = x(1 - x)“. To illustrate the technique we shall give two different saturation results for P, . Choosing p(x) = 1, the saturation class of the Gheney-Sharma operators P,, with respect to the relation 2tn f l)l P,(f,
X) -f(x)i
d M + 41)
consists precisely of all f with absolutely continuous f’, for which j -2txf’(x)
+ x(1 - X)2s”(x); < M,
a.e.
(6.4)
422
LORENTZ
AND
SCHUMAKER
On the other hand, if we choose p(x) = x(1 - x)* e-Q(“),where
then w2 = 1 and l/w, = e*(“). We conclude that the saturation class of P, relative to the relation 2(n + I)] P,(f, x) --f(x)/
< x(1 - x)” e-+)M
+ o(l)
(6.5)
consists of all functions f such thatf’e” is absolutely continuous and belongs to the classical Lipschitz class Lip, 1. In the special case t = 0, the second result above yields a theorem of Watanabe and Suzuki [22] for the Meyer-Konig and Zeller operators. The saturation results for the operators of Szasz,and of Baskakov, proved in [8] and [21] by other methods, also follow from Theorem 4.2. The Bernstein polynomials &(.f, x) satisfy relation (6.1) with A, = ~1, 01= 0 and /I(x) = x(1 - x)/2. The choice p(x) = &x(1 - X) leads to the theorem of Lorentz [13] mentioned in the introduction. Other choices of p lead to new saturation theorems. For example, taking P(X) = 1 and Df(x) = +x(1 - X)f “(x), we obtain
n I UL 4 -f(x>l
G MT
O
(6.6)
if and only iff’ is absolutely continuous and If”(x)\ < 2M/[x(l -x)]
a.e.
Still another example of a sequenceof positive linear operators satisfying an asymptotic relation of the form (6.1) with (y.# 0 can be found in Mtihlbach [ 161. For functions f(x, JJ) of two variables, there exist natural definitions of their Bernstein polynomials Bn(,f; x, JJ)for the case of the square, 0 < x, y < 1, and of the triangle; x > 0, JJ> 0, x + y < 1. Compare [12, p. 51, formulas (12) and (13), respectively]. Stancu [20, 191 has obtained for the two casesthe relation
where @Yx, v) =
X(I
-
2
X)
ay
ax2+
~(1 - ~4 azf 2
ay2
(6.8)
423
SATURATION OF POSITIVE OPERATORS
for the square and
Df(x , y) = xc1 -X) 2
Tf
=+xY&+
YU -Y> a?-
2
-@
(6.3)
for the triangle. The differential operators (6.8) and (6.9) are elliptic, and results of our Section 5 apply. (We note that in [19] Stancu assumes only the existence and continuity of the derivatives appearing in (6.91, which is not enough, instead of fg I?).
REFERENCES 1. V. G. AMEL'KOVIC, A theorem converse to a theorem of Voronovskaja type- Teuu. Funk&i, FcmkGonal. Anal. i Priloien. 2 (1969, 67-74. 2. B. BAJANSKI AND R. BOJAN&, A note on approximation by Bernstein polynom.ials. Bull. Amer. Math. Sot. 70 (1964), 675-677. 3. P. L. BUTZER AND H. BERENS, Semigroups of Operators and Approximation, Springer-
Verlag, Berlin, 1967. 4. E. W. CHE~~~EY AND A. SHARMA,Bernstein power series, Canad. J. Math. 16 (19&Q, 241-253. 5. R. CO~RANT AND D. HILBERT, “Methods of Mathematical Physics,” Vol. II, Interscience, New York, 1962. 6. K. DE LEEUW, On the degree of approximation by Bernstein polynomials, J. d’Ana!. 7 (1959), 89-104. 7. L. J. DELUCA, Algebraic approximation and saturation classes, Ph.D. dissertation, Syracuse University, June 1966, unpublished. 8. K. IKENO AND Y. SUZUKI, Some remarks on saturation problem in the local approximation, Tdhoku Math. J. 20 (1968), 214233. 9. S. J. KARL~N, “Total Positivity,” Vol. 1, Stanford University Press, Stanford, CA, 1968. 10. S, J. KARLIN AND W. J. STUDDEN, “Tchebycheff Systems,” Interscience: New York, 1966. 11. S. KARLKN AND Z. ZIEGLER, Iteration of positive approximation operators, J. Appraximation Theory 3 (1970), 310-339.
12. 6. G. LORENTZ, “Bernstein Polynomials,” University of Toronto Press, Toronto, Canada, 1953. 13. G. 6. LORENTZ, Inequalities and saturation classes of Bernstein polynomialb, [n “On Approximation Theory,” pp. 200-207, Proc. Conference Oberwolfach, 1963, Birkhauser Verlag, Base1 and Stuttgart, 1964. 14. C. A. MICCHELLI, The saturation classes and iterates of Bernstein operators, to appear in .I. Approximation Theory. 15. G. M~ACH, Operatoren vom Bernsteinschen Typ, J. Approximation Theoq? 3 (1970), 274-292. 16. G. M&LBACH, Uber das Approximationsverhalten gewisser positiver linearer Operatoren, Dissertation, Hanover, W. Germany, 1969, 93 pp. 17. G. M~JHLBACH, A recurrence formula for generalized divided differences and some applications, to appear in J. Approximation Theory. 64Q/5/4-6
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AND SCHUMAKER
18. R. SCHNABL, Zum globalen Saturationsproblem der Folge der Bernsteinoperatoren, Acta Math. Szeged, to appear. 19. D. D. STANCLJ, Some polynomials of two variables of type Bernstein and some of their applications, Dokl. Akad. Nauk SSSR 134 (1960), 48-52. 20. D. D. STANCLJ,Evaluation of the remainder term in approximation formulas by Bernstein polynomials, Math. Conzg. 17 (1963), 270-278. 21. Y. SUZUKI, Saturation of local approximation by linear positive operators of Bernstein type, T6floku Math. J. 19 (1967), 429-453. 22. Y. SUZUKI AND S. WATANABE, Some remarks on saturation problem in the local approximation. T6koko Math. .I. 21 (1969), 65-83. 23. S. WATANABE AND Y. SUZUKI, Approximation of functions by generalized MeyerKiinig and Zeller operators. BUN. Yamagnta Uniu. (Nat. Science) 7 (1969), 123-128.