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11Вестник Сыктывкарского университета. Сер Л. Вып. 7. 2007
УДК 517.98
ON THREE FORGOTTEN RESULTS OF S.KREIN, N. BOGOLYUBOV AND V. GURARI WITH APPLICATIONS TO BERNSTEIN OPERATORS
W. P. Odyniec, M. P. Prophet
1. Introduction
The results of M. Frechet in 1934 about the largesteigenvalue of a stochastic matrix [6] attracted attentionto positive linear operators with norm 1. The study of compact linearoperators with stochastic kernel by N.M. Krylov and N.N. Bogolyubovduring the 1930's ( [9], [10]) was generalized by S. Krein and N.N. Bogolyubov a decade later in [2]. These results contributed to the 1968 paper [8] of M. Krasnoselski in which the problem of determining minimal-normshape-preserving projections was present. Unfortunately, many of these papers are practically unknown, as they were published in the Ukrainian language. On the other hand, recent developments in thetheory of minimal shape-preserving projections have been made using methodsthat are independent of Krasnoselski's work (see [3] and [11]. In this paper, we attemptto connect these two directions by studying the (so called) Bernstein operators.
Let C[0,1] be the space of real-valued continuous functions and N be the set of positive integers. Let n e N. By the Bernstein polynomials we mean the particular family of n-th polynomials given by the formula
where 0 < x < 1 and / e C[0,1]. The operator which send seach / e C[0,1] to the Bernstein polynomial Bn(f,x) will becalled the Bernstein operator (or Bn for brevity). See [13] for details concerning Bernstein polynomials.
Thus Bn is a linear operator mapping C[0,1] onto the subspace Pn of polynomials of degree less than or equal to n. Itis well-known that the
© Odyniec W.P., Prophet M.P., 2007.
(i)
polynomials xk(l — x)n~k, k = 0, form a basis for the space Pn.
Additional well-knowncharacteristics of the Bernstein polynomials translate immediatelyinto properties enjoyed by the Bernstein operators; in the following, we list (as lemmas) some of these properties.
LEMMA 1. Let e„(t) = tu, where z/ > 0. Let ne N. Then
Bn(e0) = e0, Bn(e1) = eu and Bn(e2)(x) = e2(x) + -x(l - x).
Th
LEMMA 2. Let f e C[0,1] such that f(x) J^cx + b. Then for each n G N,
Bn(f) ± f-
LEMMA 3. For each f e C[0,1], we have
\\Bn(f)\\ < ll/ll
where \\f\\ = supiG[0,i] 1/(^)1- Moreover, ||i?n|| = 1 for each n e N. 2. Forgotten Results
We begin with the 1947 result by Bogolyubov and Krein. Let T be acompact linear operator with norm 1 in Banach space E.
THEOREM 1 ([2], see also statistical ergodic theorems in [4]). The
sequence of operators Tj, where j e N and
= (2)
^ 71 = 1
converge in the operator norm to the compact linear operator T^ such that
TTo, = TooT = Tl=T00 (3)
DEFINITION 1. Let E be a partially ordered Banach space (with the partial ordering denoted by >). If K C E is aconvex set, closed under nonnegative scalar multiplication, then wecall K a cone of E. Let P : E —>> Eq be a projection ontosubspace Eq. We say P is a shape-preserving projection if
PK C K. (4)
In this setting, the elements of cone K are said to have shape(in the sense of K).The positive elements of E are those x such that x > 0. Notethat theset of all positive element of E forms a cone. We say operator T is a positive operator if Tx > 0 whenever x > 0. An element x e E is an invariant element of T if Tx = x.
In [2] we also find the following result:
THEOREM 2. If T is a positive compact linear operator in the partially ordered Banach space E then the set of all invariant elements of T form afinite-dimensional sub space E0 of E with shape-preservingprojection P0 = Too : E Eq (if dim Eq > 0). Here theelements with shape are the positive elements of E.
The following corollary is an immediate consequence of applying the above results to the Bernstein operators.
COROLLARY 1. Fix n e N and let B denote the n-th degree Bernstein operator (i.e. B — Bn). Then
Poo = lim Bj
j^oo
is a shape-preserving projection onto 2-dimensional subspace Eq (generated byeo and e\) and ||Poo|| = where
BJ = 7ZB"-
^ n=0
Proof. The shape in this theorem is positivity. The existence of P^ follows from Theorem 1; the fact that P0 isshape-preserving onto the span of e0 and ei follows from Lemma land Theorem 2. The norm equal 1 claim is a consequence of Lemma 3. ■
THEOREM 3 ([8]). Let E i be an s-dimensional (s > 1) subspace of C[0,1]. Let
U = {xe C[0,1] | ||x|| < l},
the unit ball of C[0,1]. There exists a norm 1 projection P : C[0,1] —>> E\ if and only if the section UHE is a polytope with 2s faces (where the dimension of each face is s — 1).
From Theorem 3 and Corollary 1 we obtain:
COROLLARY 2. The unit sphere of subspace Eq (from Corollary 1) is a centrally symmetric quadrilateral.
3. Applications and Extensions
We apply and extend the results of the previous section in the following notes.
NOTE 1. Let denote the canonical dual operator to Bn; i.e., foreach ip e (C[0,1])* we have (B*(<p))(f) = <p(Bn(f)). Because C[0,1] is a partially ordered vector space,there exists a norm-1 element, such that for each non-zero h e C[0,1] the set
Note that we say non-zero functional cp > 0 whenever tp(f) > 0 for/ > 0. COROLLARY 3. // is a simplex.
This result is a consequence of the Theorem 2 from [2]. Wecan also prove it directly, using properties of Bn. Indeed, sincei?n is a positive compact linear operator, it follows that is as well (see for example [5]). We also have ||i?*|| = \\Bn\\ = 1- Consequently we have that the image of (C[0,1])* under is a finite-dimensional subspace Dn (see [5]). Let D C Dn be the set of all functionals from (C[0,1])* which areinvariant elements of B*. Select for D a basis so that thenonnegative elements D+ of D have positive coordinates (thusD+ is the positive orthant of Dn). Thus we see that /i* is simply the intersection of D+ with the hyperplane of allfunctionals ip such ^{u) — 1 and, therefore, is a simplex.
NOTE 2. In Corollary 1 the dimension of the subspace of invariant elements of Bn was 2. Other dimensions of Eq = T^C^O, 1]) can be obtained using different positive compactoperators acting on C[0,1]. For example, if F is a continuousfunction on the square [0,1; 0,1] and T is the operator defined by
where x e C[0,1], then T will be a compact operator. It is wellknown ([5]) that
if we require F(s,t) > 0 then we have ||T|| = 1. In this case wecall F a stochastic kernel. Furthermore, if F = 1 then ||T|| = 1 and Tf(s) is constant
Au = {t > 0 | -tu < h <tu} has inf Au ^ 0. We use u to define the set // asfollows:
^ = {cp € (C[0,1])* I <p>0, ¥>(«) = 1, <pU) = B*(ip(f))}.
for each / e C[0,1]. Hence dim^0 = 1 where E0 = Too(C[0,1]). From here, for example, wecan use tensor products (see[12]), to build shape-preserving projections onto arbitraryfinite-dimensional subspaces.
NOTE 3. In this paper we considered the Bernstein operators. Interesting results can be obtained by studying analogous operators. For example,from [1] we find the family of operators
П г
«./)(») = £ /(=f)
x(l — X)
m=0
2 n
V n
n )
xm(l-x)r
and from [14] we find the Kantorovich polynomials given by
Kn(f,x) = B'n(F,x)
where / is an integrable function on [0,1] and F(x) = J0 /(i) dt.
Литература
1. Bernstein S. Complement a Particle de E. Voronovskaya "Determination de la forme asymptotique de Г approximation des functions par les polynomes de M. Bernstein" // Dokl. Acad, of Sei. USSR. T. 2 Щ. 1932. P. 86 92.
2. Bogolyubov N.N., Krein S.G. The positive compact operators, (Ukrainian) // Translation of Institute of Mathematics of Acad, of Sei. Ukr.SSR. 1947. №. Kiev. P. ISO 139.
3. Chalmers B.L.,Prophet M.P. Minimal shape-preserving projections onto Пn// Numer. Fund. Anal, and Optimiz. 18. 1997. P. 507 520.
4. Dunford N., Schwartz J.T. Linear operators. Part 1. Interscience Publishers. New-York London. 1958.
5. Dieudonne J. Foundations of Modern Analysis. Acad. Press. New York. 1960.
6. Frechet M.M. Sur Failure asymptotique des deusties itereas dans le Probleme des probabilities en chaine// Bull. Soc. Math. France. 62. 1934. P. 68 83.
7. Kantorovich L.V., Vulikh B.Z., Pinsker A.G. Functional Analysis in Partially-Ordered Spaces, (Russian) // GITTL. Moscow-Leningrad. 1950.
8. Krasnoselski M.A. A spectral property of linear compact operators in the space of continuous functions, (Russian) // The Problems of Mathematical Analysis of Complicated Systems. №2. 1968. Voronezh. P. 68 71.
9. Krylov N.M., Bogolyubov N.N. On the work of the Chair of Mathematical Physics in the domain of nonlinear mechanics, (Ukrainian) // The memoirs of the Chair of Mathematical Physics of Acad, of Sci. Ukr.SSR III. 1937. Kiev. P. 5 39.
10. Krylov N.M., Bogolyubov N.N. The repetition of iteration with variable parameter, (Ukrainian) // The memoirs of the Chair of Mathematical Physics of Acad, of Sci. Ukr.SSR III. 1937. Kiev. P. 191
200.
11. Lewicki G., Prophet M.P. Minimal multi-convex projections, Studia Math. // 178. 2007. №2. P. 99 124.
12. Lewicki G., Prophet M.P. Shape-preserving projections in tensor product spaces, in preparation.
13. Lorentz G.G. Bernstein Polynomials. Toronto: University of Toronto Press. 1953.
14. Videnski V.S. The Bernstein Polynomials, (Russian). LGPI. Leningrad. 1990.
Российский Государственный Педагогический Университет им. Герцена Университет Северной Айовы
(Cedar Falls, IA, USA) Поступила 20.11.2007
