We obtain some Korovkin type theorems for the space \(C(X)\), where \( X \) is a compact metric space (Theorems 2 and 3).
The results are applied to the case when \(X\) is a compact subspace of a prehilbertian space and we obtain bounds for the difference \( \| B_{n}(f)-f \| \), where \(B_{n}\) is the Bernstein-Lototsky-Schnabl operator.
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AN ABSTRACT KOROVKIN TYPE THEOREM AND APPIICATIONS
D. ANDRICA* and C. MUSTATTA*
Received: January 26, 1989
REZUMAT. - 0 teoremă abstractă de tip Korovkin şi aplicafil. In lucrare se obtin teoreme de tip Korovkin pentru spațiul C(X)C(X), unde XX este un spaţiu metric compact (Teoremele 2 și 3). Se aplică rezultatele obținute pentru cazul cind XX este o submulțime compactă a unui spațiu prehilbertian şi se dau delimitări ale diferentei ||B_(n)(f)-f||\left\|B_{n}(f)-f\right\|, unde B_(n)B_{n} este operatorul lui Bernstein-LototskySchnabl.
The well known Korovkin's theorem (see e.g. [1]) asserts that if (L_(n))_(n)oint_(1)\left(L_{n}\right)_{n} \oint_{1} is a sequence of positive linear operators, acting from C[a,b]C[a, b] to C[a,b]C[a, b] and such that (L_(n)(e_(k)))_(n >= 1)\left(L_{n}\left(e_{k}\right)\right)_{n \geqslant 1} converges uniformly to c_(k)c_{k}, for k=0,1,2k=0,1,2, where c_(k)(t)=t^(k)c_{k}(t)=t^{k}, i in[a,b]i \in[a, b], then the sequence (L_(n)(f))_(n >= 1)\left(L_{n}(f)\right)_{n \geqslant 1} converges uniformly to ff, for every f∈∈C[a,b]f \in \in C[a, b].
This theorem was extended and generalized in many directions. One direction is to replace the above mentioned system of test functions by other systems of functions, which led to the theory of so called Korovkin subspaces. Another direction is to consider functions defined on more general compact spaces than the interval [a,b][a, b], first of all on compact subsets of R^(m)\mathbf{R}^{m}.
The aim of this paper is to give Korovkin type theorems for the space C(X)C(X), where XX is a compact metric space. As application, supposing that XX is a compact convex subset of a Hilbert space, one obtains evaluations of the order of approximation by the Bernstein - Lototsky - Schnabl operator, similar to those given in [4].
If ( X,dX, d ) is a compact metric space, denote by C(X)=C(X,R)C(X)=C(X, \mathbf{R}) the space of all real-valued continuous functions defined on XX and by Lip ( XX ) the subspace of C(X)^(:)C(X)^{:}formed of all real-valued Lipschitz functions defined on lambda\lambda. Equiped, as usually, with the uniform norm ||f||=s u p{|f(x)|:x in X},f∈∈C(X)\|f\|=\sup \{|f(x)|: x \in X\}, f \in \in C(X), the space C(X)C(X) is a Banach space.
Our first result is a density theorem:
theorem 1. The subspace Lip ( XX ) is dense in C(X)C(X), with respect to the uniform norm.
Proof. The assertion of the theorem will follow from the Stone-Weicstrass theorem if we shall show that Lip (X)(X) is a subalgebra of C(X)C(X) containing the constant functions and separating the points of XX.
If f,g in Lip(X)f, g \in \operatorname{Lip}(X) then
for all x,y in Xx, y \in X, where K_(f)K_{f} and K_(g)K_{g} are Lipschitz constants for ff and gg, respectively. Therefore f^(**)g in Lip(X)f^{*} g \in \operatorname{Lip}(X) and since Lip(X)\operatorname{Lip}(X) is a subspace of C(X)C(X) it follows that Lip (X)(X) is a subalgebra of the algebra C(X)C(X).
As the constant functions are obviously in Lip (X)(X) to finish the proof we have only to show that the algebra Lip(X)\operatorname{Lip}(X) separates the points of XX. For x,y in X,x!=yx, y \in X, x \neq y let f:X rarrRf: X \rightarrow \mathbf{R} be defined by f(z)=d(z,y),z in Xf(z)=d(z, y), z \in X. Then
which shows that ff is in Lip (X),f(y)=d(y,y)=0(X), f(y)=d(y, y)=0 and f(x)=d(x,y) > 0f(x)=d(x, y)>0. Theorem is proved.
A Markov operator LL on C(X)C(X) is a positive linear operator L:C(X)longrightarrow C(X)L: C(X) \longrightarrow C(X) such that L(c_(0))=e_(0)L\left(c_{0}\right)=e_{0}, where e_(0)(x)=1,x in Xe_{0}(x)=1, x \in X, i.e. LL preserves the constant functions.
In the following we shall need the following simple lemma:
I.EMMA 1. If LL is a Markov operator acting on C(X)C(X) then ||L||=1\|L\|=1.
Proof. Taking into account the positivity of LL and applying LL to the inequalities - ||f||*e_(0) <= f <= ||f||*e_(0)\|f\| \cdot e_{0} \leqslant f \leqslant\|f\| \cdot e_{0}, we obtain - ||f||*e_(0) <= L(f) <= ||f||*e_(0)\|f\| \cdot e_{0} \leqslant L(f) \leqslant\|f\| \cdot e_{0}, so that ||L(f)|| <= ||f||\|L(f)\| \leqslant\|f\|, for all *f in C(X)\cdot f \in C(X). As ||L(c_(0))||=||c_(0)||=1\left\|L\left(c_{0}\right)\right\|=\left\|c_{0}\right\|=1 it follows ||L:=1\| L:=1. Lemma is proved.
If (L_(n))_(n >= 1)\left(L_{n}\right)_{n \geqslant 1} is a sequence of Markov operators acting on C(X)C(X), let
for all x in Xx \in X and n=1,2,dotsn=1,2, \ldots.
Our first Korovin type theorem is the following:
THEOREM 2. Let (L_(n))_(n >= 1)\left(L_{n}\right)_{n \geqslant 1} be a sequence of Markov operators acting on C(X)C(X). If (alpha_(n)(x))_(n >= 1)\left(\alpha_{n}(x)\right)_{n \geqslant 1} converges to zero, uniformly with respect to x in Xx \in X, then (L_(n)(f))_(n >= 1)\left(L_{n}(f)\right)_{n \geqslant 1} conorges uniformly to ff, for all f in C(X)f \in C(X).
Proof. Let f in Lip(X)f \in \operatorname{Lip}(X) and let K_(f) >= 0K_{f} \geqslant 0 be a Lipschitz constant for ff, i.e.
for all x in Xx \in X. Applying to these inequalities the operator L_(n)L_{n} and taking into account the positivity of L_(n)L_{n} and the notations (1), one obtains:
for all x in Xx \in X. Since, by the hypothesis of the theorem the sequence (alpha_(n)(x))_(n >= 1)\left(\alpha_{n}(x)\right)_{n \geqslant 1} tends to zero, uniformly for x in Xx \in X, the inequality (2) implies that (L_(n)(f))_(n > 1)\left(L_{n}(f)\right)_{n>1} tends uniformly to ff.
By Theorem 1 the space Lip(X)\operatorname{Lip}(X) is deuse in C(X)C(X) with respect to the uniform norm on C(X)C(X) and by Lemma 1,||L_(n)||=1,n=1,2,dots1,\left\|L_{n}\right\|=1, n=1,2, \ldots so that by the
Banach-Steinhaus theorem, the sequence (L_(n)(f))_(n >= 1)\left(L_{n}(f)\right)_{n \geqslant 1} tends uniformly to ff, for all f in C(X)f \in C(X). The theorem is proved.
THEOREM 3. Let (L_(n))_(n >= 1)\left(L_{n}\right)_{n \geq 1} be a sequence of Markov operators acting on C(X)C(X). If beta_(n)(x)\beta_{n}(x) is defined by (1) and the sequence (beta_(n)(x))_(n >= 1)\left(\beta_{n}(x)\right)_{n \geqslant 1} tends to zero, uniformly fw. _(5)^(5){ }_{5}^{5} respect to x in Xx \in X, then the sequence (L_(n)(f))_(n >= 1)\left(L_{n}(f)\right)_{n \geqslant 1} tends uniformly to ff, for all ( in C(X)\in C(X).
If f in Lip(X)f \in \operatorname{Lip}(X) then, furthemore
for all x in Xx \in X. Taking into account the notations (1), it follows that the sequence (alpha_(n)(x))_(n >= 1)\left(\alpha_{n}(x)\right)_{n \geqslant 1} converges to zero, uniformly for x^(˙)in X\dot{x} \in X, provided that the sequence (beta_(n)(x))_(n >= 1)\left(\beta_{n}(x)\right)_{n \geq 1} converges to zero uniformly for x in Xx \in X. The first assertion of the theorem follows now from Theorem 2.
The inequality (2), obtained in the proof of Theorem 2, implies
for all f in Lip(X)f \in \operatorname{Lip}(X). By the inequality (4), ||alpha_(n)|| <= sqrt(||beta_(n)||)\left\|\alpha_{n}\right\| \leqslant \sqrt{\left\|\beta_{n}\right\|}, so that
which ends the proof of the theorem.
Now, let HH be a real pre-Hilbert space with inner product (:.,\langle.,.:).Fort in H\rangle . For t \in H fixed let the function c_(t):H rarrRc_{t}: H \rightarrow \mathbf{R} be defined by c_(t)(x)=(:x,t:),x in Hc_{t}(x)=\langle x, t\rangle, x \in H, and let e:H rarrRe: H \rightarrow \mathbf{R} be defined by e(x)=(:x,x:)=||x||^(2),x in He(x)=\langle x, x\rangle=\|x\|^{2}, x \in H.
theorem 4. Let XX be a compact subset of the pre-Hilbert space HH and let (L_(n))_(n >= 1)\left(L_{n}\right)_{n \geqslant 1} be a sequence of Markov operators acting on C(X)C(X). If (L_(n)(e))_(n >= 1)\left(L_{n}(e)\right)_{n \geqslant 1} converges uniformly to ee and the sequence (L_(n)(e_(x);x))_(n >= 1)\left(L_{n}\left(e_{x} ; x\right)\right)_{n \geq 1} converges to c(x)c(x), uniformly for x in Xx \in X, then the sequence (L_(n)(f))_(n rarr1)\left(L_{n}(f)\right)_{n \rightarrow 1} converges uniformly to ff, for all f in C(X)f \in C(X).
Taking into account the hypotheses of the theorem it follows that the sequence (beta_(n)(x))_(n >= 1)\left(\beta_{n}(x)\right)_{n \geqslant 1} converges to zero uniformly for x in Xx \in X, and Theorem 4 follows from Theorem 3.
Remark. If f in Lip(X)f \in \operatorname{Lip}(X) then
where a_(n)(x)=L_(n)(e;x)-e(x)a_{n}(x)=L_{n}(e ; x)-e(x) and b_(n)(x)=L_(n)(e_(x);x)-e(x)b_{n}(x)=L_{n}\left(e_{x} ; x\right)-e(x), for x in Xx \in X and n=1,2,dotsn=1,2, \ldots.
COROLLARY 1. (Korovkin's theorem). If (L_(n))_(n >= 1)\left(L_{n}\right)_{n \geqslant 1} is a sequence of Markov opcrators acting on C[a,b]C[a, b] such that L_(n)(e_(1))rarr"u"e_(1),L_(n)(e_(2))rarr"u"e_(2)L_{n}\left(e_{1}\right) \xrightarrow{u} e_{1}, L_{n}\left(e_{2}\right) \xrightarrow{u} e_{2}, where e_(1)(x)=xe_{1}(x)=x and c_(2)(x)=x^(2),x in[a,b]c_{2}(x)=x^{2}, x \in[a, b], then (L_(n)(f))_(n >= 1)\left(L_{n}(f)\right)_{n \geqslant 1} converges uniformly to ff, for all f in C[a,b]f \in C[a, b].
Proof. In Theorem 4 take H=R,X=[a,b]H=\mathbf{R}, X=[a, b] and the inner product be the usual multiplication in R,(:x,y:)=x*y\mathbf{R},\langle x, y\rangle=x \cdot y. Then e(x)=x^(2)=e_(2)(x),e_(1)(x)==t*x=t*e_(1)(x)e(x)=x^{2}=e_{2}(x), e_{1}(x)= =t \cdot x=t \cdot e_{1}(x) and L_(n)(e_(t);x)=t*L_(n)(e_(1);x)L_{n}\left(e_{t} ; x\right)=t \cdot L_{n}\left(e_{1} ; x\right). By hypothesis L_(n)(e)=L_(n)(e_(2))rarr"u"e_(2)=eL_{n}(e)=L_{n}\left(e_{2}\right) \xrightarrow{u} e_{2}=e. The corrollary will follow from Theorem 4 if we show that L_(n)(e_(x);x)rarrx^(2)L_{n}\left(e_{x} ; x\right) \rightarrow x^{2} uniformly for x in[a,b]x \in[a, b]. By hypothesis L_(n)(e_(1))rarr"n"e_(1)L_{n}\left(e_{1}\right) \xrightarrow{n} e_{1}, so that if epsi > 0\varepsilon>0 is given, there exists n_(epsi)in Nn_{\varepsilon} \in N such that |L_(n)(e_(1);x)-x| < epsi//M\left|L_{n}\left(e_{1} ; x\right)-x\right|<\varepsilon / M for all n >= n_(epsi)n \geqslant n_{\varepsilon} and all x in[a,b]x \in[a, b], where M=max(|a|,|b|)M=\max (|a|,|b|). Consequently |L_(n)(e_(2);x)-tx|=|t|\left|L_{n}\left(e_{2} ; x\right)-t x\right|=|t|. - |L_(n)(e_(1);x)-x| < epsi\left|L_{n}\left(e_{1} ; x\right)-x\right|<\varepsilon, for all n >= n_(epsi)n \geqslant n_{\varepsilon} and all xx and tt in [a,b][a, b]. In particular for l=xl=x, one obtains |E_(n)(e_(x);x)-x^(2)| < epsi\left|E_{n}\left(e_{x} ; x\right)-x^{2}\right|<\varepsilon, for all n >= n_(c)n \geqslant n_{c} and all x in[a,b]x \in[a, b], which shows that the sequence (L_(n)(e_(x);x))_(n >= 1)\left(L_{n}\left(e_{x} ; x\right)\right)_{n \geqslant 1} converges to e(x)e(x), uniformly for x in[a,b]x \in[a, b]. The corollary is proved.
If L_(n)=B_(n)L_{n}=B_{n}, where B_(n)B_{n} denotes the Bernstein polynomial operator defined by
B_(n)(f;x)=sum_(k=0)^(n)((n)/(k))(1-x)^(n-k)x^(k)f((k)/(n)),x in[0,1],f in C[0,1],B_{n}(f ; x)=\sum_{k=0}^{n}\binom{n}{k}(1-x)^{n-k} x^{k} f\left(\frac{k}{n}\right), x \in[0,1], f \in C[0,1],
then
B_(n)(e_(1);x)=e_(1)(x)" and "B_(n)(e_(2);x)=e_(2)(x)+(e_(1)(x)-e_(2)(x))/(2).B_{n}\left(e_{1} ; x\right)=e_{1}(x) \text { and } B_{n}\left(e_{2} ; x\right)=e_{2}(x)+\frac{e_{1}(x)-e_{2}(x)}{2} .
for all f in Lip[0,1]f \in \operatorname{Lip}[0,1].
Applications. 1^(@)1^{\circ}. In the Hilbert space R^(m)\mathbf{R}^{m} consider a compact convex set XX with nonvoid interior. For f inC^(1)(X)f \in C^{1}(X) (the space of all real-valued continuously differentiable functions on XX ) and u inR^(m)u \in \mathbf{R}^{m}, denote by grad f(u)\nabla f(u) the gradient vector of ff at the point uu, i.e.
lemma 2. If f inC^(1)(X)f \in C^{1}(X) then f in Lip(X)f \in \operatorname{Lip}(X) and K_(f)=max_(u in X)||grad f(u)||K_{f}=\max _{u \in X}\|\nabla f(u)\|.
Proof. Let x,y in X,x!=yx, y \in X, x \neq y. The mean value theorem implies the existence
of a point u in Xu \in X (which is an internal point of the segment joining xx and yy ) such that
corollary 2. If (L_(n))_(n >= 1)\left(L_{n}\right)_{n \geqslant 1} is a sequence of Markov operators acting on C(X)C(X), where XX is a compact convex subset of R^(m)\mathbf{R}^{m} with non-void interior, then
for all f inC^(1)(X)f \in C^{1}(X).
Proof. By Lemma 2, the inequality (7) is a consequence of the inequality (3) (see also (1) for the definition of beta_(n)\beta_{n} ). 2^(@)2^{\circ}. The Bernstein-Lototsky-Schnabl operator. If XX is a compact space, SS is a subspace of C(X)C(X) such that c_(0)in Sc_{0} \in S (remind that c_(0)(x)=1,x in Xc_{0}(x)=1, x \in X ), LL is a Markov operator on C(X)C(X) and xx is a point in XX then a Radon probability measure v_(x)v_{x} on XX is called an L(S)L(S) - representing measure for xx if
L(f;x)=int_(x)fdy_(x)L(f ; x)=\int_{x} f d y_{x}
for all f in Sf \in S.
Suppose from now on that XX is a compact convex subset of a pre-Hilhert space HH and let A(X)A(X) be the space of all real-valued continuous atine functions defined on XX. Let V=(V_(n))_(n >= 1)V=\left(V_{n}\right)_{n \geqslant 1} be a sequence of Markov operators on C(X)C(X) and let M(V)={v_(x,n):n >= 1,x in X}M(V)=\left\{v_{x, n}: n \geqslant 1, x \in X\right\} be a set of Radon probability measures on XX such that v_(x,n)v_{x, n} is an V_(n)(A(X))V_{n}(A(X)) - representing measure for xx, for all x in Xx \in X and n=1,2,dotsn=1,2, \ldots. Suppose further that the family M(V)M(V) is such that the functions E_(n):X rarrRE_{n}: X \rightarrow \mathbf{R} defined by E_(n)(x)=v_(x,n)(e),x in XE_{n}(x)=v_{x, n}(e), x \in X, are continuous for all n=1,2,dotsn=1,2, \ldots. Let P=(p_(n,j))_(n,j >= 1)P=\left(p_{n, j}\right)_{n, j \geqslant 1} be a lower triangular stochastic matrix i.e. an infinite matrix such that p_(n,j) >= 0p_{n, j} \geqslant 0 for all n,j >= 1n, j \geqslant 1,
sum_(j=1)^(n)p_(n,j)=1" and "p_(n,j)=0\sum_{j=1}^{n} p_{n, j}=1 \text { and } p_{n, j}=0
for all j > nj>n. If rho=(rho_(n))_(n >= 1)\rho=\left(\rho_{n}\right)_{n \geqslant 1} is a sequence of continuous functions rho_(n):X rarr[0,1]\rho_{n}: X \rightarrow[0,1], n=1,2,dotsn=1,2, \ldots, define
v_(x,n,0)^((V))=rho_(n)(x)v_(x,n)+(1-rho_(n)(x))epsi_(x)@V_(n),v_{x, n, 0}^{(V)}=\rho_{n}(x) v_{x, n}+\left(1-\rho_{n}(x)\right) \varepsilon_{x} \circ V_{n},
where epsi_(t)\varepsilon_{t} denotes the Dirac measure on XX centered at t in Xt \in X. I.et also pi_(n,P)::X^(n)rarr X\pi_{n, P}: : X^{n} \rightarrow X be defined by
for (x_(1),x_(2),dots,x_(n))inX^(n)\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in X^{n}.
The Bernstein-Lototski-Schnabl operator with respect to M(V),PM(V), P and rho\rho is defined by
B_(n)(f;x)=int_(x^(n))f@pi_(n,p)d⨂_(1 <= j <= n)v_(x,j)^((V))B_{n}(f ; x)=\int_{x^{n}} f \circ \pi_{n, p} d \bigotimes_{1 \leqslant j \leqslant n} v_{x, j}^{(V)}
for all x in Xx \in X and all f in C(X)f \in C(X). It follows that B_(n)B_{n} is a Markov operator on C(X)C(X) and straightforward calculations (see [5]) show that
for all x in Xx \in X. Here, the functions e_(y),c:X rarrRe_{y}, c: X \rightarrow \mathbf{R} are defined as above by e_(y)(x)==(:x,y:)e_{y}(x)= =\langle x, y\rangle and c(x)=(:x,x:),x in Xc(x)=\langle x, x\rangle, x \in X.
As a consequence of the general convergence theorems one obtains the following result:
uniformly jor x in Xx \in X, then (B_(n)(f))_(n > 1)\left(B_{n}(f)\right)_{n>1} converges uniformly to ff, for all f in C(X)f \in C(X). If f in I,ip(X)f \in I, \mathrm{ip}(X) then furthermore:
and the condition (10) of the theorem implies that (B_(n)(e;x))_(n,21)\left(B_{n}(e ; x)\right)_{n, 21} converges to e(x)e(x), uniformly for x in Xx \in X. The equality (8) gives for y=x,B_(n)(e_(x);x)=(:x,x:)=e(x)y=x, B_{n}\left(e_{x} ; x\right)=\langle x, x\rangle =e(x), for all x≐Xx \doteq X and n=1,2,dotsn=1,2, \ldots. The hypotheses of Theorem 4 are all fulfilled and, consequently, the sequence (B_(n)(f))_(n >= 1)\left(B_{n}(f)\right)_{n \geqslant 1} converges uniformly to f_(2)f_{2}. for all f in C(X)f \in C(X).
The equalities (5) (for L_(n)=B_(n)L_{n}=B_{n} ), (8) and (9) give:
It follows that the delimitation (11) is a consequence of the delimitation (3) from Theorem 3.
COROLLARY 3. If v_(x,j)=v_(x)v_{x, j}=v_{x} for j=1,2,dotsj=1,2, \ldots and rho_(j)(x)=1,x in X,j==1,2,dots\rho_{j}(x)=1, x \in X, j= =1,2, \ldots then the condition (10) from Theorem 5 reduces to
where E:X rarrRE: X \rightarrow \mathbf{R} is defined by E(x)=v_(x)(c),x in XE(x)=v_{x}(c), x \in X.
Proof. The first assertion of the Corollary follows from the following delimitation for the expression involved in the condition (10), for rho_(j)(x)=1\rho_{j}(x)=1 and v_(x,j)(e)=v_(x)(e)=E(x)v_{x, j}(e)=v_{x}(e)=E(x) :
