0N A KOROVKIN THEOREM 0F UNIFORM CONVERGENCE

0N A KOROVKIN THEOREM 0F UNIFORM CONVERGENCE

By

Ky6ichi YosmNAGA and Syuji TAMuRA

(Received Oct. 11, 1975)

The materials in the present paper are elementary. Among the Korovkin theorems relating to the approximation theory, the following is a typical one [3, p. 14].

THEoREM K. Let the sequence {L.(f)(x)} of positive linear operators satisfy the condition that

lim L.(1)(x)=1, n-co

lim L,(t)(x)=x, n-co

lim L.(t2)(x)=x2 n-co

uniformly with respect to x in the intervat [a, b]. Then for any bounded function f(t) of a real variabte, continuous at every point in [a, b], it hotds likewise that

lim L.(f)(x)=:f(x) n-co

uniformly with respect to x in [a, b].

In order to extend Theorem K so as to be applicable to a family of real-valued func- tions on a given topological space as well as on the real number field, some ambiguity rela- tive to the domain ofthe operator must be refined. Such attempts are shown us by several authors [1, p. 67, p. 123], [4, p. 7], and so far as the present writers know, the domain and the range of the operator L. are restricted to the normed space of continuous functions on a compact Hausdorff space.

The purpose of the present paper is to give an extension of Theorem K as faithfully as possible in that the domain of the operator L. is a family of functions on a uniformizable Hausdorff space X and the range is a family of functions on a given compact subspace

YofX.

Section 1 is devoted to the preliminaries. The assumptions on which we shall start

the arguments are given here, and some preparatory lemmata are proved. Our main results

are offered in Section 2. There, Theorem K is extended to Theorem 1 and some already

known results are obtained as its corollaries. Section 3 is a supplement. It is remarked

that a theorem (Theorem 2) of Korovkin type relative to the convergence of a sequence of positive linear functionals may be given as a particular case of Theorem 1.

Most of the notations and the terminologies concerning the topological space in this paper are those of J. L. Kelley [2].

1. Preliminaries

Until otherwise stated, the assumptions imposed upon us throughout the paper are as follows.

Let X be a Hausdorff space provided with the uniform structure (X, cZ(), where ar is the filter of the vicinities containing the diagonal A =={(x, x); xeX} in XÅ~X, and let

Ybe a compact subspace of X. :dik' is a given vector space over the real number field R composed of a family of real-valued functions defined on X. We assume -0r contains the constant-valued function 1 and therefore Rctat is supposed. Let, for each n=1,

2,..., L. be a linear positive operator of -0Z' into RY, i.e., .sP' gf(x).L.(f)(y)ERY satisfies Ln(f+ g) == L.(f) + L. (g), L.(ctf) = ct L.(f) for any f, g G fO ' and for any ct e R (linear) and it holds that f(x)2O implies L.(f) (y)2O (positive).

Let further a certain real-valued function F(x, y) with domain X Å~ Ybe given and sup- pose that F satisfies the following conditions i), ii), iii) and iv) :

i) F(x, y) ;}tO on XÅ~Yand F(y, y) =O for each yE Y,

ii) F,Ef`T for each yE Y, where the function F,(x) on X is defined by F,(x)=F(x,

y),

iii) for each ye Y, F,(x) is continuous with respect to x at each point in Y,

iv) 6(U) :infF(x, y)ÅrO for each UGav, where inf is taken for all (x, y)EXxY -u= uc n (x Å~ y).

The final assumption we are given is the following.

a) There exist yi, y2e Y, yily2 such that F,,(x), F,,(x) are bounded function of xGX, and it holds that

lim L.(F,,) (y) = F(Y, Yi) , n-co

lim L.(F,,) (y) == F(Y, Y2) , n-co

lim L.(F,)(y)=O n -+ co

uniformly with respect to ye Y.

REMARK 1. The assumption a) is clearly satisfied by the next condition.

b) For each yE Y, F,(x) is a bounded function ofxEX, and it holds that lim L.(F,) (y') = F(y', y)

n-co

uniformly with respect to y, y' e Y.

We begin by proving the following

LEMMA 1. Let Vbe an open subset ofXxYcontaining Ay=An(XxY). Then

it is possible to see that V=) U n (X Å~ Y) for some UE cZr.

PRooF. Assume the contrary: Vts? U n (X Å~ Y) for each UE ar, U= U- i, and we shall lead a contradiction. Putting Au=(Un(XxY))nVC7EÅë and letting Bu be the projection of AucXÅ~Yon Y, we may observe that the family Y=={Bu; UEcZ(, U

=U-i} of subsets of Ybecomes a filter base in Y. Owing to the compactness of Y, one finds a cluster point yo of the filter generated by Y, and therefore yo e Bu for each U E cZr,

U= U'i. Consequently one obtains yuEBu so that yuE U(yo) for every UGcZr, U=U-i.

Since yuEBu means that (xu, yu)eAu for some xuEX, it follows that yuE U(xu), i.e., xuEU(yu). This proves xueUoU(yo) for each UEcZr, U=U-' and consequently the convergence xu-Åryo in X and so the convergence (xu, yu).(yo, yo)EAy in XÅ~ Yis ob- tained. Because of (xu, yu) EAucVC, it results that (xo, yo) e VC=Vc and this is a con- tradiction. This completes the proof.

LEMMA 2. For some integer no, it holds that {L.(1)(y);nl}l no, yG Y} is a bounded subset of R.

PRooF. For the points yi, y2EYgiven in the assumption a), let us take UoGdZf in such a waythat (yi, y2) G Uo. Choose UeciY, U=U-`, such that UoUcU, and ob- 1

serve that F(x, y,)+F(x, y2)l}i6(U) for every xEX. This proves 6(u) (F,,(x)+F,,(x)) År 1 and therefore one obtains

1

(Ln(Fy i)(Y) + Ln(Fy 2) (Y )) ;}t Ln(1)(Y) ;}l O

6(U)

for each yGYand n= 1, 2,.... Because of the uniform convergence of lim L.(Fy ,) (Y) = Fy i(Y)

n-co and

lim L.(Fy,)(Y)=Fy2(Y) n-co

with respect to y e Y, and owing to the assumption iii), it is not diMcult to get the desired conclusion. This completes the proof.

LEMMA 3. Let f(x, y) be a real-vatued and bounded function deLtined on XxY

and assume that it is continuous at each diagonal point (y, y)eAy. Let further f,ege

and f,(y)==O for each yE Y, where f, is the function of xEX deLfined by f,(x) =f(x, y).

Then it holds that

lim L.(f,)(y)=O n-co

uniformly with respect to yG Y.

PRooF. Given eÅrO, we put N={(x,y);(f(x, y)lÅqe,xeX,yGY}. Owing to

the continuity off(x, y) at (y', y')EAy, we may find an open neighbourhood V(y') of y' in X so that

lf(x, y)l -lf(x, y) ---f(y', y')IÅqs for each (x, y)e V(y')Å~(V(y')n Y). Putting V- v (V(y') Å~ (V(y') n Y)) , Y'EY

we may infer that Vis an open subset ofXÅ~ Ycontained in N and that Ayc V. Lemma 1 allows us to get some U G `a( satisfying U n (X x Y) c V. Letting M= sup lf(x, y)1 Åq co, it is not diMcult to see

M If(x, y)I s6+

F, (x) 6(U)

for every (x, y)EXÅ~ Y. Therefore it follows, from the inequality -6h 6(Mu) F,(x) Kfl(x)s6+ 6(Mt) F, (x) ,

that

-6Ln(1)(Y)h 6(M u) Ln(Fy)(Y) f{: Ln(fy)(Y) S: 8Ln(1)(Y)+ 6(M u) Ln(I7y)(Y) , and consequently, by virtue of theinequality OgL.(1)(y)gC (n2no) for some C obtained from Lemma 2, and in accordance with the assumption a) we may find some ni in such a way that

-sc- 6(Mu) s f{{ L.(f,)(y) f{g 6C+ 6(Mus-s

js true for each n;}ii ni and for each yE Y. This bompletes the prooÅí

2. Main results and some applications

In this section an extension of Theorem K is proved and as its corollaries some applica- tlons are glven.

J

THEoREM 1. Letf(x) be a bounded real-valuedfunction deLfined on X and continuous at each point in Y. iffe ,9i', then it holds that

lim L.(f)(Y) =f(Y) n-co

uniformly wi,th respect to ye Y.

PRooF. As in the proof of Lemma 2, we take Uoe`Zf and UecZr, U :U-i, jn such a way that (yi, y2) e\ Uo and UoUcUo, where yi, y2 E Yare given in the assumption a).

Putting G(x) =F(x, y,)+F(x, y2) and f(y)

g(x,y) =f(x)- -G(y) G(x)

for xEX, yE Y, we may observe that GEjORC' and that OÅqj(U) f{; G(x) f{gC for some CÅrO and for each xEX. Therefore it follows that g,Eta' and g,(y)=O for each yE Y. In order to see that g(x, y) is continuous at each (y', y')eAy, it is enough to note that the functionsf(x),f(y), G(x) and G(y) are all continuous at each y'EYand that G(y)26(U) for each yE Y. By making use ofM=sup lf(x)I.Åq co, it is now seen that g(x, y) is bounded:

M

C. Thus Lemma3 is applicabie to g(x, y) and it results that lg(x, y)l sM+

S(U)

f(y)

.1'LII} (Ln(f)(Y)- G(y) Ln(G)(Y))=i'LM. Ln(gy)(y)==O

uniformly with respect to yEY. Since the assumption a) tells us that lim L.(G)(y) n-co =6(y) uniformly with respect to yE Y, it is now a simple matter to get the desired con-

clusion: Iim L.(f)(y)==f(y) uniformly with respect to ye Y. This completes the proof.

n-oo

For the particular case where X= Y, we give the following

CoRoLLARy 1. (Korovkin-Bohman-Lorentz [4, p. 7]) Let X be a compact Haus- dorff space and let fi()c),...,f.(x) be continuous real-valued funetions deLfined on X

satisfying the condition that there exist continuous real-valued functions ai(x),..., a.(x)

defined onXsuch that putting x

P(x, y) = ai(y)fi (x) + • • • + a.(iy)f.()c)

forxeX, yEX, it holds that P(x, y)2i:O and P(x, y)==O if and only if x= y. Let L.,

n == 1, 2,..., be a positive linear operator of C(X) into itself, where C(X) is the vector space of real-valued continuous functions on X, and assume that

lim L.(fi)(x)=fi(x) n-co

uniformly with respect to xEX, i=1,.,,, m. Then for eachfG C(X) it holds that

lim L.(f) (x) =f(x) n-co

uniformly with respect to xeX.

PRooF. By taking .S2'=C(X),X= Yand F(x, y)=:P(x, y), it is not diMcult to see that the assumptions in Theorem 1 are all satisfied. We here note only that

m

J'LM. Ln(Py)(X)=: i.E,.), "i(Y)Ji.M. Ln(fi)(X)

m

= i= , ai(Y)fi (X) = P,(x)

holds true uniformly with respect to x, yeX and consequently the condition b) is now satisfied. This completes the proof.

Theorem K is now refined as follows.

CoRoLLARy 2. Let .2`JZn'e be a vector space over R composed ofafamily of real-valued functions deLfined on R and assume that RcyO"e . Let, for each n =1, 2,..., L. be a linear

positive operator of ge into R[a,b], the vector space of real-valued functions defined on the elosed interval [a, b]. Suppose that a bounded non-decreasing function q)Ege be given such that (p2E,s7' and q)(x)=xfor xel, where I is an open interval containing [a, b], and assume further that

lim L.(1) (y) == 1, n-co

lim L.(q) (y) = Y, n-co

lim L.(q2)(y)=y2 n-co

uniformly with respect to ye[a, b], Then for any boundedfE,2`ki', continuous at each point of [a, b], it holds that

lim L.(f)(Y)==f(Y) n-co

uniformly with respect to yE [a, b].

PRooF. Letting

F(x, y) ==y2-2yq(x)+q2(x) = (Y -- q(x))2,

we may observe that F(x, y)2O and that F,(x) is continuous with respect to x at each point

in [a, b]. F,G.SZ' and F(y, y)=:O for each yG[a, b] is clear and owing to the condition

satisfied by op it is not diMcult to see that

6(ct) == infF(x, y) År O

for each ctÅrO, where inf is taken for (x, y), lx-yl;)ct and xER,ye[a, b]. Finally it holds that

lim L.(F,) (y ') = y2 -- 2yy' + y' 2 = F,(y ')

n-co

uniformly with respect to y, y' E [a, b], namely the condition b) is fu1fi11ed. Thus taking X =R and Y :[a, b], the statement is clear from Theorem 1. This completes the proof.

The periodic version of Theorem K will be obtained as follows.

CoRoLLARy 3. Let ,:f"o be a vector space over R composed ofafamily of real-valued periodic functions of period 1 on R and assume that Rc,9i'. Let, for each n=1, 2,..., L. be a linear positive operator of -`T into R[",b], the real vector space of real-valued functions deLfined on the closed interval [a, b]. Suppose that cos2nxe-0', sin27txGx`Zi'

and that

lim L.(1)(y)=1, n-co

lim L.(cos2nx)(y)=cos2zy, n-co

lim L.(sin 2n.x) (y) = sin 2zy n-co

uniformly with respect to ye[a, b]. Then for any bounded fege, continuous at each

point of [a, b], it holds that

lim L.(f)(y)"=f(Y) n-co

uniformly with respect to ye [a, b].

PRooF. Let je be the 1-dimensional torus T=RIZ and let 9be the image of [a, b]

under the canonical mapping Rgx-ÅrÅíERIZ==j?. Define f(Åí), for any fege, by the

AA identityf(Åí)=f(x) and denote ,:`Zk'=:{f; fG ,9a'}. SZ' is a vector space over R composed of a

A family of real-valued functions defined on )? and Rc,9i'. The proof that follows is divided

. mto two cases.

Case 1 : OÅq b -- a Åq 1. Letting Åí.(fA) (yA) = L.(f) (y) for every fe ge, y E [a, b], n=: 1, 2,..., it is not diMcult to see that Åí. is a linear positive operator of ,2;}' into Rr. Putting

fi(Åí, ") =2 sin2z(x -- y)

== 1 - cos 2zx cos 2zy - sin 2nx sin 2zy

for xeR, ye [a, b], we may observe that P satisfies the assumptions i), ii), iii) and iv).

AA

We also see that the condition b) is fu1filled by F and {L.}. Thus Theorem 1 tells us that

AAA

h'M Ln(f)(") =f(") n-co

uniformly with respect to "e 9, which turns out to lim L,(f) (y) =f(Y)

n-co

uniformly with respect to y E [a, b] as desired.

Case 2: 1f{gb--a. Divide the interval [a, b] into a suitable number of subintervals [ci-i, ci], i=1,..., m, eo==a, c.==b, such that OÅqci-ci-iÅq1, and it is enough to apply Case 1 for each subinterval. This completes the proof.

3. Supplement

If, in Section 1, all the assumptions except a) are assumed and if a) is replaced by the next assumptlon :

a') Yconsists of only one point yo: Y ={yo}, and it holds that lim L.(1)(Yo)=1,

n-co

lim L,(F,,) (Yo) = O, n-co

then the linear operator L. may be regarded as a linear functional and so the following theorem of Korovkin type relative to the convergence of a sequence of positive linear functionals is obtained, and at the same time it may be seen that the space X need not be uniformizable in such a case.

THEoREM 2. Let X be a Hausdorff space and let -a:' be a given veetor space over R composed of a family of real-valued functions deLfined on X and assume that Rc,20-Z'e.

Let, for each n =1, 2,..., Åqb. be a linear positive functional deLfined on -`2:' and suppose that afunction F(x) on X with thefollowing properties is given:

i') F(x) 2}i O for each xEX and F(yo) -- O for some yo E X, ii') FE3czz:-,

iii') F(x) is continuous atx=yo,

iv') 6(U) = infF(x)ÅrO for each neighbourhood U of .y,, where inf is taken for all Xe Uc,

a') F(x) is a bounded function qfxGX and it holds that liM Åën(1) == 1,

n-oo

ljm Åë.(F)=O.

n-co

Then for any boundedfG -`T', continuous at x=yo, it holds that

lim di.(f)=f(Yo)•

n-oo

PReeF. Thc pfeof of this theorem is ebtaiRed by a simplified modificatieR of that of Theorem 1. Lemma 1 is needless and Lemma2 is trivial. It is not diMcult to see directly that Lemma 3 is modified'as follows.

LEMMA 3'. Let a bounded junction f(x)Ekark be continuous at the point x=yo and assume thatf(ye)=O, Then it holds that

lim di.(f) =O.

g-ee

The proof of the theorem is now obtained by applying Lemma 3' to the function g(x) defued by

g(x) -f(x) --f(yo) • This completes the proof.

ReferenÅëes

{ l ] E. W. CHENEy, Introductiento appreximatieg theory, McGraw-Hill, New \ork, l 966.

[2] J.L.KELLEy, Generaltopology, D.VanNostrand,Princeton,l955.

[3] P.P.KoRovKIN, Linear operators and approximation theory, Fizmatgiz, Moscow, 1959 (RgssiaA)===ERglish translaticn, KindgstaR Pgblisking Corp. (IRdia), Delhi, 196e.

[4] G.G. LoRENTz, Approximation of,functions, Hoit, Rinehart and Winston, New York, l966.

Department of Computer ScieRce

Kyushu Jnstitute of Technology