ELSEVIER Journal of Computational and Applied Mathematics 83 (1997) 5549

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

On certain Banach spaces in connection with interpolation theory

Francisco Prrez Acosta Departamento de An(disis Matemdtico, Universidad de La Laguna, 38271 La Laguna, Spain

Received 27 February 1996; received in revised form 14 March 1997

Abstract

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.

Finally, certain inequalities for the minimax series of a product are obtained which permit us to define a product over the space (C0,S). With such a product (Co,S) is a Banach Algebra.

Keywords: Interpolation polynomials; Error series; Minimax series; Convergence; Banach spaces; Banach Algebras

AMS classification: 65D05; 46B99; 46J10

1. Introduction

Let ~ = {xSxl n), . ,x~ n), ..(n)~..(n) • " xi ~-~j , (i#J)},~=0 a sys tem o f interpolation nodes on [a,b]. For each f E C[a, b], let us consider the interpolation polynomial o f degree n at most, denoted by Pn(f, r) such that

pn(f,r)(x~n))= f(x~ n)) ( j = 0 , 1 , . . . , n ) .

It is well known that the Lagrange interpolation formula for Pn(f, r) is

n

Pn(f , "r)(X)= E f(x~n))g ")(x)' j=0

where Lj(x) are the fundamental Lagrange polynomials given by the expression

nn(x) Lj(x) = H,(x) ---- H (x - xJ n))

A.j~(n ) ~ lTt t~(n ) ) ln l A, j j = 0 (x

and H'n(x ) denotes the usual derivative o f I-In(x).

0377-0427/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PII S 0377-0427(97)00060-5

56

Furthermore,

Ilp.(f)ll~ <~A.llfl[~, V f EC[a,b],

where A, is the Lebesgue constant given by

A, = ~j:0 ILj(x)} "

F. P~rez Acosta/Journal of Computational and Applied Mathematics 83 (1997) 5549

and

Do(z) = { f EMo " T( f ) < c~}.

Only one fixed set z is considered in the first three sections, so that the • can be omitted there. It is a simple matter to ckeck the next proposition:

This constant is the smaller one satisfying the above inequalities for all continuous functions on [a,b].

Set

T(z)( f ) = ~ Ilf - pk(f, z)l[~. k=0

Clearly, T(z)(f) could be infinity for some functions f . If T(z)(f) < oo then Pk(f, z) converge uniformly to f on [a, b] as k tends to infinity.

A condition which implies the convergence of T(z)(f) in terms of the Lebesgue's constants and the minimaxs of f ,

E k ( f ) = inf [ I f - PII~, k = 0 , 1 , 2 , . . . pE Ilk

is the following:

lim Ak+lEk+l(f) < 1, k - ~ AkEk(f)

where /-/k denotes the set of all polynomials of degree less than or equal to k. This fact is a consequence of the well-known estimate

I I / - Pk(f ,z) l l~ ~<(1 + Ak)Ek(f).

Let us introduce the following definitions.

Definitions. Let Y0 be a fixed point in [a, b]. We will deal with the following spaces:

Mo= { f EC[a,b] : f ( yo )=O}

F. Pkrez AcostalJournal of Computational and Applied Mathematics 83 (1997) 5549 57

Proposition 1. (Do, T) is a normed vector space.

In the same way,

Proposition 2. (Do, II- II ~) is dense in (Mo, [I. II ~ )

Proof. By the Walsh Theorem [2 p. 121], there exists for each n a polynomial p, such that

Pn(Yo) = f(Yo)

and

1 I I f - Pnll~ <--. n

Then

{p ,}CD0 and {p,}l l l t°~fasn---~oo. []

In this paper we will characterize the convergence in the space Do and we will show that (Do, T) is a Banach space.

Some results in order to characterize the convergence in the norm T will be given in Section 2. In this sense, the following definition of convergence of series will play an important role. Namely, let {f}j~j be a family of functions in Do.

We say that the series {T(J))}j~j converges uniformly in J iff for each e > 0 there exists a k0(e)6 [~ such that

I I f - pk(£)ll~ < e, (VjeJ) . k >~ ko

In Section 3 we prove that (Do, T) is a Banach space. On the other hand, for a continuous function on [a, b], and for each k = 0, 1,2, . . . , we consider

the minimax:

Ek( f ) = inf I l f - p l l ~ pE/T,

and the minimax series (3O

S ( f ) = ~_, Ek(f) . k=0

We introduce the space Co which is constituted for all functions f E M o such that S ( f ) < co. S is a norm on Co and it is already known that (Co, S) is a Banach space, for details see [4]. However, another proof of this result is given in Section 4. More precisely, we show that the space Co is the sum of all the spaces Do obtained from systems of interpolation nodes on [a, b]. This is one of the main result in this work. It establishes a connection between the best polynomial approximation and interpolation polynomials.

In Section 5, an important result is shown, namely, that Co is a Banach Algebra with respect to a conveniently defined product.

58 F. Pkrez Acosta/Journal o f Computational and Applied Mathematics 83 (1997) 5 5 4 9

2. Some results concerning to the convergence in the norm T

Proposition 3. I f {f~} ~ f then there exists ( c , ) c ~ such that 9, =fn~-Cn converges uniformly to f on [a, b].

Proof. Let e > 0. Then there exist nee [~ such that T ( f ~ - f ) < e,(Vn >/n~), and set c, = P o ( f - f,). Then

IIf,-f+Cnll <<.T(fn-f)< , (Vn

i.e.,

g n = f n + C n II IJ~ f . []

Corollary 4. I f {f~} CD0 and {f~) ~ f , f EDo then {fn} II I1~ f on [a,b].

The next theorem characterizes the convergence in the space (Do, T). This convergence is stronger than the uniform convergence.

Theorem 5 (Characterization of convergence). {f~} cD0 is convergent to f EDo in the norm T iff (i) {f,} converge uniformly to f on [a,b],

(ii) The series {T(f~)}~= 1 converges uniformly in ~.

Proof. "=~": (i) It follows from Corollary 1. (ii) Let e > 0 . Then there exists n0(e)E ~ such that

OQ

T(f~ - f ) = ~ - ~ Ilfn - f - pk(f~ - f ) [ l ~ <~, Vn>~no(e). k=O

On the other hand, for f l - f , T ( f : - f ) < cxD, there exists ml E N such that

o~

IIf - f - Pk(f~ - f)ll < k >~m:

For f2 - f , T ( f 2 - f ) < cxD, there exists a m2 E N such that

IIA - f " p k ( f 2 - f)ll < e. k >~m2

Repeating the argument, one obtain that for f n 0 _ 1 - - f , T ( f ~ o _ 1 - f ) < :xD, there exists m,0_l E N such that

OQ

IIf o- - f - pk(f~o-~ - f ) [ [~ < e. k >~ m,o - i

F Pbrez AcostalJournal of Computational and Applied Mathematics 83 (1997) 5549 59

IIA - f l l o ~ <

Then

Taking v = max(ml,mz,...,m,o_l), we have

~ l l f . - f - p k ( f ~ - f ) l [~ <~, (VnEN), k >~v

that is, the series T ( f . - f ) converges uniformly in ~. This is equivalent to say that the series T(f~) converges uniformly in [~.

" ~ " : First, observe that

Ilf. - f - pk(f~ - f)[[o~ ~<( 1 + Ak)llf. - f l l ~ ,

and let e > 0. Then by (ii) the series T(f . - f ) converges uniformly in [~, which implies that there exists ko(e)E [~ such that

I l f . - f - p k ( f . - f ) [ [ ~ < e , (VnE~) . k~>ko

Then

T(fn - f ) = ~ [[f~ - f - Pk(f~ - f)[Io~ + ~ IIf. - f - pk(f. - f ) l l ~ k<ko k~>ko

k<ko

On the other hand, by (i) there exists no E N such that

Ek<ko(1 + A k ) ' (Vn>>.no).

T ( f ~ - f ) < (k~< ( l + A k ) ) [lf~- f l [ o ¢ + e < e + e = 2 ~ , (Vn>~no). []

Theorem 6 (Characterization of Cauchy sequences). {f~} c Do is a Cauchy sequence with the norm Ti f f

(i) {f~} is a Cauchy sequence in the uniform norm, (ii) The series {T(fz - f~ )}~= l converges uniformly in ~2.

Proof . "=~". (i): Let {f~} be a Cauchy sequence in (Do, S), then for each e > 0 there exists v(e)E such that

T ( f m - f ~ ) < e, (Vm, n>~v(e)).

Then

Ilfm - f~ - po( f . - f . ) l l - - [[fm -- f~ -- Cm,.ll~ 4 T ( f . - f~) < ~.

for x = Yo,

I(/m - fn)(Y0) -- Po(f,, -- f~)l = Icm,.I ~ I[/m - f~ - cm,. l l~ < ~.

60 F. P~rez Acosta/Journal o f Computational and Applied Mathematics 83 (1997) 55~59

Thus,

ICm,,I < e , (Vm, n>-v(e ) ) .

Therefore,

IIf" - Al[o~ ~< Ilf,. - f ~ - c " , . l l ~ + Ic",nl < ~ + ~ .

(ii) Let e > O. Then,

1 Vm, n>~n~, T ( f m -- fn) < ~8

and set J = {(re, n): O<<.m,n<~n~}. On the other hand, for each j = ( m , n ) E J , T ( f " - f . ) < co, then there exists kj

Ilf,. - L - Pk(fm -- f~)lloo < ½e. k~>kj

I f r = m a x { n ~ , k ( m , n ) : ( m , n ) E J } , then Case 1: I f m, n >/n~, then

IIf" - f . - pk ( f , . - f ~ )N~ <~ ~ IIf" - J. - p k ( f " - f . ) l l~ < ½~ < ~. k >~r k ~n~

Case 2: I f m,n<<.n~, and j = (re, n ) then

1 Y~ IIf,. - f . - p k ( f , . - f.)[[oo <~ Y ~ Ilfm - L - p k ( f " - f . ) l l~ < ~ < 5. k>~r k>~ky

Case 3: If m~n~, n <n~, and j = (n,n~) then

Y ~ [if,. - f . - P k ( f " - f.)l[o~ ~< Y~ II/m - f . ~ - p k ( f " - f.~)ll~ k ) r k >~r

+ ~ I lL- L - P(f.~ - f~)l[o¢ k >~r

k >~n~

1 1

k>~kj

Case 4: I f m < n~, and n/> n~. This case is similar to ease 3. Hence, anyway

liT" - f . - Pk(fm -- f . ) l l~ < e, Vm, n. k>-r

" ~ " : Let e > 0. From (ii), there exists koE N satisfying,

1 (Vm, n). IlL - f . - P k ( f " - f . ) l l~ < ~ , k>~k0

F. Pkrez Acosta/Journal of Computational and Applied Mathematics 83 (1997) 5549

On the other hand, by (i) there exists n~ E [~ such that

Ilfm - f.ll~ < 2~k<ko(1 + Ak)' Vm, n~n~.

Consequently,

T(fm - f~) = ~ IlL - fm - - pk(f . - fm)ll~ + Y~ IIf. - fm - - pk(f . - f ) l l ~ k~>ko k~>ko

( ) , 1 , < ~- -~( l+Ak) I I f . - / l l ~ + ~ < ~ + ~ - E , Vm, n>n~. [] k~>ko

Remark 7. If {f.} is a Cauchy sequence in (D0,T) then {f.} is bounded in the norm T.

61

3. The Banach space (Do, T)

A central result of this paper is to show that the space (D0,T) is a Banach space. In order to make this, some other previous results are needed.

Proposition 8. I f {f.} CDo, f , IEl_~ f , and T(f~)~<M(M > 0 ) n = 1,2 . . . . , then T( f )~<M and

f EDo.

Proof. Let P k ( f ) be the interpolation polynomial of f and Pk(f . ) the kth interpolation polynomial of f . on Pk, then

IIf - p ~ ( f ) l l ~ ~< IIf - fnll~ + IIf~ -- Pk(L)I I~ + }{Pk(f~) - p k ( f ) l l ~

~< IIf - f . l l~ + IIf. - pk( f . ) l l~ +Akllf - f . l l~

= (1 + Ak) l l f - fnll~ + IlL - pk(f.)}l~.

Let e > 0. For each k there exists nk(nk >~nk-l) satisfying

I I f - f . l l ~ < 2k+ l (Ak+l ) , (Vn>nk) .

Then

8 IIf - p k ( f ) l l ~ ~< ~ + IIf~ - pk(f~)ll~, (Vn > nk).

62 F. Pdrez Acosta/Journal of Computational and Applied Mathematics 83 (1997) 55-69

Hence, m

~ l l f - pk(f)lloo ~ ~ 2---~-r + E IIA- p~(f.)lloo k=O k=O k=O

+ IIA - pk(L)llo~ k=0 k=O

=T(fn) + e~<M + e, ( n > n z ) ,

which implies that m

[ i f - pk( f ) [ [~ ~<M + e, k=O

Thus, one has

I I / - pk(f)ll~ ~<M, k=0

(VmE ~,Ve > 0).

VmE~, and T ( f ) ~<M. []

Theorem 9. (Do, T) is a Banach space.

Proof. Let {fn} be a Cauchy sequence in (Do, T). Thus, by Theorem 6 it follows that {f~} is a Cauchy sequence in (Mo, [I I1~), which is a Banach space. Then, there exists f E M o such that

f~ ii tl~ f . By Remark 7, there exists a positive constant M satisfying that T(f~)~<M. Taking into account

Proposition 8, T ( f )~<M < c¢. Then f E D o . Now will prove that the series T(f~) converges uni- formly in ~.

i Vm, n >>. no. In particular, Let e > 0 be. Then there exists noE [~ such that T ( f ~ - fro)< ie

T(fn - f~o) < ½e,

and T(f~o) < c¢. Thus, there exists koE N such that

E IIAo- Pk(Ao)ll~ < ½e. k~>ko

On the other hand,

IlL - Pk(L)II~ ~< Ilf~ - L o - pk(f~ - f~o)[[~ + IlAo - Pk(Lo)II~. Hence,

1 1 (Vn>~no). IIf~ - pk(L)l l~ ~<T(A - Ao) + ~ IIAo - pk(f~o)H~¢ < i ~ + i e = e,

Put J = { 1,2 . . . . , no - 1}. For each j E J there exists kj such that

k>~kj

F Pdrez AcostalJournal of Computational and Applied Mathematics 83 (1997) 5549

then, taking r = max{k0,kl,...,k,0_l }, it follows that

Z I1£. - pK£)lloo < e, (Vj). k>~r

Hence, by Theorem 5, f, T f and (D0,T) is a Banach space. []

63

4. The relation between the minimax series and the interpolatory series

A space of type (Do, T) is called a space of interpolation series. Each system of interpolation nodes,

r (n) (,) . X~nn),(X~")7~X} ")) ( i # j ) ; n = 0 , 1 , 2 , ..} T = ~[X 0 , X 1 , . . ,

on [a, b] determine a Banach space. Let J be the set of all interpolation systems nodes on [a, b]. We establish the next result.

Proposition 10. (a) S( f )<~T(z) ( f ) and D0(z)C Co (V'cEJ), (b) S ( f ) = min,ej T(z)(f) and Co = U~ej D0(z).

Proof. (a) This result follows from

Ek(f)<~[lf--Pk(f)[Ioo (k=O, 1,2,...),

and S( f )<~T(z)( f ) (VzEJ) . If fCDo(z) then fCMo and T ( , ) ( f ) < o o; hence, S ( f ) < c ¢ , i.e., f ECo.

(b) Let f c Co. It is well known that the best polynomial approximation is given by interpolation polynomials. Let -c be a system of interpolation nodes of the best approximation polynomials. Then

Ek(f) = Ilf - pk(f)lloo, (k = 0, 1,2,3, . . .)

and T ( z ) ( f ) = S( f ) , hence, f EDo(z). []

The next result gives a new proof of the fact that (Co, S) is a Banach space. We define the space U~cJ D0(z) as the set of all functions f E Co such that there exists

{"ci}i°~=lCJ and f~ ,EDo(z i ) ( i cN)

oo oo

with f=~--~.f~, and y~T(z i ) ( f , , )<oc , i = 1 i = 1

and we set

T ( f ) = inf T('ci)(f~,)" i = 1

64 F. Pdrez AcostalJournal of Computational and Applied Mathematics 83 (1997) 55~9

T, oG There exists { i}i=l C J and f~, EDo(zi) (iE ~) with

oo } f = ~-'~f~, and y~ T('ci)(f~,) < c~ .

i=1 i=1

Proposition 11. (U~cjDo(v),T) is a Banach space.

Proof. It is (i) T(f)~>O; that

easily checked that T satisfies the axioms of a norm in UzcjDo('~). In fact oc oo T ( f ) = 0 implies that for all e > 0 and { ~};=l c J , there exists {f*e,k}~=l,k=l such

o o

f = Y~. f~,,k i=1

T(zi)(Zi,k)=O(iEt~,kEt~), t h en f~ , , k=0 ( iE~ ,kE t~ ) and f = 0 .

(ii) If 2 = 0 then T ( 2 f ) = T(0) = I IT(f)- If 2 # 0 then f = ~i~1 T(zi)(f~, ) if and only if 2 f = Ei°~__l 2T(z;)(f~,), and

T ( 2 f ) = inf ~ T(z~ )(2f~, ) = inf 12[ ~ T(z~)(f~, ) i=1 i=l

O<3

- - I , t l i n f y~. T ( ' c i ) ( f T i ) = I 2 I T ( f ) .

i=1

(iii) For each e > 0, there exist {fzi} and {g,,} such that

f = g= i=1 i=1

1 T(zi)(f~,) < T ( f ) + ½~, ~ T(zi)(g~,) < T(g) + ~. i=1 i=l

and o o

i=1

Then

T(zi)(f~,,k) < ~. k=l i=l

Since e was arbitrary,

T(zi)(f~,k) = O. k=l i=1

Thus,

F. P~rez AcostalJournal of Computational and Applied Mathematics 83 (1997) 5549 65

Then

oo oc o~

T ( f + 9)<.% ~ T(z~)(f~, + g~,)<~ ~ T(z~)(f¢,) + ~_,T(z~)(g~,)<~T(f) + T(g) + e. i = l i=1 i=1

Since e > 0 was arbitrary

T ( f + g) <~ T ( f ) + T(g).

Let us prove that (U, ~j D0(Q, T) is complete. It is sufficient to prove that every absolutely convergent series is convergent. See, for example, [3, p. 21].

Let ~ ,~ l T ( f~ )< c¢; for all nE t~, there exists {f,,,}i~t such that

o~

s . = i=1

and

< r ( f , ) + i = t

Then

T(z,)(f~,,,) = T(zi)(f¢,,,) ~< T(f , ) + ~-g < oo. n = l i=1 i = l n = l n = l n = l

Hence, for all zi we get

~--~ T(zi)(f~,,.) < c¢, n = l

Do(r;) is a Banach space. Then there exists g,, CDo(zi) such that

n = l

Hence,

i=1 i=1 n = l

then

o o

g=~-~g¢ , cUDo(Q. i=1 " r E J

66

Finally, we prove that g = )--]-,~1 f~:

N oo N o~ N oo

n=l i=1 n=l i=1 n=l i=l

=~-~.g,,- fi,,~= g z , - f* , ,n • i=1 i=1 n=l i=1

F P~rez AcostalJournal of Computational and Applied Mathematics 83 (1997) 55~59

) =ET('~i) f.,,~ <~ f¢,,. <~ ~ ~--~T(zg)(f¢,,~), i=1 \ n = N + l i=1 n =N+l n =N+l i=1

which tends to zero as N tends to infinity by the convergence of ~ n ~ l T ( f n ) . [ ]

Theorem 12. Co - - U~EjDo( 'C) and S ( f ) ---- T ( f ) , ( f E C o ) .

Proof. By proposition 10, algebraically Co = U, E/D0(z). Let f E Co. If f = ~ l f, with f, ED0(Ti), then

(x) oo

S(f)<~ y ~ S(fi,)<~ ~-~ T(T~)(fi,). i=1 i=1

Hence, S ( f ) <~ T( f ) . On the other hand, if f E C o there exist z E J such that fEDo('C) and S ( f ) = T(~)(f). It is a

consequence of the called Alternation Theorem in [1, p. 75]. Then

r ( f ) <~ T(~)(f) ~< S( f ) .

Thus,

S ( f ) =- T( f ) . []

Corollary 13. (C0,S) is a Banach space.

Proof. Note that Co is the Banach space given by Co = U~ejD0(z), sum of the Banach spaces Do(z), "cEJ. []

The space Co obtained using the minimax series to define its norm, is the sum of all the in- terpolatory spaces Do(z). This fact is interesting because the interpolation is linear and the best approximation is not linear in general.

Hence,

T g - E f ~ ~< ~--~T(Ti) OT i - - EfTi,n n=l i=1 n=l

F. P~rez AcostalJournal of Computational and Applied Mathematics 83 (1997) 55-69 67

5. The Banach Algebra (C0,S)

In this section we establish some inequalities involving minimax series which allows us to define a product in Co giving that (Co, S) is a Banach Algebra.

Lemma 14. I f f has a zero in [a,b], then Ilfll~ <<.2s(f).

Proof. Note that E o ( f ) = IIf- cll~ <~S(f), if f ( t ) = 0, for any tE[a,b]. Then

Icl = I f ( t ) - cl ~< IIf - ello~ <.S(f).

Hence,

Ilflloo ~< IIf - cllo~ + Icl <.%s(f) + s ( f ) = 2 S ( f ) . [ ]

Lemma 15. I f pk is the best uniform approximation to f on Hk then (a) IlPklt~ ~<211f[l~, (b) IlPkll~ <<.S(f) + [If lion.

Proof. (a) Ilpkll~ ~ l l f - p , llo~ + I l f l l~ ~ l l f l l ~ + I l f l l~ = 2[ [ f l l~ . (b) Ilpkllo~ ~< I l f - pkllo~ + I l f l l~ <.s( f ) + I l f l l~ . []

Theorem 16 (Inequalities for the product). (a) S(fg)<<.2[llfll~S(g) + [Igll~S(f) + s(f)S(g)], (b) s(fg)<~ gllfl l~S(g) + 2llgll~s(f), (c) s(fg)<<.2llfll~S(9) + 4llgll~S(f), (d) S(fg)<~3[llfll~s(g) + Ilgllo~s(f)].

Proof. (a) Let Pk and qk be the best approximations o f f and g on Hk, respectively. Then

E2k(fg) ~< I[fg -- Pkqkllo~ <~ [[fg - f qk + f qk - Pkqkllo~

I l f (g - qk) + ( f - Pk)qkllo~ ~ Ilfllooll0 - qkllo~ + I l f - pklloollqkllo~

< IlfllooEk(g) -4- Ek(f)llqkll~

IlfllooEk(g) + Ek(f)(llgllo~ + S(g)).

Hence,

f~O

~-~E2k(fg) <~ ]lfllooS(g) + S(f)(llgll~ + S(g)) k=O

= [[fllooS(g) + [[g[[~S(f) + S(f)S(g) .

6 8

On the other hand,

E2k+l(fg) <~E2k(fg),

Thus,

F. P6rez Acosta/Journal of Computational and Applied Mathematics 83 (1997) 5549

O O O ~

E2k(fg). k=0 k=0

S(fg) <~2[[If[l~S(g) + Ilgllo~s(f) + s(f)S(g)] .

(b) Arguing in a similar way and taking into account that

IIq~llo~ ~<211011~,

we obtain

E2k(f9) << . Ilf[[~Ek(g) + Ek(f)llqkll~ <~ Ilfllo~Ek(g) + 2HgllooEk(f).

Then O O

Z E2k(fg) <<. Ilfllo~s(o) + 2llgll~S(f), k=0

and O 0 (X3

E2k+,(fg) <-% ~ E2k(fg) <<. Ilfll~S(g) + 211gllooS(f). k=O k=O

Hence,

S(fg) <~211fllo~S(g) + 411gllo~S(f). (c) Interchange f and g in (b). (d) S(fg)<~211fll~S(g) + 41[gll~S(f) and S(Tg)~<211fllooS(g) + 411gllooS(f). Thus,

2S(fo)<<. 6[llfllooS(g) + IIoIIo~S(f)]. []

Corollary 17. I f f and g have a zero on [a, b] then

S( fg) <.% IOS(f )S(g).

Proof. By Lemma 14, it follows that Ilfll~ ~<2s(f), Iloll~ ~<2s(0). Then

S(fg) <~ 2[llfllo~S(g) + Iloll~S(f) + S(f)S(g)]

<~ 2[2S(f)S(g) + 2S( f )S(g) + S(f)S(g)] = IOS(f)S(g). []

Remark 18. It would be desirable to have an estimate of the type S(f9)<.KS(f)S(g) , for all con- tinuous functions f and g with a zero on [a, b], where K is best possible. This extremal problem is open.

F. Pdrez Acosta/Journal of Computational and Applied Mathematics 83 (1997) 55~59 69

Let 7 = inf{ cER+: S(fg)<~cS(f)S(g), for all f , gECo}. Then 7~< 10.

Theorem 19. (Co, S) is a Banach algebra with the product f .g = &.

Proof. Note that S( f .g )= ~S(fg)<~S(f)S(g). []

Final remark. The Banach Algebra (Co, S) is a commutative Banach Algebra without unit, the only constant function that belongs to Co is the function identically zero.

References

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vol. 54, American Mathematical Society, Providence, RI, 1982. [6] P.J. Laurent, Approximation et optimisation, Hermann, Paris, 1972. [7] F. P6rez acosta, P. Gonzfilez Vera, Approximation and convergence in the norm induced by the minimax series,

Preprint, 1995, submitted.