Comparison of inequalities for the minimax series
via Tchebychev polynomials
F. Perez Acosta∗, J.M. Almira Picazo1
Department of Mathematical Analysis, La Laguna University, 38271-La Laguna, Tenerife, Spain
Received 25 October 1997; received in revised form 25 March 1998
Abstract
In this paper several remarkable inequalities for the product with minimax series are considered and are compared via the valuation of these inequalities on Tchebyshev’s polynomials. c 1998 Elsevier Science B.V. All rights reserved.
Keywords:Banach algebras; Inequalities for the product; Minimax series; Polynomial approximation; Tchebychev poly-nomials
1. Introduction
The minimax series of a continuous function f∈C[a; b] is dened by S[a; b](f) =P∞
n=0En(f)[a; b],
whereEn(f)[a; b] is the error of best approximation of fby algebraic polynomials of degree6n. This
series appears in the norm of the Besov space B1
∞;1[a; b] via the identity kfkB1
∞;1[a; b]=kfk∞;[a; b]+
S[a; b](f) (cf. [3]); and has been considered by the authors in several papers (cf. [1–4]).
The main goal of this paper is to consider several remarkable inequalities with the formS(fg)6F
(kfk∞;[a; b];kgk∞;[a; b]; S[a; b](f); S[a; b](g)) and compare these inequalities using the Tchebychev
poly-nomials {Tn}, for which it is known that, taking [a; b] = [−1;1], kTnk∞= 1 and S(Tn) =n.
2. Inequalities for the product
Inequalities of the form
S[a; b](fg)6(kfk∞;[a; b]S[a; b](g) +kgk∞;[a; b]S[a; b](f)) +S[a; b](f)S[a; b](g); f; g∈X (2.1)
∗Corresponding author.
1Supported by Gobierno Autonomo de Canarias.
have been proved for several choices of (; ) and for several classes of functions X⊂B1
∞;1[a; b].
The inequality (2.1) is called the (; )-inequality for the product. For example, the (3;0);(2;2) and (1;4)-inequalities for the product are valid on all B1
∞;1[a; b]: We will include the proof of these
inequalities to be selfcontained although they appear in [4, 5].
Proposition 2.1. The (3;0) and (2;2)-inequalities for the product are valid on all B1
∞;1[a; b]: Proof. Let pk and qk be the best approximations of f and g; respectively, on k: Then
E2k(fg)[a; b]6kfg−pkqkk∞;[a; b]
6kf(g−qk) +qk(f−pk)k∞;[a; b]
6kfk∞;[a; b]Ek(g)[a; b]+kqkk∞;[a; b]Ek(f)[a; b]:
But kqkk∞;[a; b]6kgk∞;[a; b]+S[a; b](g): Hence
E2k(fg)[a; b] 6 kfk∞;[a; b]Ek(g)[a; b]+Ek(f)[a; b](kgk∞;[a; b]+S
[a; b](g));
X∞
k=0E2k(fg)[a; b] 6 kfk∞;[a; b]S
[a; b](g) +S[a; b](f)(
kgk∞;[a; b]+S[a; b](g))
= kfk∞;[a; b]S[a; b](g) +
kgk∞;[a; b]S[a; b](f) +
∞S[a; b](f)S[a; b](g):
On the other hand, E2k+1(fg)[a; b]6E2k(fg)[a; b] for all k: Hence
S[a; b](fg)62[kfk∞;[a; b]S[a; b](g) +kgk∞;[a; b]S[a; b](f) +∞S[a; b](f)S[a; b](g)]
and the (2;2)-inequality has been proved.
Proceeding in a similar form and taking into account that kqkk∞62kgk∞ we obtain
E2k(fg)[a; b] 6 kfk∞;[a; b]Ek(g)[a; b]+ 2kgk∞;[a; b]Ek(f)[a; b];
X∞
k=0E2k(fg)[a; b] 6 kfk∞;[a; b]S
[a; b](g) + 2
kgk∞;[a; b]S[a; b](f):
Hence
S[a; b](fg) 6 2
kfk∞;[a; b]S[a; b](g) + 4
kgk∞;[a; b]S[a; b](f);
S[a; b](fg) 6 2kgk∞;[a; b]S[a; b](f) + 4kfk∞;[a; b]S[a; b](g):
Thus
S[a; b](fg) 6 3[kgk∞;[a; b]S[a; b](f) +kfk∞;[a; b]S[a; b](g)]
which is the (3;0)-inequality for the product.
Denition 2.2. Let X⊂B1
∞;1[a; b] be a class of functions with S[a; b](f)¡∞. We dene
(X) ={(; )∈R2: the (; )-inequality which are valid on X}:
It is clear that (X) is a convex set (e.g., (2:5;1) =1
2[(3;0) + (2;2)]∈(B 1
∞;1[a; b])) and that if
(; )∈(X) then (; ) + [0;+∞)2⊂(X).
Theorem 2.3. (1) (0; )∈= (B1
∞;1[a; b]) for all ¿0.
(2) (0;6)∈(C0[a; b]); where C0[a; b] ={f∈B1
∞;1[a; b]:∃x0∈[a; b]; f(x0) = 0}: Proof. We may assume that [a; b] = [0;1]: Let ¿0 and let f(x) =x+c, (c¿0). Then
S[0;1](f2)¿E
0(f2)[0;1]=12 +c;
S[0;1](f)2=1 4:
It is clear that there exits some c¿0 such that 1 2 +c¿
1
4. The rst claim has been proved.
Set S[a; b]
m (f) = Pm
K=0(E2k(f)[a; b]+E2k+1(f)[a; b]). It is clear that S[a; b](f) = limm→∞Sm[a; b](f). On
the other hand, E2k+1(f)[a; b]6E2k(f)[a; b] for all k∈N, so that Sm[a; b](f)62 Pm
k=0E2k(f)[a; b] for all
m. The idea to prove the second claim is take bounds for E2k(fg)[a; b] when f; g∈ C0[a; b].
We remember that if f; g∈ C0[a; b] then, denoting by pk and qk the best approximations to f and g, respectively, with polynomials of degree at most k, then
kfk∞;[a; b] 62E0(f)[a; b]; |q0|6E0(f)[a; b];
kqkk∞;[a; b] 6Ek(g)[a; b]+kgk∞;[a; b]
so that
E0(fg)[a; b] = E0(f(g−q0) +q0f)[a; b]
6 E0(g)[a; b]kfk∞;[a; b]+|q0|E0(f)[a; b]
6 2E0(g)[a; b]E0(f)[a; b]+E0(g)[a; b]E0(f)[a; b]
= 3E0(g)[a; b]E0(f)[a; b];
E2k(fg)[a; b]6kfg−pkqkk∞;[a; b] =kf(g−qk) +qk(f−pk)k∞;[a; b] 6Ek(g)[a; b]kfk∞;[a; b]+kqkkEk(f)[a; b]
62E0(f)[a; b]Ek(g)[a; b]+ (Ek(g)[a; b]+kgk∞;[a; b])Ek(f)[a; b]
62(E0(f)[a; b]Ek(g)[a; b]+Ek(f)[a; b]Ek(g)[a; b]+E0(g)[a; b]Ek(f)[a; b]):
Hence
S[a; b]
m (fg)66E0(f)[a; b]E0(g)[a; b]
+ 4 E0(f)[a; b]
m X
k=1
Ek(g)[a; b]+E0(g)[a; b]
m X
k=1
Ek(f)[a; b]+
m X
k=1
Ek(f)[a; b]Ek(g)[a; b] !
:
66S[a; b]
m (f)S
[a; b]
m (f):
Corollary 2.4. (Cx
0[a; b]; S
[a; b]) is a Banach algebra with the product f◦g=1
6fg; where
Cx
0[a; b] ={f∈B
1
∞;1[a; b]=f(x0) = 0}. Proof. It follows from Theorem 2.2.
Theorem 2.5. (0;4)∈(C+
0[a; b]); where C0+[a; b] ={f∈B∞1 ;1[a; b]:f¿0 on [a; b] and ∃x0∈
[a; b]; f(x0) = 0}:
Proof. Remember that if f∈C0[a; b] then kfk∞;[a; b]62E0(f). Hence
E2k(fg)[a; b]6kfk∞;[a; b]Ek(g)[a; b]+Ek(f)[a; b](kgk∞;[a; b]+Ek(g)[a; b])
62E0(f)[a; b]Ek(g)[a; b]+ 2E0(g)[a; b]Ek(f)[a; b]+Ek(f)[a; b]Ek(g)[a; b];
E2k+1(fg)[a; b]+E2k(fg)[a; b]
62(2E0(f)[a; b]Ek(g)[a; b]+ 2E0(g)[a; b]Ek(f)[a; b]+Ek(f)[a; b]Ek(g)[a; b])
64(E0(f)[a; b]Ek(g)[a; b]+E0(g)[a; b]Ek(f)[a; b]+Ek(f)[a; b]Ek(g)[a; b]):
Hence
Sm[a; b](fg)6E0[a; b](fg) +E1[a; b](fg)
+ 4 E0[a; b](f)
m X
k=1
Ek[a; b](g) +E0[a; b](g)
m X
k=1
Ek[a; b](f) +
m X
k=1
Ek[a; b](f)Ek[a; b](g)
!
:
But for f; g∈C+
0[a; b] the inequality
E0[a; b](fg) =1
2kfgk∞;[a; b] 6 1
2kfk∞;[a; b]kgk∞;[a; b] = 2E [a; b] 0 (f)E
[a; b] 0 (g)
holds. Hence
Sm[a; b](fg)64Sm[a; b](f)Sm[a; b](g)
for all m∈N. The proof follows taking limit for m→ ∞.
Corollary 2.6. (1;4)∈(B1
∞;1[a; b]). Proof. Letf; g∈B1
∞;1[a; b] and setF=f−min{f(t):t∈[a; b]}, G=g−min{g(t): t∈[a; b]}:Then
F; G∈C+
0[a; b] and we can apply Theorem 2.5. Hence
4S[a; b](f)S[a; b](g) = 4S[a; b](F)S[a; b](G)
¿S[a; b](FG) =S[a; b](fg)
−(minf)g−(ming)f)
¿S[a; b](fg)
− |minf|S[a; b](g)
Thus
(; )-inequalities for the product have been used to
• Obtain bounds of S[a; b](f) for several special functions f. • Prove theoretical properties of the Besov space B1
∞;1[a; b] as an algebra.
One other inequality for the product which is valid on B1
∞;1[a; b] is the square root’s inequality
(cf. [7] for a proof)
S(fg)62[kfk∞;[a; b]S[a; b](g) +kgk∞;[a; b]S[a; b](f) +qkfk∞;[a; b]kgk∞;[a; b]S[a; b](f)S[a; b](g)]:
(2.2)
It is also possible to obtain inequalities for the product which depend on the smoothness properties of the functions.
Proof. From the well known theorem of D. Jackson which relates the errors of best polynomial approximation of a continuous function to its moduli of continuity it follows that (cf. [2] for a proof)
3. Comparison of inequalities for the product
Set [a; b] = [−1;1] and let us denote byTn thenth Tchebyshev’s polynomial of the rst kind. Then
Ek(Tn)[−1;1]= 1 ifk¡n andEk(Tn)[−1;1]= 0 if k¿n. Hence S[−1;1](Tn) =n. Between the classical sets
of orthogonal polynomials,{Pn(d)}∞n= 0, {Tn}∞n= 0 is the only for which it is already known the exact
value of S[−1;1}(P
n(d)). Hence it is quite natural to use this information to compare inequalities
for the product.
Denition 3.1. Let
S[−1;1](fg)6F
1(S[−1;1](f); S[−1;1](g);kfk∞;[−1;1];kgk∞;[−1;1]); (3.1)
S[−1;1](fg)6F2(S[−1;1](f); S[−1;1](g);kfk∞;[−1;1];kgk∞;[−1;1]) (3.2)
be two inequalities for the product. We say that
• (3.1) is T-better than (3.2) if F1(n; n;1;1)6F2(n; n;1;1); ∀n
• (3.1) is better than (3.2) in the Tchebychev’s sense if F1(n; m;1;1)6F2(n; m;1;1); ∀n; m
We will use the following notation:
(3:1)T
7→(3:2) means ‘(3:1) is T-better than(3:2)’
(3:1)T
=(3:2) means ‘(3:1)7→T(3:2) and (3:2)7→T(3:1)’
Remark 3.1. We will only compare inequalities for the product of the form (3:1) in this paper. For example, the Jackson’s type inequalities are not of this form and will not be compared here, although it is clear that Denition 3.1 could be extended to consider inequalities which depend also on the derivative of the functions f; g and then use that the Tchebychev’s polynomials satisfy (cf. [8] for a proof)
Tn(k)
∞;[−1;1] =
n2(n2−12)· · ·(n2−(k−1)2) Qk
s= 1(2s−1)
:
Theorem 3.1. The following assertions are equivalent:
(1) The (1; 1)-inequality is better than the (2; 2)-inequality in the Tchebychev’s sense.
(2) (1; 1)-inequality T
7→ (2;2)-inequality.
(3) 1−260 and 2(1−2) + (1−2)60.
Proof. The (1; 1)-inequality is better than the (2; 2)-inequality in the Tchebychev’s sense if and
only if
(1−2)(n+m) + (1−2)nm60 for all n; m∈N: (3.3)
Suppose that (3:3) holds and set n=m. Then 2(1−2) + (1−2)n60 for alln∈N and it follows
that
We use this to write (3:3) in the form
1−2
2−1 6min
nm
n+m: n; m∈N
=1
2: (3.5)
The equivalence between the rst and third assertion follows from (3:4) and (3:5). On the other hand, we observe that the minimum min{nm=(n+m): n; m∈N} is attained on the diagonal n=m
and the equivalence (2)⇔(3) follows.
Corollary 3.2. Let X be a class of functions contained in B1
∞;1[a; b]. Then T
7→ is an order relation
on (X).
Proof. (1;2)4(1; 2) if and only if 1−260 and 2(1−2) + (1−2)60 is an order relation
on R2.
Corollary 3.3. (1) For all t∈[0;1]; (3;0) T
7→t(3;0) + (1−t)(0;6):
(2) (3;0)T
7→(2:5;1)7→T(2;2)7→T(1;4)7→T (0;6):
Proof. Set (1; 1) = (3;0) and (2; 2) =t(3;0) + (1−t)(0;6) = (3t;(1−t)6). Then
1−2= −(1−t)660 and 2(1−2) + (1−2) = 0
and the rst assertion follows. Second assertion is also clear.
Proposition 3.4. (1) Square root’s inequality T
= (3;0)-inequality
(2) The square root’s inequality is better than the (3;0)-inequality in the Tchebychev’s sense.
Proof. The rst assertion is trivial. For the second assertion it is enough to observe that 2√nm6n+
m for all n; m. Hence 2(n+m+√nm)62(n+m) +n+m= 3(n+m):
References
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[2] E.W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966.
[3] Z. Ditzian, V. Totik, Remarks on Besov spaces and best polynomial approximation, Proc. Amer. Math. Soc. 104 (1988) 1059–1066.
[4] N. Hayek, F. Perez Acosta, Boundedness of the minimax series of some special functions, Rev. Acad. Canaria de Ciencias VI (1) (1994) 119–127.
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