Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:
Journal of Computational and Applied Mathematics 99 (1998) 319–325
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
c 1998 Elsevier Science B.V. All rights reserved.
the valuation of these inequalities on Tchebyshev’s polynomials.
Keywords: Banach algebras; Inequalities for the product; Minimax series; Polynomial approximation; Tchebychev polynomials
1. Introduction
The minimax series of a continuous function f ∈ C[a; b] is de ned by S [a; b] (f) = ∞
n=0 En (f)[a; b] ,
where En (f)[a; b] is the error of best approximation of f by algebraic polynomials of degree 6 n. This
1
1
series appears in the norm of the Besov space B∞;
[a; b] = kfk∞; [a; b] +
1 [a; b] via the identity kfkB∞;
1
[a; b]
S (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 form S(fg) 6 F
(kfk∞; [a; b] ; kgk∞; [a; b] ; S [a; b] (f); S [a; b] (g)) and compare these inequalities using the Tchebychev polynomials {Tn }, for which it is known that, taking [a; b] = [−1; 1], kTn k∞ = 1 and S(Tn ) = n.
P
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);
∗
1
Corresponding author.
Supported by Gobierno Autonomo de Canarias.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 6 6 - 6
f; g ∈ X
(2.1)
320 F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325
1
have been proved for several choices of ( ; ) and for several classes of functions X ⊂ B∞;
1 [a; b].
The inequality (2.1) is called the ( ; )-inequality for the product. For example, the (3; 0); (2; 2) and
1
(1; 4)-inequalities for the product are valid on all B∞;
1 [a; b]: We will include the proof of these
inequalities to be selfcontained although they appear in [4, 5].
1
Proposition 2.1. The (3; 0) and (2; 2)-inequalities for the product are valid on all B∞;
1 [a; b]:
Proof. Let pk and qk be the best approximations of f and g; respectively, on k : Then
E2k (fg)[a; b] 6 kfg − pk qk k∞; [a; b]
6 kf(g − qk ) + qk (f − pk )k∞; [a; b]
6 kfk∞; [a; b] Ek (g)[a; b] + kqk k∞; [a; b] Ek (f)[a; b] :
But kqk k∞; [a; b] 6 kgk∞; [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=0
E2k (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] 6 E2k (fg)[a; b] for all k: Hence
S [a; b] (fg) 6 2[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 kqk k∞ 62kgk∞ we obtain
E2k (fg)[a; b] 6 kfk∞; [a; b] Ek (g)[a; b] + 2 kgk∞; [a; b] Ek (f)[a; b] ;
X∞
k=0
E2k (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 2 kgk∞; [a; b] S [a; b] (f) + 4 kfk∞; [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.
1
[a; b]
De nition 2.2. Let X ⊂ B∞;
(f)¡∞. We de ne
1 [a; b] be a class of functions with S
(X ) = {( ; ) ∈ R2 : the ( ; )-inequality which are valid on X }:
We will usually identify (X ) with the set of ( ; )-inequalities which are valid on X .
F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325 321
1
It is clear that (X ) is a convex set (e.g., (2:5; 1) = 21 [(3; 0) + (2; 2)] ∈ (B∞;
1 [a; b])) and that if
2
( ; ) ∈ (X ) then ( ; ) + [0; +∞) ⊂ (X ).
1
Theorem 2.3. (1) (0; ) ∈= (B∞;
1 [a; b]) for all ¿0.
1
(2) (0; 6) ∈ (C 0 [a; b]); where C0 [a; b] = {f ∈ B∞
;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 ) ¿ E0 (f2 )[0;1] = 21 + c;
S [0;1] (f)2 = 14 :
It is clear that there
exits some c¿0 such that 21 + c¿ 41 . The rst claim has been proved.
Pm
[a; b]
Set Sm (f) = K=0 (E2k (f)[a; b] + E2k+1 (f)[a; b] ). It is clear that S [a; b] (f) =Plimm→∞ Sm[a; b] (f). On
the other hand, E2k+1 (f)[a; b] 6 E2k (f)[a; b] for all k ∈ N, so that Sm[a; b] (f) 6 2 mk=0 E2k (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] 6 2E0 (f)[a; b] ;
|q0 | 6 E0 (f)[a; b] ;
kqk k∞; [a; b] 6 Ek (g)[a; b] + kgk∞; [a; b]
so that
E0 (fg)[a; b] = E0 (f(g − q0 ) + q0 f)[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] 6 kfg − pk qk k∞; [a; b] = kf(g − qk ) + qk (f − pk )k∞; [a; b]
6 Ek (g)[a; b] kfk∞; [a; b] + kqk kEk (f)[a; b]
6 2E0 (f)[a; b] Ek (g)[a; b] + (Ek (g)[a; b] + kgk∞; [a; b] )Ek (f)[a; b]
6 2(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
Sm[a; b] (fg) 6 6E0 (f)[a; b] E0 (g)[a; b]
+ 4 E0 (f)[a; b]
m
X
Ek (g)[a; b] + E0 (g)[a; b]
k=1
6 6Sm[a; b] (f)Sm[a; b] (f):
The proof follows on taking limits for m → ∞.
m
X
k=1
Ek (f)[a; b] +
m
X
k=1
!
Ek (f)[a; b] Ek (g)[a; b] :
322 F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325
Corollary 2.4. (Cx0 [a; b]; S [a; b] ) is a Banach algebra with the product f ◦ g = 61 fg; where
1
Cx0 [a; b] = {f ∈ B∞;
1 [a; b]=f(x0 ) = 0}.
Proof. It follows from Theorem 2.2.
1
Theorem 2.5. (0; 4) ∈ (C0+ [a; b]); where C0+ [a; b] = {f ∈ B∞;
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] 6 2E0 (f). Hence
E2k (fg)[a; b] 6 kfk∞; [a; b] Ek (g)[a; b] + Ek (f)[a; b] (kgk∞; [a; b] + Ek (g)[a; b] )
6 2E0 (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]
6 2(2E0 (f)[a; b] Ek (g)[a; b] + 2E0 (g)[a; b] Ek (f)[a; b] + Ek (f)[a; b] Ek (g)[a; b] )
6 4(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) 6 E0[a; b] (fg) + E1[a; b] (fg)
+4
E0[a; b] (f)
m
X
Ek[a; b] (g)
k=1
+
E0[a; b] (g)
m
X
Ek[a; b] (f)
k=1
+
m
X
Ek[a; b] (f)Ek[a; b] (g)
k=1
!
:
But for f; g ∈ C0+ [a; b] the inequality
b]
b]
(f)E [a;
(g)
E0[a; b] (fg) = 21 kfgk∞; [a; b] 6 21 kfk∞; [a; b] kgk∞; [a; b] = 2E [a;
0
0
holds. Hence
Sm[a; b] (fg) 6 4Sm[a; b] (f)Sm[a; b] (g)
for all m ∈ N. The proof follows taking limit for m → ∞.
1
Corollary 2.6. (1; 4) ∈ (B∞;
1 [a; b]).
1
Proof. Let f; g ∈ B∞;
1 [a; b] and set F = f−min{f(t): t ∈ [a; b]}, G = g−min{g(t): t ∈ [a; b]}: Then
+
F; G ∈ C0 [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) − (min f)g − (min g)f)
¿ S [a; b] (fg) − |min f| S [a; b] (g) − |min g| S [a; b] (f):
F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325 323
Thus
S [a; b] (fg) 6 4S [a; b] (f)S [a; b] (g) + |min f| S [a; b] (g) + |min g| S [a; b] (f);
S [a; b] (fg) 6 4S [a; b] (f)S [a; b] (g) + kfk∞; [a; b] S [a; b] (g) + kgk∞; [a; b] S [a; b] (f):
( ; )-inequalities for the product have been used to
• Obtain bounds of S [a; b] (f) for several special functions f.
1
• Prove theoretical properties of the Besov space B∞;
1 [a; b] as an algebra.
1
One other inequality for the product which is valid on B∞;
1 [a; b] is the square root’s inequality
(cf. [7] for a proof)
S(fg) 6 2[kfk∞; [a; b] S [a; b] (g) + kgk∞; [a; b] S [a; b] (f) +
q
kfk∞; [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.
Proposition 2.7. Let f ∈ C (k) [−1; 1]; k ¿ 2. Then
S [−1; 1] (f) 6 k kfk∞; [−1; 1] +
(k)
1
k
f
:
∞; [−1; 1]
2k k!(k − 1)
(2.3)
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)
En (f)[−1; 1] 6
Hence
(k)
1
k
f
∞; [−1; 1]
k
2 (n + 1) · · · (n + 1 − (k − 1))
S [−1; 1] (f) 6 k kfk∞; [−1; 1] +
= k kfk∞; [−1; 1] +
for all n ¿ k:
∞
(k)
1
k X
f
∞; [−1; 1]
k
2 n=k (n + 1) · · · (n + 1 − (k − 1))
(k)
1
k
f
:
∞; [−1; 1]
k
2 k!(k − 1)
Corollary 2.8 (Jackson’s type inequalities). Let f; g ∈ C (k) [−1; 1]; k ¿ 2. Then
S
[−1; 1]
k
(k)
(k−s)
k 1 X
1
g
f
(fg) 6 k kfk∞; [−1; 1] kgk∞; [−1; 1] + k
∞; [−1; 1]
∞; [−1; 1]
2 k − 1 s=0 s!(k − s)!
(2.4)
Proof. It follows from (2:3) and the Leibnitz’s rule.
324 F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325
3. Comparison of inequalities for the product
Set [a; b] = [−1; 1] and let us denote by Tn the nth Tchebyshev’s polynomial of the rst kind. Then
Ek (Tn )[−1; 1] = 1 if k¡n and Ek (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
[−1; 1}
(Pn (d )). Hence it is quite natural to use this information to compare inequalities
value of S
for the product.
De nition 3.1. Let
S [−1; 1] (fg) 6 F1 (S [−1; 1] (f); S [−1; 1] (g); kfk∞;[−1; 1] ; kgk∞; [−1; 1] );
S
[−1; 1]
(fg) 6 F2 (S
[−1; 1]
(f); S
[−1; 1]
(3.1)
(g); kfk∞; [−1; 1] ; kgk∞; [−1; 1] )
be two inequalities for the product. We say that
• (3.1) is T -better than (3.2) if F1 (n; n; 1; 1) 6 F2 (n; n; 1; 1); ∀n
• (3.1) is better than (3.2) in the Tchebychev’s sense if F1 (n; m; 1; 1) 6 F2 (n; m; 1; 1);
(3.2)
∀n; m
We will use the following notation:
(3:1) T (3:2) means ‘(3:1) is T -better than(3:2)’
7→
(3:1) T (3:2) means ‘(3:1) T (3:2) and (3:2) T (3:1)’
7→
=
7→
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 De nition 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)
(k)
T
n
∞; [−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 ( 2; 2 )-inequality.
7→
(3) 1 − 2 6 0 and 2( 1 − 2 ) + ( 1 − 2 ) 6 0.
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 )nm 6 0
for all n; m ∈ N:
(3.3)
Suppose that (3:3) holds and set n = m. Then 2( 1 − 2 ) + ( 1 − 2 )n 6 0 for all n ∈ N and it follows
that
1 − 2 6 0:
(3.4)
F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325 325
We use this to write (3:3) in the form
1 − 2
nm
6 min
: n; m ∈ N
2 − 1
n+m
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.
1
Corollary 3.2. Let X be a class of functions contained in B∞;
1 [a; b]. Then T is an order relation
7→
on (X ).
Proof. ( 1; 2 )4( 1 ; 2 ) if and only if 1 − 2 6 0 and 2( 1 − 2 ) + ( 1 − 2 ) 6 0 is an order relation
on R2 .
Corollary 3.3. (1) For all t ∈ [0; 1]; (3; 0) T t(3; 0) + (1 − t)(0; 6):
7→
(2) (3; 0) T (2:5; 1) T (2; 2) T (1; 4) T (0; 6):
7→
7→
7→
7→
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)6 6 0
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 nm 6 n+
m for all n; m. Hence 2(n + m + nm) 6 2(n + m) + n + m = 3(n + m):
References
[1] J.M. Almira, F. Perez Acosta, Minimax series in variable intervals, Preprint.
[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.
[5] F. Perez Acosta, Boundedness of minimax series from values of the function and its derivative, J. Comput. and Appl.
Math. 81 (1997) 101–106.
[6] F. Perez Acosta, On certain Banach spaces in connection with interpolation theory, J. Comput. App. Math. 83 (1997)
55–69.
[7] F. Perez Acosta, Inequalities for the minimax series of a product, Rev. Acad. Canaria de las Ciencias VIII (1996)
113–133.
[8] T.J. Rivlin, The Tchebychev Polynomials, Pure and Applied Mathematics, Wiley, New York, 1974.
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
c 1998 Elsevier Science B.V. All rights reserved.
the valuation of these inequalities on Tchebyshev’s polynomials.
Keywords: Banach algebras; Inequalities for the product; Minimax series; Polynomial approximation; Tchebychev polynomials
1. Introduction
The minimax series of a continuous function f ∈ C[a; b] is de ned by S [a; b] (f) = ∞
n=0 En (f)[a; b] ,
where En (f)[a; b] is the error of best approximation of f by algebraic polynomials of degree 6 n. This
1
1
series appears in the norm of the Besov space B∞;
[a; b] = kfk∞; [a; b] +
1 [a; b] via the identity kfkB∞;
1
[a; b]
S (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 form S(fg) 6 F
(kfk∞; [a; b] ; kgk∞; [a; b] ; S [a; b] (f); S [a; b] (g)) and compare these inequalities using the Tchebychev polynomials {Tn }, for which it is known that, taking [a; b] = [−1; 1], kTn k∞ = 1 and S(Tn ) = n.
P
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);
∗
1
Corresponding author.
Supported by Gobierno Autonomo de Canarias.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 6 6 - 6
f; g ∈ X
(2.1)
320 F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325
1
have been proved for several choices of ( ; ) and for several classes of functions X ⊂ B∞;
1 [a; b].
The inequality (2.1) is called the ( ; )-inequality for the product. For example, the (3; 0); (2; 2) and
1
(1; 4)-inequalities for the product are valid on all B∞;
1 [a; b]: We will include the proof of these
inequalities to be selfcontained although they appear in [4, 5].
1
Proposition 2.1. The (3; 0) and (2; 2)-inequalities for the product are valid on all B∞;
1 [a; b]:
Proof. Let pk and qk be the best approximations of f and g; respectively, on k : Then
E2k (fg)[a; b] 6 kfg − pk qk k∞; [a; b]
6 kf(g − qk ) + qk (f − pk )k∞; [a; b]
6 kfk∞; [a; b] Ek (g)[a; b] + kqk k∞; [a; b] Ek (f)[a; b] :
But kqk k∞; [a; b] 6 kgk∞; [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=0
E2k (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] 6 E2k (fg)[a; b] for all k: Hence
S [a; b] (fg) 6 2[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 kqk k∞ 62kgk∞ we obtain
E2k (fg)[a; b] 6 kfk∞; [a; b] Ek (g)[a; b] + 2 kgk∞; [a; b] Ek (f)[a; b] ;
X∞
k=0
E2k (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 2 kgk∞; [a; b] S [a; b] (f) + 4 kfk∞; [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.
1
[a; b]
De nition 2.2. Let X ⊂ B∞;
(f)¡∞. We de ne
1 [a; b] be a class of functions with S
(X ) = {( ; ) ∈ R2 : the ( ; )-inequality which are valid on X }:
We will usually identify (X ) with the set of ( ; )-inequalities which are valid on X .
F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325 321
1
It is clear that (X ) is a convex set (e.g., (2:5; 1) = 21 [(3; 0) + (2; 2)] ∈ (B∞;
1 [a; b])) and that if
2
( ; ) ∈ (X ) then ( ; ) + [0; +∞) ⊂ (X ).
1
Theorem 2.3. (1) (0; ) ∈= (B∞;
1 [a; b]) for all ¿0.
1
(2) (0; 6) ∈ (C 0 [a; b]); where C0 [a; b] = {f ∈ B∞
;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 ) ¿ E0 (f2 )[0;1] = 21 + c;
S [0;1] (f)2 = 14 :
It is clear that there
exits some c¿0 such that 21 + c¿ 41 . The rst claim has been proved.
Pm
[a; b]
Set Sm (f) = K=0 (E2k (f)[a; b] + E2k+1 (f)[a; b] ). It is clear that S [a; b] (f) =Plimm→∞ Sm[a; b] (f). On
the other hand, E2k+1 (f)[a; b] 6 E2k (f)[a; b] for all k ∈ N, so that Sm[a; b] (f) 6 2 mk=0 E2k (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] 6 2E0 (f)[a; b] ;
|q0 | 6 E0 (f)[a; b] ;
kqk k∞; [a; b] 6 Ek (g)[a; b] + kgk∞; [a; b]
so that
E0 (fg)[a; b] = E0 (f(g − q0 ) + q0 f)[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] 6 kfg − pk qk k∞; [a; b] = kf(g − qk ) + qk (f − pk )k∞; [a; b]
6 Ek (g)[a; b] kfk∞; [a; b] + kqk kEk (f)[a; b]
6 2E0 (f)[a; b] Ek (g)[a; b] + (Ek (g)[a; b] + kgk∞; [a; b] )Ek (f)[a; b]
6 2(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
Sm[a; b] (fg) 6 6E0 (f)[a; b] E0 (g)[a; b]
+ 4 E0 (f)[a; b]
m
X
Ek (g)[a; b] + E0 (g)[a; b]
k=1
6 6Sm[a; b] (f)Sm[a; b] (f):
The proof follows on taking limits for m → ∞.
m
X
k=1
Ek (f)[a; b] +
m
X
k=1
!
Ek (f)[a; b] Ek (g)[a; b] :
322 F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325
Corollary 2.4. (Cx0 [a; b]; S [a; b] ) is a Banach algebra with the product f ◦ g = 61 fg; where
1
Cx0 [a; b] = {f ∈ B∞;
1 [a; b]=f(x0 ) = 0}.
Proof. It follows from Theorem 2.2.
1
Theorem 2.5. (0; 4) ∈ (C0+ [a; b]); where C0+ [a; b] = {f ∈ B∞;
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] 6 2E0 (f). Hence
E2k (fg)[a; b] 6 kfk∞; [a; b] Ek (g)[a; b] + Ek (f)[a; b] (kgk∞; [a; b] + Ek (g)[a; b] )
6 2E0 (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]
6 2(2E0 (f)[a; b] Ek (g)[a; b] + 2E0 (g)[a; b] Ek (f)[a; b] + Ek (f)[a; b] Ek (g)[a; b] )
6 4(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) 6 E0[a; b] (fg) + E1[a; b] (fg)
+4
E0[a; b] (f)
m
X
Ek[a; b] (g)
k=1
+
E0[a; b] (g)
m
X
Ek[a; b] (f)
k=1
+
m
X
Ek[a; b] (f)Ek[a; b] (g)
k=1
!
:
But for f; g ∈ C0+ [a; b] the inequality
b]
b]
(f)E [a;
(g)
E0[a; b] (fg) = 21 kfgk∞; [a; b] 6 21 kfk∞; [a; b] kgk∞; [a; b] = 2E [a;
0
0
holds. Hence
Sm[a; b] (fg) 6 4Sm[a; b] (f)Sm[a; b] (g)
for all m ∈ N. The proof follows taking limit for m → ∞.
1
Corollary 2.6. (1; 4) ∈ (B∞;
1 [a; b]).
1
Proof. Let f; g ∈ B∞;
1 [a; b] and set F = f−min{f(t): t ∈ [a; b]}, G = g−min{g(t): t ∈ [a; b]}: Then
+
F; G ∈ C0 [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) − (min f)g − (min g)f)
¿ S [a; b] (fg) − |min f| S [a; b] (g) − |min g| S [a; b] (f):
F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325 323
Thus
S [a; b] (fg) 6 4S [a; b] (f)S [a; b] (g) + |min f| S [a; b] (g) + |min g| S [a; b] (f);
S [a; b] (fg) 6 4S [a; b] (f)S [a; b] (g) + kfk∞; [a; b] S [a; b] (g) + kgk∞; [a; b] S [a; b] (f):
( ; )-inequalities for the product have been used to
• Obtain bounds of S [a; b] (f) for several special functions f.
1
• Prove theoretical properties of the Besov space B∞;
1 [a; b] as an algebra.
1
One other inequality for the product which is valid on B∞;
1 [a; b] is the square root’s inequality
(cf. [7] for a proof)
S(fg) 6 2[kfk∞; [a; b] S [a; b] (g) + kgk∞; [a; b] S [a; b] (f) +
q
kfk∞; [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.
Proposition 2.7. Let f ∈ C (k) [−1; 1]; k ¿ 2. Then
S [−1; 1] (f) 6 k kfk∞; [−1; 1] +
(k)
1
k
f
:
∞; [−1; 1]
2k k!(k − 1)
(2.3)
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)
En (f)[−1; 1] 6
Hence
(k)
1
k
f
∞; [−1; 1]
k
2 (n + 1) · · · (n + 1 − (k − 1))
S [−1; 1] (f) 6 k kfk∞; [−1; 1] +
= k kfk∞; [−1; 1] +
for all n ¿ k:
∞
(k)
1
k X
f
∞; [−1; 1]
k
2 n=k (n + 1) · · · (n + 1 − (k − 1))
(k)
1
k
f
:
∞; [−1; 1]
k
2 k!(k − 1)
Corollary 2.8 (Jackson’s type inequalities). Let f; g ∈ C (k) [−1; 1]; k ¿ 2. Then
S
[−1; 1]
k
(k)
(k−s)
k 1 X
1
g
f
(fg) 6 k kfk∞; [−1; 1] kgk∞; [−1; 1] + k
∞; [−1; 1]
∞; [−1; 1]
2 k − 1 s=0 s!(k − s)!
(2.4)
Proof. It follows from (2:3) and the Leibnitz’s rule.
324 F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325
3. Comparison of inequalities for the product
Set [a; b] = [−1; 1] and let us denote by Tn the nth Tchebyshev’s polynomial of the rst kind. Then
Ek (Tn )[−1; 1] = 1 if k¡n and Ek (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
[−1; 1}
(Pn (d )). Hence it is quite natural to use this information to compare inequalities
value of S
for the product.
De nition 3.1. Let
S [−1; 1] (fg) 6 F1 (S [−1; 1] (f); S [−1; 1] (g); kfk∞;[−1; 1] ; kgk∞; [−1; 1] );
S
[−1; 1]
(fg) 6 F2 (S
[−1; 1]
(f); S
[−1; 1]
(3.1)
(g); kfk∞; [−1; 1] ; kgk∞; [−1; 1] )
be two inequalities for the product. We say that
• (3.1) is T -better than (3.2) if F1 (n; n; 1; 1) 6 F2 (n; n; 1; 1); ∀n
• (3.1) is better than (3.2) in the Tchebychev’s sense if F1 (n; m; 1; 1) 6 F2 (n; m; 1; 1);
(3.2)
∀n; m
We will use the following notation:
(3:1) T (3:2) means ‘(3:1) is T -better than(3:2)’
7→
(3:1) T (3:2) means ‘(3:1) T (3:2) and (3:2) T (3:1)’
7→
=
7→
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 De nition 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)
(k)
T
n
∞; [−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 ( 2; 2 )-inequality.
7→
(3) 1 − 2 6 0 and 2( 1 − 2 ) + ( 1 − 2 ) 6 0.
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 )nm 6 0
for all n; m ∈ N:
(3.3)
Suppose that (3:3) holds and set n = m. Then 2( 1 − 2 ) + ( 1 − 2 )n 6 0 for all n ∈ N and it follows
that
1 − 2 6 0:
(3.4)
F. Perez Acosta, J.M. Almira Picazo / Journal of Computational and Applied Mathematics 99 (1998) 319–325 325
We use this to write (3:3) in the form
1 − 2
nm
6 min
: n; m ∈ N
2 − 1
n+m
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.
1
Corollary 3.2. Let X be a class of functions contained in B∞;
1 [a; b]. Then T is an order relation
7→
on (X ).
Proof. ( 1; 2 )4( 1 ; 2 ) if and only if 1 − 2 6 0 and 2( 1 − 2 ) + ( 1 − 2 ) 6 0 is an order relation
on R2 .
Corollary 3.3. (1) For all t ∈ [0; 1]; (3; 0) T t(3; 0) + (1 − t)(0; 6):
7→
(2) (3; 0) T (2:5; 1) T (2; 2) T (1; 4) T (0; 6):
7→
7→
7→
7→
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)6 6 0
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 nm 6 n+
m for all n; m. Hence 2(n + m + nm) 6 2(n + m) + n + m = 3(n + m):
References
[1] J.M. Almira, F. Perez Acosta, Minimax series in variable intervals, Preprint.
[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.
[5] F. Perez Acosta, Boundedness of minimax series from values of the function and its derivative, J. Comput. and Appl.
Math. 81 (1997) 101–106.
[6] F. Perez Acosta, On certain Banach spaces in connection with interpolation theory, J. Comput. App. Math. 83 (1997)
55–69.
[7] F. Perez Acosta, Inequalities for the minimax series of a product, Rev. Acad. Canaria de las Ciencias VIII (1996)
113–133.
[8] T.J. Rivlin, The Tchebychev Polynomials, Pure and Applied Mathematics, Wiley, New York, 1974.