approximation, optimization and mathematical economics || trends in hölder approximation

'!rends in Holder Approximation

Jorge Bustamante and Miguel Antonio Jimenez

Universidad Autonoma de Puebla Apartado Postal J-27, Colonia San Manuel, Puebla, C. P. 72571, Pue., Mexico E-mails:[email protected]@fcfm.buap.mx

Abstract. It is presented a survey of the main known results related with qualitative and quantitative trigonometric or algebraic polynomial approximation in Holder metric.

Keywords: Lipschitz (Holder) functions, best approximation, K-functionals, modulus of smoothness, Fourier series, semigroup of operators, homogeneous Banach spaces, Bernstein polynomials.

AMS classification: 41A17 , 41A25, 41A35, 41A65, 42AIO.

1 Introduction

In many mathematical problems one needs smooth functions but differentiability sometimes could be a severe restriction. Then we weaken the hypothesis by considering Lipschitz or Holder functions. Consequently these functions have been extensively used in different branches of mathematics. Many properties of their approximation by algebraic or trigonometric polynomials in the usual LP or sup norm, are well known since many years ago. However, the research on approximation in Lipschitz or Holder norm is mostly recent. The aim of this paper is just to present in an organized way some of the main results obtained in that research. Reasons of space do not allow us to focus attention to other related properties of Lipschitz spaces such as topological and algebraic structures.

We begin by introducing the following preliminaries. Let (X, d) be a metric space containing at least one accumulation point,

f : X -+ IR. a function and 0; > 0 a real number. For each 6 > 0 define

()o: (f, 6) := sup ""YEX,O<d(""y)~o

I f(x) - f(y) I d(x,y)o:

(1)

and ()o:(f) := sUPo>o()o:(f,o). It is clear that ()o:(f) is well defined (finite or infinite) .

If (Z, dz ) is another metric space and f : X -+ Z is a function, we can define a similar expression by considering the fraction dz (f(x), f(y)) / d(x, y)o:.

Following Mirkil [35], we say that f : X -+ IR. satisfies a Holder (Lipschitz) condition of order 0;, 0; E (0,1] (0; = 1), if ()o:(f) < 00. This classification remarks the deep difference between the results for 0; < 1 and 0; = 1. However, M. Lassonde (ed.), Approximation, Optimization and Mathematical Economics© Physica-Verlag Heidelberg 2001

82 J. Bustamante and M. A. Jimenez

for simplicity of redaction, the family of all these functions is denoted here by the same symbol Lipo:X while sometimes we will not set any difference between Lipschitz or Holder concepts.

Notice that Lipo:X c C(X), where C(X) is the space of all continuous functions on X. If (X,d) is a compact space, then the set Lipo:X becomes a Banach space by considering the norm

Ilfllo: := IIflloo + (}o:(f), (2)

where II 0 1100 denotes the uniform norm.

Before to introduce historical developments and references on the subject treated in this paper it is obvious but worthwhile to point out that they could be limited by our knowledge.

In 1957, in order to study a certain class of integro-differential equations, A. 1. Kalandiya [31] presented the first known results related with approximation in Holder norm.

Theorem 1. (Kalandiya) Fix a E (0,1] and f E Lipo:[O,I]. Suppose that for n = 1,2",,; there exist algebraic polynomials Pn of degree not greater that n such that IIf - Pnlloo :::; KdnO:. Then, for 0 < 2{3 < a, one has (}{3(f - Pn ) :::; K 2/no:- 2{3.

In 1983, N. 1. Ioakimidis [26] presented a simplified proof of this theorem and further, an improvement in [27]. Ioakimidis was motivated by the fact that Theorem 1 had been used by Elliot-Paget [18], Tsamasphyros-Theocaris [56], Chawla- Kumar [13] and N. 1. Ioakimidis [28], in studying the convergence of quadrature rules for Cauchy type principal value integrals and the quadrature methods for the corresponding singular integral equations. For more information about this kind of applications see [53].

Taking into account the classical Jackson theorem about approximation of continuous functions by polynomials and Theorem 1, one finds that each function f E Lipo:[-I, 1] can be well approximated by algebraic polynomials in Lip{3[-I, 1] for 0 < {3 < a/2. Notice this does not imply that the algebraic polynomials are dense in Lip{3[-I, 1], because Lip{3[-I, 1] \ Lipo:[-l, 1] "" 0. Moreover, for (3 E [1/2,1), Theorem 1 does not give any information about approximation by algebraic polynomials. As we will see below, the algebraic (trigonometric) polynomials are not dense in Lipo:[-I,I] (Lipo:(27r), the HOlder space of 27r-periodic real functions). This last remark explains at least one reason for which many contributions in the study of polynomial approximation in the HOlder metric 110 "0" only deal with the approximation of functions in Lipo:_<.

Trends in Holder Approximation 83

2 Summation of Fourier Series

First, we analize the approximation of periodic functions by trigonometric polynomials. In fact, the use of the translation operator

Thl(x) := I(x + h), (3)

simplifies the problem and allows us to introduce some generalizations. If I is a 21l"-periodic function, we have (Ja(fj 8) = sup IIThl - 11100/1 h la,

0<lhl~6

where II 0 1100 is taken in IR. If E is a Banach space and, for each h, Th : E -t E is a linear bounded 0-

perator, we introduce abstract HOlder spaces in a natural form by considering

(JE( -,) IIThl - IIIE (JE() (JE(-,) a Ij U := sup I hi' a I := sup a Ij U • O<lhl~o a 0>0

(4)

Usually we ask the family (Th ) to satisfy some conditions and the simplest examples are given in the theory of semi-group of operators that we will review further.

In this section we only consider the usual translation defined by (3), but also we analize the L~1I" spaces of periodic integrable functions, 1 ~ p < 00.

Thus 11 0 lip refers to L~1I" if 1 ~ p < 00 and to C211" if P = 00. For simplicity

we write (J~ := (J~~" and (Ja := (J~2", in (4). The corresponding spaces of Lipschitz functions are denoted by Lip~(21l") and LiPa(21l").

In order to have ()~ (f) < 00, it is necessary that

IIThf - filE --7 0 (h -t 0). (5)

A remarkable paper in the history of characterizing functions f in a given space E, for which (5) holds, is due to Plessner [40J. Condition (5) holds for every I in E = L~1I"' 1 ~ P < 00, or E = C211"' but fails for E being any of the Lipschitz spaces Lip~(21l") (1 ~ P ~ 00) and this property is very important as we shall discover immediatelly.

Let us denote the family of all trigonometric polynomials of degree not greater than n by Tn.

If Q E (0,1), 1 ~ p ~ 00 and f E Tn, then

(J~(f, h) --+ 0, h ---t O. (6)

It means that if Q E (0,1) and 1 ~ p ~ 00, a function I E Lip~(21l") needs to satisfies (6) in order to be the limit of a sequence of polynomials in the Holder norm. This motivates the following definition of new spaces.

For Q E (0,1) and 1 ~ p ~ 00, denote by lip~(21l") the space of all functions I E Lip~(21l") for which (6) holds and including Q = 1, denote

p~ := {f E Lip~(21l") : h -t Thf is continuous}.

The links that connect these ideas are the solutions to the following implicit Plessner problems:


Theorem 2. (Shilov) For 0 < a :::; 1 and 1 :::; p :::; 00, the closure of the trigonometric polynomials in Lip~ (27r) is exactly P~.

Indeed, Shilov result appeared in [54] in a more general framework.

Theorem 3. (Hardy-Littlewood, [25]) For 1 < p < 00, Pi coincides with the space of all antiderivatives of Lg" functions. The space Pi is the set of all functions of bounded variation and Pf coincides with C~".

Finally, from a very general paper of Mirkil [35], we choose

Theorem 4. (Mirkil) For a E (0,1) and 1:::; p:::; 00, P~ = lip~(27r).

From Theorems 3 and 4 we obtain that the trigonometric polynomials are dense in lip~(27r) (1:::; P :::; 00 and 0 < a < 1). Notice that this result cannot be obtained from Theorem 1. Moreover, it can be proved that the Banach spaces LiPo:(27r) are not separable.

A natural approach to approximation by trigonometric polynomials is to analyze the partial sums of the Fourier series as an operator defined on Lip~(27r). The first result that we know in this direction is due to Prossdorf [51] in 1975. He proved that, for each a E (0,1), there exists 1 E liPo:(27r) such that IISn(f) - 1110: --1--+ 0, where Sn denotes the partial sums of the Fourier series. The same result is obtained by proving that IISnllo: '" In n, where IISnllo: denotes the norm of Sn acting from lip",(27r)) into lip",(27r). From this last estimate follows (see [6]):

Theorem 5. For a E (0,1) and f E lipo:(27r) we have

()",(fi l/n)lnn-+ 0 => IISnf - fll", -+ o. One of the first results related with the quantitative theory for periodic

functions is also due to Prosdorff. It is inspired in a theorem of Alexits for the uniform approximation.

Theorem 6. (Prossdorf, [51]) Fix a E (0,1] and f3 E (0, a). If f E Lip",(27r), then

if 0 < a < 1

if a = 1.

This result was a starting point for other works. In particular several papers analized the convergence in the Holder norm of different summation processes related with the Fourier series. For instance:

Theorem 7. (Stypinski, [55]) Fix a E (0,1] and f3 E (O,a). Let A := (An) be a sequence of integer numbers such that A1 = 1 and 0 :::; An+1 - An :::; 1. For 1 E Lip",(27r) define

1 n-1 Vn(A,fi x ):= .An L Sk(fi X ).

k=-n-An

Trends in HOlder Approximation 85

Then

if 0 < a < 1

if a = 1.

A similar result due to Leindler for generalized Holder classes can be found in [33]. R. N. Mohapatra and P. Chandra analyzed different summation processes in the spirit of Theorem 6 (see [10], [11], [36], [37] and [38]).

However, many of the results above are implied by the short but remarkable paper [34] of Leindler, Meir and Totik, in 1985. In fact, the following theorem states that most of results on HOlder norms that deal with a sequence of convolution operators from C2rr into C2rr , are reduced to the relatively simple problem of approximation in the sup-norm.

Theorem 8. (Leindler-Meir-Totik) For a sequence (An) of convolution operators from C2rr into C2rr , and each f E C2rr , one has

( 1) 2w(f,8) IIAnl - IlIcp :S IIAnl - 11100 1 + -(1/) + (1 + IIAnl!) sup (8)'

cp n 0<6:9/n cP

where cp is any increasing positive real function on (0,00) and for each function g E C2rr ,

II II II II IIT6g - glloo g cp:= g 00 + sup (.) .

6>0 cp u

Recall that, for I E L~rr (p ~ 1) and a sequence (vn) on integers such that o ~ Vn ~ nand n ~ AVn for a constant A, the strong mean of the Fourier series of I is defined by

Gorzenska, Leswiewiecz and Rempulska have obtained different results related with the strong approximation of Fourier series in Holder norms (see [23] and [24]). For other results in this section see [21], [22], [36], [46] and [49].

3 Best Trigonometric Approximation

The best approximation in Lipschitz spaces has been considered among 0-

thers by J. Prestin in different papers ([42]-[48]). In particular he has used de la Vallee Poussin sums to describe the best approximation in Lipschitz


spaces and to obtain estimates for special approximation processes and he analyzed problems of approximation and interpolation (also called simultaneous approximation) in Holder norms. In these problems he approximates a function f together with its derivatives of certain orders by a polynomial and the derivatives of the polynomial. We need some notations.

Let E be one of the spaces C21f or L~1f (1 ~ p < 00) of 27l'-periodic functions. For 0: E (O,IJ and r = 0,1"", we denote by Er,o: the class of functions f with the following property: There exists a 27l'-periodic (r - 1)times absolutely continuous function g, with g(r) E E (g E X in the case r = 0), f = 9 a.e. in X and B~(g(r)) < 00. The norm in Er,o: is defined by

r

Ilfllp,r,o: := L Ilg(k) lip + B~(g(r)). (7) k=O

The analogous small HOlder spaces are defined in the following way:

Er,o: := {f E Er,o: : B~(g(r), 15) --T 0, as 15 --T O}.

For f E Er,o: the best trigonometric approximation of order r is defined by

E~(J,Er,o:):= inf Ilf-tilpro:' tErn ' ,

Theorem 9. (Prestin, [42]) Fix 1 ~ p ~ 00 and integers r, m. Moreover, fix positive reals 0, f3 such that r + f3 ::; m + 0 and 0 ::; r ::; m. Then, there exists a positive constant C such that lor I E Em,o: and n = 1,2, ...

II, in addition, I E Em,o:, 0 < 0::; 1, then E~(J,Er,f3) = o(nr+f3 - m-o:), n --T 00.

The result above is used to obtain estimates for special approximation processes as in Kalandiya theorem. For instance:

Theorem 10. (Prestin [42]) Under the conditions of Theorem 9, if one has Ilf - tllp ~ Cn E!:(J, t), then Ilf - tllp,r,)3 ~ Cn(n + ly+f3-m-O:lIg(m)lIp,o:'

J. Prestin also used theorem 9 to characterize some functional classes by applying K-functional methods. We only list two of the fourteen equivalent conditions given in Theorem 6 of [42J.

Theorem 11. (Prestin) Assume that 0 < r + f3 < m + 0: and 0 < 0: < 1. Then f E Em,o: and En (J,Er,)3) = O(nr+f3 - m-o:) are equivalent.


Other estimates for approximation of periodic functions by Fourier series and interpolatory polynomials in HOlder spaces are given in [52J. These results were generalized by J. Prestin in [44J. In particular Prestin gave explicit values for the constants appearing in the Holder norm results.

Some of the estimates given by Prestin can be improved if we use the function (J~(f,t5) defined in (4). In fact, J. Bustamante, D. Mocencahua and C. A. Lopez used ideas of [IJ and [5] in [8] to extend some of Prestin results. The first theorem of these extensions is related with approximation processes.

Theorem 12. [8] Fix a E (0,1), a non negative integer r, 1 ~ p ~ 00 and let E be one of spaces L~7T' Let Ln : E -+ Tn be a sequence of operators and suppose that there exists a positive constant C such that for n = 1,2, ... and every f E L~7T' IIf - Lnlllx ~ Cw:+! (f,~), where w:+! is the usual modulus of smoothness of order r + 1 in E. Then, there exists a positive constant D such that, for each I E EP,a and n ~ N

III - Lnlllp,r,a ~ D(J~(f(r), ~),

where 11·llp,r,a is defined by (7) and (J~ is defined by (4).

It follows from Theorem 9 that, if a, (3 E (0, 1), r, m are positive integers with r+(3 < m+a and I E C~,a, then En(f,Er,{3) = O(nr+{3-m-a), where En(f, Er,(3) is the best approximation by trigonometric polynomials of I in the norm of Er,{3. Notice that this result involves different HOlder spaces. If we suppose that the functions (Ln) in Theorem 12 are the Jackson operators, then Theorem 12 is an improvement of Theorem 9: it gives an estimate of the rate of convergence and it holds for functions in Er,{3 \ Em,a.

Many facts related with periodic functions can be studied in the more general framework of the Theory of Semi-groups of Operators. Fix a Banach space E and denote by L(E) the family of all continuous linear operators from E to E. An equibounded semi-group of operators is a map T : 1R+ -+ L(E) such that: (i) T(O) = I (I denotes the identity operator)j (ii) For s, t E 1R+, T(s + t) = T(s)T(t)j (iii) For each lEE, limt--to+ T(t)1 = I, where the limit is considered in the norm of Ej (iv) There exists a positive constant M such that, for t ~ 0, IIT(t)1I ~ M. Denote by D(A) the family of all fEE for which the limit AI:= lim u-1[T(u) - 1]1 exists in the norm of E. A

u-tO+ is called the infinitesimal generator of the semi-group. For r > 1, the linear operators Ar and the subspaces D(Ar) are defined inductively. We refer to [9] for these notions and others related with semi-groups of operators.

To each a E (0,1) we associate a Holder space Ea by considering all elements lEE such that sUPs>o s-all(T(s) -I)/IIE < 00. Define an abstract version of the spaces liPa[O, 211'] (0 < a < 1) by

Ea := {f E E : lim sup s-all(T(s) - I)flIE = O}. h-tOo<s~h


Also, define ()o:(f) := sups-O:II(T(s) - I)fIIE. Then Eo: is a Banach space 8>0

under the norm II f 110::=11 filE +()o:(f). Moreover Eo: is the closure of D(A) in Eo: and for each f E Eo: and h 2: 0, II T(h)f 110:::; M II f 110: (see [9], p. 163).

There are at least two different forms of measuring the smoothness in Eo:. Denote 1N := {I, 2, 3, ... }. For r E 1N the usual modulus of smoothness of order r of f E Eo: is defined by

w~(f, t):= sup (11[T(h) - Ir fllo:). (8) O<h~t

and the lipschitzian modulus of smoothness of order r is defined by

()~(f, t):= sup h-O:II(T(h) - I)r filE. (9) O<h~t

A development of (8) shows that (9) is simpler than (8). Let us consider the following Petree's K -functional defined for every r E

1N, t > ° and f E Eo: by

Theorem 13. (Bustamante-Jimenez, [5]) For every r E 1N fixed, there exist positive constants C1 and C2 such that, for each f E Eo: and t E (0,1]'

and

Corollary 1. For each r E 1N, there exist constants C1 and C2 (and if r > 1 also a constant C3 ) such that, if f E Eo: and t E (0,1]' then

w~ (f, t) ::; C 1 ()~(f, t) ::; C2W~ (f, t 1-0:/ r ) ::; C3W~-1 (f, t).

Theorem 14. (Bustamante-Jimenez) Let E be one of the spaces C27r or L~7r (1 ::; p < 00). For each n E 1N and f E liPo:(21l") (0 < 0: < 1), let En(f) be the best approximation of f by elements of Tn in liPo:(21l"). Then for each r E 1N, there exist positive constants Cr,l and Cr,2 such that, if n E 1N and f E liPo:(21l"), then

Cr,l En(f) ::; ()~(f, lin) ::; ~~: {llfllEa + ~ kr- a- 1 Ek (f) } .

For other results in this section see [15], [29] and [45].


4 Best Algebraic Approximation

Let C[ -1, 1] be the Banach space of all real continuous functions 1 defined on [-1,1]' equipped with the sup norm. By lIn, n E lN, we denote the linear space of real algebraic polynomials of degree at most n. Let En(f) :=

inf 111 - Plloo be the best algebraic approximation of lout of IIn. For PEIIn

X E [-1,1], we denote 'P(x) := Vl- x2 •

For 1 E C[-I, 1] and h > 0 the symmetric divided difference LliJ(x) is defined by

r

Llhl(x) := ~) -ly-j C) l(x + (~ - j)h), (10) j=O

whenever x ± rh/2 E [-1,1] and the usual modulus of smoothness of order r is defined here by wr(f, t) := sUPhE{O,t] sUPxEI{r,h) I LliJ(x) I, where J(r,u) := {z En: z ± (ru)/2 E [-1, I]}.

It is well known that, for a function 9 E C27r and a real (3 E (0,1), the assertions En(g) = O(n-/3) and wl(g,c5) = 0(15/3) are equivalent. However, if we consider functions 1 E C[-I, I] and the modulus wl(f,t), such a result does no hold for the best algebraic approximation in C[ -1, 1]. This later phenomenon was first recognized by S. N. Nikolskii.

Let 1 E C[-I, 1] and t > O. Define

w:(f, t):= sup sup I Ll~{x)hl(x) I hE{O,t] xEJ(r,h)

(11)

where J(r, h) := {z E [-1,1] : z ± (rh'P(z))/2 E [-1, I]}. It is known that En(f) = O(n- f3 ) and w'!(f,t) = O(tf3 ) are equivalent whenever (3 E (O,r) ([16], p.83).

Given 0: E (0,1), define the Holder class with respect to 'P as the linear space lip~[-I, 1] of all functions 1 E C[-I, 1] such that

lim w'" (f, t) = ° t-+O+ ta

The space lip~[-I, 1] is equipped with the norm I1ll1a := 1111100 +sup w'f'~!,h) h>O

and the modulus of smoothness Bf,a(f, t):= sup w,:~!,h). hE{O,t]

Notice that, for each n E lN, lIn C lip~[-I, 1]. This allows us to define, for 1 E lip~[-I, 1], the best algebraic approximation in Holder norm by En a(f) := Eft a(f):= inf 111 - Plla. , , PEIIn

Theorem 15. (Bustamante-Castaneda, [4]) Fix 0: E (0,1), and a positive integer r. Then, there exist positive constants Cr ,1 and Cr ,2 such that, for


every I E lip~[-l, 1] each n > r and every t E (0, I),

Cr,l En,a(J) ~ ()t,a(J, ~) ~ Cr,2 (~r-a L kr- a- 1 Ek,a(J). 09~n

As a consequence, for a function I E liPa[-l, 1] and (3 E (O,r - a) the equations En,a(J) = O(n-/3) and ()f,a(J) = O(t/3) are equivalent.

Let us denote by W r the family of all functions 9 E C[-l, 1], such that, on each closed interval [a,b] c (-1,1), 9 is r-times continuously differentiable. We consider two seminorms in W r defined by

( I 9 Iwr:= sup Ilg(r)(X)II). xE(-l,l)

Fix a E (0,1]. For I E liPa[-l, 1] and t > 0, the associated K-functionals are defined by

Kra(J,t):= inf {II/-glla+tlglwr} , , gEW r <P

K;,a(J, t) := g~w,r {III - gila + t I 9 Iw; +t(2r-a)/(r-a) I 9 Iwr} .

The relation between the moduli of smoothness and the K-functionals is given below.

Theorem 16. For each positive integer r and each a E (0,1) there exist positive constants C1 and C2 (which depend on r) such that, for every f E

liPa[-l, 1] and t E (O,l/r]

J. Prestin and A. Stosiek [50] have discussed estimates for non periodic functions in Holder norms with varying order of difference and they used the results to estimate linear approximation processes. In particular they analyzed the Lagrange interpolation. In [47] and [50] the main results are presented by comparing the approximation order in different Holder norms.

5 Other Approximation Processes

A remarkable theorem related with approximation by Bernstein polynomials in Holder norm is due to B. M. Brown, D. Elliot and D. F. Paget.

Trends in HOlder Approximation 91

Theorem 17. [2] If Bn(f; x) is the Bernstein polynomial of order n of f E LiPa[O,I] (0 < a ~ 1), then Oa(Bn(f)) ~ Oa(f).

J. Bustamante and M. A. Jimenez used the theorem above to prove:

Theorem 18. [7] If a E (0,1) and f E lipa[O, 1], then IIBn(f) - fila ~ 0, where (Bn(f)) is the sequence of Bernstein polynomials of f·

For analogous results relating second order modulus of continuity, Bernstein polynomials and Lipschitz classes, see [20].

Concerning direct and converse theorems related with approximation by linear splines, let us consider the triangular infinite matrix S defined by the points

m xn,m = 2n' n = 1,2,3"", m = -2n,··· ,2n. (12)

For n = 1,2,"" let Sn be the family of all functions f E C[-I, 1] such that, on each open interval (Xn,k, xn,k+d (-2n ~ k ~ 2n -1), f is a real algebraic polynomial of degree not greater than 1.

For each positive integer n and f E liPa[-I, 1], define

En,a(f):= inf Ilf - gila. gESn

Theorem 19. (Bustamante-Mocencahua-Lopez, [8]) There exist positive constants C and D such that, for each f E liPa[-I, 1] and n = 1,2"",

1 D { ~ Ek,a (f) } CEn,a(f) ~ Oa(f,;;:) ~ nl-a IIflia + ~ ka .

Corollary 2. Fix a E (0,1). For a function f E liPa[-I,I] andf3 E (O,I-a) the equations En,a(J) = O(~) and Ow,a(J, t) = O(t/3) are equivalent.

The exponential Borel's mean of the Fourier series of a periodic function 00

f E C21r is define by Bp(J; x) = exp( -p) L: Sn(J; x)pn In!, (p > 0) (do not n=O

confuse with Bernstein operators). In [12], P. Chandra improved a result of [37], as follows.

Theorem 20. [12] Let ° < f3 < a ~ 1 and fix f E LiPa,21r' Then

IIBp(J) - fll/3 = O(p-(a-/3)/2(Iog).B la.

Let us recall that the Poisson-Cauchy and Gauss- Weierlrass singular integrals of a function f are given by

1r

Q(J, ()(x) := ((/rr) ! f(x + t)/(t2 + (2)dt, (13) -1r


and 11"

W(J,()(x):= (1/.,J;() J !(x+t)exp(-t2/()dt. (14)

A. Khan [32] and R. N. Mohapatra et al. [39], considered error bounds of functions ! E Lp[O, 21r] by using Poisson-Cauchy and Gauss-Weierstrass singular integrals. Some estimates for the approximation of these singular integrals, in Lip,8 norms, were presented further by Mohapatra- Rodriguez [39]. Finally these estimates were improved by J. Bustamante as follows:

Theorem 21. [3] Let 0 < f3 < 1 fixed. For each f E h,8 and ( E (0,1]' if Q(J, () and W(J, () are given by (13) and (14), then

11"

Ilf - Q(J, () 11,8 ~ 21~JI,8 ( + ~cp(J, () + ~6 ( J ~({, ~; dt

(

and Ilf - W(J, ()II,8 ~ Ilfll,8( + 8cp(J, v'().

Let us review the paper of Jimenez [30], where the usual definition of liPa(21r) spaces, 1 ~ p < 00, is modified in order to obtain homogeneous Banach spaces. Consider in IR. the pseudometric

\Ix, y E [0, 21r), \Ij, k E 'll, d(x + 2j1r, y + 2k1r) = min{1 x - y I, 21r- I x - y I}.

Then define for f E LP(21r) , 0: > 0 and x, y E IR.

{(J(X) - !(y))/d(x,y)a j

Fa(x,y) := o· ,

if x -::J: y mod (21r)

if x = y mod (21r).

The spaces B~ are defined by the functions! E U(21r) such that Fa! E LP((21r)2) and normed by Ilfllp,a := (1Ifll~ + IlFa(J)II~)P . Since na>lB~ is reduced to constant functions for every 1 ~ p < 00, we restrict ourselves to the bound 0: ~ 1.

Theorem 22. [30] For every 1 ~ p < 00 and 0 < 0: ~ 1, the space B~ is a homogeneous Banach space and the classical Holder space Lip~(21r) is continuously embedded in B~ by the identity operator.

Corollary 3. For each f E B~ and each summability kernel of 21r-periodic continuous functions (Kn) in L~1I"' one has IIKn * f - fllp,a ----t o.

Other results in the mentioned paper deal with the best trigonometric approximation in II 0 IIp,a. But an interesting result appears with the case p = 2. In fact, B~ is a Hilbert space with inner product

'frends in Holder Approximation 93

The somewhat unexpected result is that the classical trigonometric system still is an orthogonal basis of B~ and then, given f E B~, the trigonometric polynomial of best approximation to f in the norm II 0 112,a, coincides with the polynomial of best approximation to f in the original norm 11 0 11£2 .

2" For other results related with this section see [19] and [41]

Acknowledgment. The authors are indebted to the referee for his (her) valuable suggestions.

References

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