Some equivalence theorems with K-functionals

Journal of Approximation Theory 121 (2003) 143–157 http://www.elsevier.com/locate/jat

Some equivalence theorems with K-functionals$ Feng Dai* School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia Received 15 May 2001; accepted 22 November 2002 Communicated by Zeev Ditzian

Abstract The generalized Riesz means, which were introduced by Ditzian (Acta Math. Hungar. 75 (1997) 165), are shown to be equivalent to the corresponding K-functionals in a general setting. Similar results are also obtained for the Cesa`ro means. r 2003 Elsevier Science (USA). All rights reserved. Keywords: Generalized Riesz means; Cesa`ro means; K-functionals; Strong converse inequality

1. General notations and assumptions We first introduce some notations from [3]. Let S be a nonempty set equipped with a positive measure m and let Lp ðSÞ; ð1pppNÞ denote the space of functions 1 R on S with the usual norm jjf jjp ¼ ð S jf jp dmðxÞÞp ; 1ppoN and jjf jjN :¼ ess supxAS jf ðxÞj: Suppose PðDÞ is a self-adjoint, unbounded operator on L2 ðSÞ: We make the following assumptions on PðDÞ: (i) PðDÞ has only discrete spectrum flðkÞgN k¼0 and each eigenvalue lðkÞ corresponds to a finite-dimensional eigenspace Hk : (ii) 0 ¼ lð0Þolð1Þo?olðkÞo? and lðkÞ is a polynomial in k: T 0 (iii) For some fixed pA½1; N ; Hk CLp ðSÞ Lp ðSÞ and span

[

Hk ¼ Lp ðSÞ:

k $

This work was supported by a grant from the Australian Research Council. *Corresponding author. Fax: +612-9351-4534. E-mail address: [email protected] (F. Dai).

0021-9045/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9045(02)00059-X

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Throughout the rest of this paper, the letter B always denotes the space Lp ðSÞ with pA½1; N satisfying assumption (iii) above and with jj  jj denoting the norm jj  jjp : Under the above assumptions, the projection Pk f of f AB on Hk is obviously well defined. For the formal expansion fB

N X

Pk f ;

k¼0

we define its cth order Cesa`ro means (as usual) by scN ðf Þ ¼

N X AcNk Pk f ; AcN k¼0

where Ack ¼

Gðk þ c þ 1Þ : Gðk þ 1ÞGðc þ 1Þ

We make the following additional assumption on the Cesa`ro means: (iv) For some c ¼ cðBÞAN; sup jjscN ðf ÞjjpCðc; BÞjjf jj:

ð1:1Þ

N

We remark that as pointed out in [1,3], the above assumptions are very natural and many differential operators and the expansions related to them, such as spherical harmonics and the Laplace–Beltrami operator, Jacobi expansions and the Jacobi operator, and Hermite and Laguerre expansions and their operators, satisfy these assumptions. Now let us define the fractional differential operator PðDÞa ( for a given a), in the sense of distributions, by PðDÞa f B

N X

ðlðkÞÞa Pk f :

k¼0

We write ðPðDÞÞa f ¼ f ðaÞ if PðDÞa f AB: To each operator ðPðDÞÞa is associated with a K-functional Ka ðf ; tÞ ¼ inffjjf  gjj þ tjjgðaÞ jj: gðaÞ ABg: It will be convenient to use the notation Aðf ; tÞEBðf ; tÞ; which means that there is a C40; independent of f and t; such that C 1 Aðf ; tÞpBðf ; tÞpCAðf ; tÞ: A strong converse inequality of type B ( in the sense of [4]) is a result of the type jjTt ðf Þ  f jj þ jjT t ðf Þ  f jjEKr ðf ; tÞ M

ð1:2Þ

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and a strong converse inequality of type A will be (1.2) when we can drop the second term on its left-hand side. Here fTt gt40 is a given family of continuous linear operators on B: For further details of the background information, we refer the reader to [2,5–7] and to [1,3], where many impressive results were obtained in the above setting and many interesting applications were given to some known differential operators and the expansions related to them. Except when otherwise stated, the letter C denotes a general constant depending only on the parameters indicated as subscripts, and possibly also on the space B and the operator PðDÞ: 2. Riesz means For l40; a40 and cAN; the generalized Riesz means, which were introduced in [3], are defined by   c X  lðkÞ a Rl;a;c ðf Þ ¼ 1 Pk ðf Þ: l lðkÞol It follows from [3] that under assumption (iv), sup jjRl;a;c ðf ÞjjpCjjf jj

ð2:1Þ

l40

with C independent of f : In this section, we shall prove the following theorem, which was conjectured in [3] under hypothesis (2.1): Theorem 2.1. Suppose cAN and (1.1) is satisfied. Then for l40; a40 and mAN; jjðRl;a;c  IÞm f jjEKam ðf ; lam Þ: Theorem 2.1 for c ¼ 1 is due to [3]. For cX2; a result of type B like (1.2) was obtained in [1, (3.10), p. 181]; [3, (5.7), p. 335]. For all cX1; it was shown in [3, (5.1), p. 334] that jjðRl;a;c  IÞm f jj þ lam jjðRl;a;c;m f ÞðamÞ jj EKam ðf ; lam Þ; where the operator Rl;a;c;m is defined by m

Rl;a;c;m ¼ I  ðI  Rl;a;c Þ ¼

m X k¼1

ð1Þ

k1

! m Rkl;a;c : k

ð2:2Þ

ð2:3Þ

To prove Theorem 2.1, we need the following lemmas. Lemma 2.2. Suppose cAN is as in assumption (iv) and ZAC ðcþ1Þ ðRþ Þ is of compact support. For l40; define   N X lðkÞ Z Vl ðf Þ ¼ Pk ðf Þ: l k¼0

146

F. Dai / Journal of Approximation Theory 121 (2003) 143–157

Then jjVl ðf ÞjjpCZ jjf jj; with CZ 40 independent of f and l40: Proof. Suppose supp ZC½0; a with a40 depending only on Z; and suppose lðn0  1Þpalolðn0 Þ with n0 AN: Noticing that ! ’ cþ1 k þ c c jjPk f jj ¼ W sk ðf Þ pCðk þ 1Þc jjf jj; c by assumption (ii), without loss of generality, we may assume the function lðxÞ is strictly increasing on ½0; NÞ: By (1.1) and the Abel transformation, it suffices to prove   n0 X lðkÞ jWcþ1 Z ð2:4Þ jkc pCZ : l k¼0 Let jðxÞ ¼ ZðlðxÞ l Þ: Then a straightforward computation shows that  cþ1  X lðxÞ i 1 jjðcþ1Þ ðxÞjpCZ : l ðx þ 1Þcþ1 i¼1

ð2:5Þ

Noticing that   lðkÞ ¼ jðcþ1Þ ðyk Þ Wcþ1 Z l for some yk A½k; k þ c þ 1 ; we get from (2.5)    cþ1  X cþ1 lðxÞ lðyk Þ i 1 W Z pCZ : l l ycþ1 k i¼1

ð2:6Þ

Now substituting (2.6) into the left-hand side of (2.4), taking into account the monotonicity of lðxÞ; we obtain (2.4) and complete the proof. & Lemma 2.3. Suppose l40 and Rl;a;c;m f is defined by (2.3). Then  am 1 ðamÞ m ðR f Þ l;a;c;m l pCjjðI  Rl;a;c Þ f jj; with C40 independent of l and f. Proof. By (2.3) and (2.1), it is sufficient to prove  am 1 ðamÞ m ðR f Þ l;a;c l pCjjðI  Rl;a;c Þ f jj:

ð2:7Þ

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We begin by fixing Z; a C N function of compact support, defined on R; with the properties that ZðxÞ ¼ 1 for jxjp14 and ZðxÞ ¼ 0 for jxjX12: Let   c  lðkÞ a aðk; lÞ ¼ 1  : l We decompose the operator ð1lÞam ðRl;a;c f ÞðamÞ as  am 1 ðRl;a;c f ÞðamÞ ¼ Tl1 f þ Tl2 f ; l

ð2:8Þ

where X

Tl1 f ¼

 aðk; lÞ

lðkÞol

Tl2 f

   lðkÞ am lðkÞ Z Pk f ; l l



X

lðkÞ ¼ aðk; lÞ l lðkÞol

am    lðkÞ 1Z Pk f : l

First, we will prove for i ¼ 1; 2 jjTli f jjpCjjðI  Rl;a;c Þm f jj;

ð2:9Þ

with C40 independent of l and f : For i ¼ 1; let us rewrite Tl1 f as   X lðkÞ Tl1 f ¼ aðk; lÞx Pk ðhÞ; l lðkÞol

ð2:10Þ

where xðtÞ ¼

ZðtÞtam ð1  ð1  ta Þc Þm

;

and

h ¼ ðI  Rl;a;c Þm f :

ð2:11Þ

Noticing that ZðtÞ

xðtÞ ¼ cþ

Pc1 j¼1

ð1Þj

c jþ1

! !m AC0N ðRþ Þ taj

with supp xCft: 0ptp12g; we obtain (2.9) for i ¼ 1; by Lemma 2.2, (2.1) and (2.10)– (2.11). Next, we prove (2.9) for i ¼ 2: Define   X  lðkÞ aðk; lÞ 1Z Pk ðgÞ: Ul ðgÞ ¼ l ð1  aðk; lÞÞm lðkÞol

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Below we will prove jjUl ðgÞjjpCjjgjj;

ð2:12Þ

with C40 independent of l and g: We rewrite Ul ðgÞ as Ul ðgÞ ¼ Ul1 ðgÞ þ Ul2 ðgÞ; where Ul1 ðgÞ ¼

Ul2 ðgÞ

  X  lðkÞ 1Z aðk; lÞ½1 þ maðk; lÞ Pk ðgÞ; l lðkÞol

   X  lðkÞ 1 ¼ 1Z  1  maðk; lÞ Pk ðgÞ: aðk; lÞ l ð1  aðk; lÞÞm lðkÞol

By Lemma 2.2 and (2.1), one can easily verify for i ¼ 1 jjUli ðgÞjjpCjjgjj:

ð2:13Þ

To deal with Ul2 ; we set 8 h i 1 a c < ð1  ZðtÞÞð1  ta Þc ; if 0ptp1; c m  1  mð1  t Þ a ð1ð1t Þ Þ jðtÞ ¼ :0 if tX1: Then, a straightforward computation shows Pðm1Þc ð1  ta Þ3c j¼0 Cm;c;j taj jðtÞ ¼ ð1  ZðtÞÞ ; ð1  ð1  ta Þc Þm

1 ptp1; 4

where the Cm;c;j are constants depending only on m; c and j: This clearly implies ðcþ1Þ

ðRþ Þ: Noticing that   N X lðkÞ 2 Ul ðgÞ ¼ j Pk ðgÞ; l k¼0

jAC0

by Lemma 2.2, we get (2.13) for i ¼ 2: Putting this together, we get (2.12). Now noticing that  am 1 2 Tl ðf Þ ¼ ðUl ðI  Rl;a;c Þm f ÞðamÞ ; l by Bernstein’s inequality ( see [3, (3.5), p. 330]), we get from (2.1) and (2.12) jjTl2 ðf ÞjjpCjjðI  Rl;a;c Þm f jj; which, together with (2.9) and (2.8), yields (2.7). This completes the proof. Now Theorem 2.1 is an immediate consequence of Lemma 2.3 and (2.2).

&

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3. Cesa`ro means In this section, we prove Theorem 3.1. Suppose cAN and (1.1) is satisfied. Then   1 jjf  scN ðf ÞjjEKa0 f ; ; N where a0 ¼ ðdeg lðxÞÞ1 and lðxÞ is as in assumption (ii). Throughout the rest of this section, the symbol a0 will always denote the number 1 deg lðxÞ: The proof of Theorem 3.1 is based on the following lemmas. Lemma 3.2. Suppose jðxÞ; fðxÞ are two algebraic polynomials of the same degree. Assume there exists a positive integer n0 ; such that fðxÞjðxÞ40 whenever xXn0 : For a given r40; define  N  X jðkÞ r Pk f : Tðf Þ ¼ fðkÞ k¼n 0

Then jjTðf ÞjjpCðj; f; r; n0 Þjjf jj: Proof. Let  CðxÞ ¼

jðxÞ fðxÞ

r ;

xXn0 :

Noticing that jðxÞ and fðxÞ are polynomials of the same degree, one can easily verify that  cþ2 1 ðcþ1Þ jC ðxÞjpC ; xXn0 : ð3:1Þ xþ1 Now let us define ( CðkÞ; kXn0 ; mk ¼ 0; 0pkon0 : Using Abel’s transformation c þ 1 times, taking into account (1.1), we obtain jjTðf ÞjjpCB

N X

jWcþ1 mk jkc jjf jj;

k¼0

which, by (3.1), implies the desired result.

&

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Lemma 3.3.   1 jjf  scN ðf ÞjjpCKa0 f ; ; N with C40 independent of N and f. Proof. We get the idea from [3]. Let g ¼ RlðN Þ;a ;c;1 f with RlðN Þ;a 2

0

2

0 ;c;1

defined as in

N

2 Hk and ðlðN2 ÞÞa0 BN; we get from (2.2) that (2.3). Observing that gA"k¼0

jjf  gjj þ

  1 ða0 Þ 1 jjg jjpCKa0 f ; : N N

On the other hand, by (1.1), jjscN ðf Þ  f jjp jjscN ðf Þ  scN ðgÞjj þ jjscN ðgÞ  gjj þ jjg  f jj p Cjjf  gjj þ jjscN ðgÞ  gjj: Hence, it suffices to prove that jjscN ðgÞ  gjjpC

1 ða0 Þ jjg jj: N

ð3:2Þ

From assumptions (iii) and (iv), it follows that lim jjscN ðgÞ  gjj ¼ 0;

N-N

which implies scN ðgÞ  g ¼

N X

ðsck ðgÞ  sckþ1 ðgÞÞ

k¼N

¼

N X k¼N

N

2 Ac X 1 kj jcðk þ 1Þ Pj ðgÞ: ðk þ 1 þ cÞðk þ 1Þ j¼0 Ack k  j þ 1

ð3:3Þ

Let ZAC0N ðRþ Þ be a C N function, defined on R; with the properties that ZðxÞ ¼ 1 N 2 Hk ; by (1.1) and for jxjp12 and ZðxÞ ¼ 0 for jxjX34: Then, noticing that gA"k¼0

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Lemma 3.2, we have N N   c X 2 Ac 2 j X Akj jcðk þ 1Þ kj jcðk þ 1Þ ¼ P Z ðgÞ ðgÞ P j j j¼0 Ack k  j þ 1 j¼0 Ack k  j þ 1 N N   X 2 kþ1 j ða0 Þ p C Z Pj ðg Þ j¼0 k  j þ 1 N 3     4N X j k þ 1 cþ1 ~ pC Z W N kþ1j j¼0

ðj þ 1Þc jjgða0 Þ jj: A straightforward computation shows that      cþ1 cþ1 j k þ 1 1 ~ W Z ; pC N kþ1j N

3 3 0pjp Np k: 4 4

ð3:4Þ

ð3:5Þ

Now substituting (3.5) into (3.4), taking into account (3.3), we obtain (3.2) and complete the proof. & Lemma 3.4.  a0 1 jjðscN ðf ÞÞða0 Þ jjpCjjf  scN ðf Þjj; lðNÞ with C independent of N and f. Now Theorem 3.1 follows directly from Lemmas 3.3 and 3.4 and the fact that lðNÞa0 BN: So, it remains to prove Lemma 3.4. To this end, we need some additional lemmas. Lemma 3.5. Let 8   c > < N 1  ANk ; AcN ak ¼ k > : 0;

1pkpN; kXN þ 1:

Then for i ¼ 0; 1; y; c þ 1 and 1pkp½34N þ 1;  i  iþ1 ! 1 1 ~ i ak jpC þ jW : N kþ1 Proof. We rewrite ak as ak ¼ bk þ c k ;

ð3:6Þ

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where

  ! N k c 1 1 bk ¼ ; k N !   N k c AcNk 1  c : ck ¼ k N AN

ð3:7Þ

Let jðtÞ ¼ 1t ð1  ð1  tÞc Þ: Noticing that ! c X c jðtÞ ¼ ð1Þj1 tj1 AC N ½0; N j j¼1 and bk ¼ jðNk Þ; we get for iAZþ ;  i 1 ~ i bk jpC jW ; 1pkpN: N þ1 Hence, by (3.6), it remains to show for i ¼ 0; 1; y; c þ 1;   iþ1 1 3 i ~ jW ck jpC ; 1pkp N þ 1: kþ1 4 On account of (3.7), it suffices to prove for 0pipc þ 1; !    k c AcNk 1 3 ~i 1  c ; 1pkp N þ 1: pCi W N 4 AN ðN þ 1Þiþ1

ð3:8Þ

Noticing that ~ Ad ¼ Ad1 ; W Nk Nk

d4  1;

and (see [8, p. 77, (1.18)])    kd 1 d Ak ¼ 1þO ; k Gðd þ 1Þ

ð3:9Þ

d4  1; kAN;

we get for 0pkp½34N þ 1; c ~ i ANk W AcN

8 ci A Gðcþ1Þ ðNkÞci i c ~ ¼ Gðcþ1iÞ ð1 þ OðN1 ÞÞ if 0pipc; W ANk < ANk c Nc N ¼ ¼ :0 AcN if i ¼ c þ 1: ð3:10Þ

On the other hand, it is easy to verify that for 0pkp½34N þ 1;  c ( Gðcþ1Þ 1 i kþyi;k ci if 0pipc; k Gðciþ1ÞðN Þ ð1  N Þ i ~ W 1 ¼ N 0 if i ¼ c þ 1; with 0oyi;k oi;

i ¼ 0; 1; y; c:

ð3:11Þ

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153

Now combining (3.10) and (3.11), we get for 0pipc and 0pkp½34N þ 1; !     k c AcNk Gðc þ 1Þ 1 k þ yi ci i ~ W 1  c 1  ¼ N Gðc  i þ 1ÞN i N AN  ci  ! k 1  1 þO N N  iþ1 ! 1 ¼O ; N and for i ¼ c þ 1 and 0pkp½34N þ 1; !   k c AcNk cþ1 ~ W 1  c ¼ 0; N AN which gives (3.8) and completes the proof.

&

Lemma 3.6. Suppose ak Xd40; k ¼ 0; 1; y: Then n1 ~ i1 akþj ?W ~ im akþj j: 1piu ; ju pn; 1pupmpn; ~ W p Cðd; nÞsupfjW 1 m ak i1 þ i2 þ ? þ im ¼ ng: Lemma 3.6 can be easily obtained by induction on n and using the following two identities: ~ ~ 1 ¼  W ak ; W ak ak akþ1 ! ~ ak W nþ1 1 n ~ ~ W ¼ W ak ak akþ1 ! !   ! j n X X n j 1 1 njþ1 i ji ~ ~ ~ ¼ akþj Þ W : ðW W ak akþ1þi j i j¼0 i¼0 Lemma 3.7. Let 8 c < ANk ; 0pkpN; AcN mk ¼ : 0; kXN þ 1: Then for i ¼ 0; 1; y; c þ 1 and N8 pkpN;  2   i i mk ~ W pC 1 : N 1  mk

ð3:12Þ

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Proof. First, we prove for i ¼ 0; 1; y; c;  i i 1 ~ W pC 1 ; kXN ; 1m N 8

ð3:13Þ

k

and for i ¼ c þ 1 ( 1 cþ1 cþ1 1 CðN Þ ; if N8 pkpN  c  2; ~ W p 1  mk if N  c  1pkpN þ c þ 1: CðN1 Þc ;

ð3:14Þ

Since for iX0 ~ i aNþ1þc þ ~ i ak ¼ W W

N þc X

~ iþ1 aj ; W

ð3:15Þ

j¼k

it is sufficient to prove (3.14). By (3.9), it is easy to verify that for 0pipc;  i 1 i ~ jW mk jpC ; kX0; ð3:16Þ N and for i ¼ c þ 1 ( 1 ¼ OðN1c Þ AcN cþ1 ~ W mk ¼ 0

if k ¼ N; otherwise:

ð3:17Þ

A straightforward computation shows that Ac7N 1  mk X1 

8 AcN

1 X 16

ð3:18Þ

whenever kXN8 : Now applying Lemma 3.6 with n ¼ c þ 1 and ak ¼ 1  mk ; we get, by (3.16) and (3.18),  cþ1 cþ1 1 ~ ~ cþ1 m j; W pC 1 þC max jW kþj 1pjpcþ1 1  mk N which, on account of (3.17), gives (3.14) and hence (3.13). Next, we prove for i ¼ 0; y; c þ 1; ~ i m2 jpCð 1 Þi ; jW k N As ~ i m2 W k

¼

k X

i

j¼0

j

kXN8 : ! ~ ij m W ~ jm ; W k k

(3.19) follows from (3.16) and (3.17).

ð3:19Þ

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Finally, we prove (3.12). By (3.15), it suffices to consider the case i ¼ c þ 1: We use the following identity  2  ~ cþ1 2   cþ1 W mk mk 1 2 ~ cþ1 ~ W ¼ þ m W k 1  mk 1  mk 1  mkþcþ1 !   c X c þ 1 ~ i 2 ~ cþ1i 1 þ W mk W : 1  mkþi i i¼1 We then get from (3.16), (3.17) and (3.19)  cþ1  2  cþ1 mk ~ W pC 1 : N 1  mk This gives (3.12) and completes the proof.

&

Proof of Lemma 3.4. Without loss of generality, we may assume P0 ðf Þ ¼ 0: Let ZAC0N ðRþ Þ such that ZðxÞ ¼ 1 for jxjp14 and ZðxÞ ¼ 0 for jxjX12: We decompose 1 a0 c ðlðNÞ Þ ðsN ðf ÞÞða0 Þ as  a0 1 ðscN ðf ÞÞða0 Þ ¼ TN1 ðf Þ þ TN2 ðf Þ; lðNÞ

ð3:20Þ

where TN1 ðf Þ

TN2 ðf Þ

   N  X lðkÞ a0 AcNk k :¼ Z Pk ðf Þ; c lðNÞ N AN k¼0 :¼

 N  X lðkÞ a0 Ac k¼0

lðNÞ

Nk AcN

   k 1Z Pk ðf Þ: N

We will prove for i ¼ 1; 2; jjTNi ðf ÞjjpCjjf  scN ðf Þjj: For i ¼ 1; by Berntein’s inequality and (1.1), we get   X N k k 1 Z jjTN ðf ÞjjpC Pk ðf Þ : k¼1 N N

ð3:21Þ

ð3:22Þ

Define 1 GN ðgÞ ¼

  k N X k N Z Pk ðgÞ: Ac N 1  Nk k¼1 AcN

Observe that for 1pkpN;  k !   AcNk N N c 1 1 1 c X 1 1 40: X k k N þc 2cþ1 AN

ð3:23Þ

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From Lemmas 3.5 and 3.6, it follows that 0 1    cþ2  cþ1 ! 1 1 1 ~ cþ1 B k C þ ; W @Z A pC c A N N ð1  Nk Þ kþ1 N c k AN which, by (1.1), implies 1 jjGN ðgÞjjpCjjgjj:

ð3:24Þ

Now combining (3.22)–(3.24), we obtain (3.21) for i ¼ 1: For i ¼ 2; we define    N X AcNk 1 k 2 GN ðgÞ ¼ 1  Z Pk ðgÞ: c AcNk N A N 1 k¼0 c A N

We

2 decompose GN as 2;1 2 GN ðgÞ ¼ GN ðgÞ

2;2 þ GN ðgÞ;

where 2;1 GN ðgÞ

:¼

AcNk 2 Þ AcN

N ð X k¼0

2;2 ðgÞ GN

1

AcNk AcN

 1Z

  k Pk ðgÞ; N

   N X AcNk k :¼ 1Z Pk ðgÞ: c N A N k¼0

From Lemma 3.7 and Abel’s transformation, it follows that 2;1 ðgÞjjpCjjgjj: jjGN

On the other hand, by assumption (iv) and Lemma 2.2, it is easy to verify 2;2 jjGN ðgÞjjpCjjgjj: Thus 2 jjGN ðgÞjjpCjjgjj:

Observing TN2 ðf Þ

 a0 1 ða0 Þ 2 c ¼ ðGN ðf  sN ðf ÞÞÞ ; lðNÞ

by Bernstein’s inequality (see [3, (3.5), p. 330]), we derive (3.21) for i ¼ 2: This completes the proof. & Remark 3.1. In assumption (iv) the condition ‘‘c is a positive integer’’ can be removed. Indeed, a modification of the above proofs will show that Theorems 2.1 and 3.1 remain valid with c replaced by any positive number d for which (1.1) is satisfied.

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Acknowledgments The author thank the referee for many helpful comments and Professor Wang Kunyang for his constructive suggestions. We thank Z. Ditzian for supplying us with some preprints of his excellent papers on K-functionals, which gave us a better perspective on our own results.

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