Inequalities for Trigonometric Polynomials and Some Integral Means

Journal of Mathematical Analysis and Applications 255, 147᎐162 Ž2001. doi:10.1006rjmaa.2000.7212, available online at http:rrwww.idealibrary.com on

Inequalities for Trigonometric Polynomials and Some Integral Means Hans-Bernd Knoop and Xinlong Zhou Department of Mathematics, Gerhard-Mercator-Uni¨ ersity of Duisburg, D-47057 Duisburg, Germany E-mail: [email protected] and [email protected] Submitted by A. M. Fink Received September 16, 1999

Some inequalities associated with the Laplacian for trigonometric polynomials are given, which will be applied to investigate the behavior in approximation by trigonometric polynomials in higher dimensions and the best lower and upper estimates for some linear operators. In particular, we obtain a complete characterization for the approximation behavior of the classical Jackson operator. Thus the approximation error of a function by Jackson polynomials is equivalent to a K-functional defined by the Laplacian. 䊚 2001 Academic Press Key Words: Bernstein inequality; Jackson inequality; Jackson operator; Laplacian; linear operator; K-functional.

1. INTRODUCTION Let ␲n, d be the set of trigonometric polynomials of degree F n with respect to each variable x i , i s 1, . . . , d. Denote Di [ ⭸r⭸ x i and D ␣ [ Di␣ 1 ⭈⭈⭈ Dd␣ d , where the multiindex ␣ s Ž ␣ 1 , . . . , ␣ d . has nonnegative integers ␣ i . The length of ␣ is < ␣ < s ␣ 1 q ⭈⭈⭈ q␣ d . Furthermore, denote by L p ŽT d . the L p-space of periodic functions with T s wy␲ , ␲ x, the usual norm 5 ⭈ 5 p, d , and 1 F p F ⬁. We identify L⬁ŽT d . with the space C ŽT d . of continuous periodic functions. The Sobolev space Wpr ŽT d . is given by Wpr Ž T d . [  f g L p Ž T d . : D ␣ f g L p Ž T d . , < ␣ < s r 4 . For f g L p ŽT d . the best approximation constant is defined by En Ž f . p , d [

inf

Tng ␲ n, d

5 f y Tn 5 p , d .

147 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

148

KNOOP AND ZHOU

Two basic results concerning the best approximation constants in approximation theory are the so-called Bernstein inequality and the Jackson inequality. Roughly speaking, the Bernstein inequality tells us that the increase in the L p-norm of derivatives of polynomials can be estimated by the product of the L p-norm of the polynomial and the degree of the corresponding polynomial. For example, there holds Žsee w12x. 5 D ␣ Tn 5 p , d F Cn < ␣ < 5 Tn 5 p , d ,

᭙1 F p F ⬁, Tn g ␲n , d .

Ž 1.1.

Using this inequality one can get the smoothness of the function f by the order of convergence of the sequence of EnŽ f . p, d Žsee w12x.. The above inequality is generalized with the use of the Laplacian as follows Žsee w16x for p s ⬁ and w17x for 1 F p - ⬁.: sup < ␤
5 D ␤ Tn 5 p , d F Ck n 5 ⌬k Tn 5 p , d ,

᭙1 F p F ⬁, Tn g ␲n , d . Ž 1.2.

There is another kind of inequality, which gives information on the conjugate function in the case of dimension d s 1. Let for f g L p ŽT . the conjugate function f˜ of f be defined by f˜Ž x . [ y

1

␲

H 2␲ 0

f Ž x q t. y f Ž x y t. tan Ž tr2 .

dt

Žcf. w18x.. In the case where 1 - p - ⬁ one has Žsee w18x. 5 T˜n 5 p , 1 F C p 5 Tn 5 p , 1 ,

᭙Tn g ␲n , 1 .

Ž 1.3.

The above inequality is not true for p s 1 and p s ⬁. In fact, the following estimate is in general sharp: for p s 1 or p s ⬁ one has Žsee w18, p. 12, Vol. IIx. 5 T˜n 5 p , 1 F C ln n 5 Tn 5 p , 1 ,

᭙Tn g ␲n , 1 .

On the other hand, it is known that for polynomials and their conjugate Žsee w18, p. 12, Vol. IIx.: functions one has the so-called Szego-inequality ¨ for all 1 F p F ⬁ there holds 5 T˜nX 5 p , 1 F Cn 5 Tn 5 p , 1 ,

᭙Tn g ␲n , 1 .

To generalize the above inequalities for higher dimensions let L i g be the ˜i [ Di L i . In this way we conjugate function of g with respect to x i and D ˜ ␣ s D˜1␣ 1 ⭈⭈⭈ D˜d␣ d . With these notations the Szego-inequality define D in the ¨ multivariate case is

˜ ␣ Tn 5 p , d F C␣ n < ␣ < 5 Tn 5 p , d , 5D In w10x we proved the following.

᭙Tn g ␲n , d .

149

TRIGONOMETRIC POLYNOMIAL INEQUALITIES

THEOREM 1.1. Let 1 F p F ⬁, k g ⺞, and the dimension d G 1 be fixed. Then there exists a positi¨ e constant Ck ) 0 such that for all ⑀ ) 0, all n, and all Tn g ␲n, d , sup < ␣ q ␤
˜ ␤ Tn 5 p , d F Ck 5 D ␣D

½

1

⑀

2 kq2

sup < ␣
5 D ␣ Tn 5 p , d q ⑀ n 5 ⌬k Tn 5 p , d .

5

Ž 1.4. The second basic result is the Jackson inequality. This inequality says that the best approximation constant can be estimated if the smoothness of the given function is known. The first result was pointed out by Jackson in 1912 Žsee w7x. in the case where d s 1 and p s ⬁. To consider his result let kn, r Ž t . s ␣n, r ⭈

sin 2 r Ž ntr2. sin 2 r Ž tr2 .

,

where the constant ␣ n, r is such that 5 k n, r 5 1, 1 s 1. Denote the so-called Jackson operator by Jn , r f s k n , r ) f . Jackson’s result may be formulated as 5 f y Jn , 2 f 5 p , 1 F C 2 ␻ 2 f ,

1

ž / n

, p

where ␻ 2 Ž f, ⭈ . p is the modulus of smoothness of second order for f Žsee w11x.. Hence the best approximation constant EnŽ f . p, 1 can be estimated by ␻ 2 Ž f, 1rn. p . In w8x Žfor r G 3. and w5x Žfor r s 2. we proved that there exists a positive constant Cr ) 0 such that for all 1 F p F ⬁, all f g L p ŽT ., all n G 1, and all r G 2, Cy1 r ␻2 f ,

1

ž / n

p

F 5 f y Jn , r f 5 p , 1 F C r ␻ 2 f ,

1

ž / n

. p

After more than 80 years it was shown that approximation of a function f by Jackson polynomials Jn, r f is equivalent to the second-order modulus of continuity. What is the analogy of the above result for the d-dimensional tensor product of the Jackson operator? Let K n, r be given by d

K n, r Ž t . s

Ł k n , r Ž ti . ,

is1

t s Ž t1 , . . . , t d . ,

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KNOOP AND ZHOU

and let Tn, r be the corresponding convolution operator with the kernel K n,r . Among others, we will prove THEOREM 1.2. For fixed d G 1, r G 2, and k G 1 there exists a positi¨ e constant Cr, d, k ) 0 such that for all 1 F p F ⬁, all f g L p ŽT d ., and all n G 1, 1 1 k k Cy1 F 5 Ž I y In , r . f 5 p , d F Cr , d , k K ⌬k f , , Ž 1.5. r , d , k K⌬ f , n p, d n p, d

ž /

where

K ⌬k Ž

ž /

f, ⭈ . p, d is the K-functional for f, defined by K ⌬k Ž f , t . p , d s

inf ggW p2 k ŽT d .

 5 f y g 5 p , d q t 2 k 5 ⌬k g 5 p , d 4 ,

where ⌬ is the Laplacian and I is the identity operator. For k s 1 Theorem 1.2 gives 1 1 Cy1 F 5 f y In , r f 5 p , d F Cr , d , 1 K ⌬ f , r , d , 1 K⌬ f , n p, d n

ž /

ž /

.

Ž 1.6.

p, d

Let us observe the definition of the K-functional K ⌬Ž f, ⭈ . p, d . Thus, if one replaces the Laplacian ⌬ by D ␣ Ž< ␣ < s 2. and takes the supremum over all such ␣ then the corresponding K-functional K 2 Ž f, ⭈ . p, d Žsee Section 2. is equivalent to the modulus of smoothness of second order in dimension d Žsee definition in Section 2.. The upper estimate in Ž1.6. with this modulus is known Žsee w12x.. Later we will see that in general the K-functionals K ⌬Ž f, ⭈ . p, d and K 2 Ž f, ⭈ . p, d are not equivalent. Thus in general K ⌬Ž f, ⭈ . p, d is not equivalent to the modulus of smoothness of second order. Therefore, not only the lower estimate in Ž1.6. but also the upper estimate is new. Theorem 1.2 will be proved in Section 3. In fact, we will prove a more general result Žsee Theorem 3.2., i.e., a lower and upper estimate for a large class of operators which is optimal even for each individual function in L p ŽT d .. The interest in the Jackson operator is due to its historical background. In Section 2 we investigate some general properties concerning the approximation by trigonometric polynomials.

2. APPROXIMATION BY TRIGONOMETRIC POLYNOMIALS This section is devoted to the investigation of approximation by trigonometric polynomials and the lower estimates for linear operators. We begin with the definition of other K-functionals. We define K k Ž f , t . p, d [

inf ggW pk ŽT d .

½5 f y g5

p, d

q t k sup 5 D ␣ g 5 p , d . < ␣
5

151

TRIGONOMETRIC POLYNOMIAL INEQUALITIES

It is known Žsee, e.g., w13x. that if 1 - p - ⬁ then sup 5 D ␣ g 5 p , d F C p , d , k 5 ⌬k g 5 p , d .

< ␣
Thus K 2 k Ž f, t . p, d is equivalent to K ⌬k Ž f, t . p, d . For p s 1 or p s ⬁, however, one can use the approach of w13x to show that these K-functionals are not equivalent. In general Žsee w17x for the details. we have for some C ) 0, which does not depend on f g L p ŽT d . and t, the estimate Cy1 K 2 kq1 Ž f , t . p , d F K ⌬k Ž f , t . p , d F CK 2 k Ž f , t . p , d .

Ž 2.1.

For convenience we shall give a briefly proof of these inequalities by using of the singular Weierstraß convolution integral. Let us first record some properties concerning the so-called Weierstraß convolution integral. The kernel of the singular Weierstraß integral can be written as KtŽ x. [

1

Ž 2'␲ t .

exp y

Ý

d

kg⺪ d

ž

Ý dis1 Ž 2␲ k i y x i .

2

4t

/

,

where k s Ž k 1 , . . . , k d .. Denote W Ž t. f [ Kt) f , then W Ž t .4 is a semi-group for t ) 0, i.e. W Ž t . j f s W Ž jt . f. The following properties are well-known Žsee w2x.: 5 f y W Ž t . f 5 p , d F Ct 5 ⌬ f 5 p , d ,

᭙ f , ⌬ f g Lp Ž T d .

Ž 2.2.

and for < ␣ < G 0 5 D␣ W Ž t . f 5 p, d F

C t

<␣< 2

5 f 5 p, d ,

᭙ f g Lp Ž T d . .

Ž 2.3.

Moreover, for f, D ␣ f g L p ŽT d ., D␣W Ž t. f s W Ž t. D␣ f ,

5 D␣ W Ž t . f 5 p, d F C 5 D␣ f 5 p, d .

Furthermore, by Taylor expansion and simple computation one can show that for g g Wp3 ŽT d . 5 g y W Ž t . g y ␦ Ž t . ⌬ g 5 p , d F Ct sup 5 D ␣ g 5 p , d , 3 2

< ␣
where < ␦ Ž t .< ; t, i.e., for some C⬘ ) 0 there holds C⬘y1 t F < ␦ Ž t .< F C⬘t, and the constant C does not depend on g. It follows from the last estimate and induction with respect to k that for g g Wp2 kq1 ŽT d . 5 Ž I y W Ž t . . k g q Ž y␦ Ž t . . k ⌬k g 5 p , d F Ck t

2 kq 1 2

sup < ␣
5 D ␣ g 5 p , d . Ž 2.4.

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KNOOP AND ZHOU

The definition of the K-functional and Ž2.8. imply 5 Ž I y W Ž t . . k f 5 p , d F Ck K ⌬k Ž f , 't . p , d ,

᭙ f g Lp Ž T d . .

Ž 2.5.

Now we are in the position to verify these inequalities. Proof of Ž2.1.. In the following, Ck or C will always denote positive constant, independent of n and p. In each formula the constants may take a different value. It is clear we need only to prove K 2 kq1 Ž f , t . p , d F CK ⌬k Ž f , t . p , d .

Ž 2.6.

Let us choose g s Ž I y Ž I y W Ž u.. k .Ž f . for some u ) 0. Thus, g g Wp2 kq1 ŽT d . and by Ž2.5. 5 f y g 5 p , d F Ck K ⌬k Ž f , 'u . p , d . Moreover, one has k

Ý Ž y1. j

gsy

js1

k W Ž ju . Ž f . . j

ž/

Hence, Ž2.3. and Ž2.5. imply for some 1 F k 0 F k 5 D ␣ g 5 p , d F Ck 5 D ␣ W Ž k 0 u . f 5 p , d s Ck

⬁

Ý D ␣ W j Ž k0 u. Ž I y W Ž k0 u. . f js1

s Ck

F Ck F Ck F Ck

p, d

⬁

Ý

j1s1 ⬁

Ý

k

Ý

j k sj ky1

⭈⭈⭈

j1s1

⬁

⭈⭈⭈

⬁

Ý

j k sj ky1

D ␣ W jk Ž k 0 u . Ž I y W Ž k 0 u . . f p, d

1

ž / jk k 0 u

1

5 'u Ž I y W Ž k 0 u . . 1

'u

2 kq 1 2

k

5Ž I y W Ž k0 u. . k f 5 p , d

f 5 p, d

K ⌬k Ž f , 'u . p , d .

The inequality Ž2.6. follows from the last two estimates and the definition of the K-functional. On the other hand, it is known Žsee w1, 6x. that K k Ž f, t . p, d is equivalent to the modulus of smoothness ␻ k Ž f, t . p, d , given by

␻k Ž f , t . p, d [

sup < h
½

k

Ý Ž y1. j js0

k f ⭈q jh . j

ž /Ž

p, d

5

.

TRIGONOMETRIC POLYNOMIAL INEQUALITIES

153

We conclude, therefore, K kq 1 Ž f , t . p , d F CK k Ž f , t . p , d ,

Ž 2.7.

K ⌬k Ž f , t . p , d F CK ⌬j Ž f , t . p , d .

Ž 2.8.

and for 1 F j F k,

One other fact to be considered is K ⌬j Ž f , it . p , d F i 2 j K ⌬j Ž f , t . p , d ,

᭙ i g ⺞.

Ž 2.9.

In this paper we will use K-functionals instead of moduli of smoothness to describe the approximation behavior. The following result can be found in w17x. For completeness we shall give a short proof. THEOREM 2.1. such that

For fixed d G 1, k G 1, and 1 F p F ⬁ there exists C ) 0

En Ž f . p , d F

C n2

En Ž ⌬ f . p , d ,

En Ž f . p , d F CK ⌬k f ,

1

ž /

K ⌬k Ž f , h . p , d F Ch 2 k

Ý

n

᭙ f g Wp2 Ž T d . ;

Ž 2.10.

᭙ f g Lp Ž T d . ;

Ž 2.11.

, p, d

Ž n q 1.

2 ky1

En Ž f . p , d ,

᭙ f g Lp Ž T d . .

0FnF1rh

Ž 2.12. Proof. The second inequality of Ž2.1. and the estimate in w12x Žsee pages 197᎐204 of w12x. imply Ž2.12., while Ž2.11. is a simply consequence of Ž2.10.. Thus, we need only to show Ž2.10.. It is known Žsee page 197 of w12x. that E n Ž f . p, d F CK 3 Ž f , n1 . p, d . Thus, Ž2.1 . implies E n Ž f . p, d F Cny2 5 ⌬ f 5 p, d . On the other hand, as for any Tn g ␲n, d there holds EnŽ f . p, d s EnŽ f y Tn . p, d , we conclude that En Ž f . p , d F Cny2 5 ⌬ f y ⌬Tn 5 p , d ,

᭙Tn g ␲n , d .

Assume TnU g ␲n, d a best approximation of ⌬ f, so there exists a Tn g ␲n, d such that ⌬Tn s c q TnU with c s yH TnU . Hence, the fact that H⌬ f s 0 implies < c < F H < ⌬ f y TnU < F CEnŽ ⌬ f . p, d . Therefore, En Ž f . p , d F Cny2 5 ⌬ f y ⌬Tn 5 p , d F Cny2 En Ž ⌬ f . p , d , which gives Ž2.10..

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KNOOP AND ZHOU

For applications Že.g., in computer-aided geometric design. sometimes one needs to know not only the distance between the approximation polynomial and the function which is approximated, but the uniform smoothness of this polynomial sequence in connection with this function as well. For a best approximation sequence TnU 4 of f one has Žsee w17x. 1 n

2k

1

5 ⌬k TnU 5 p , d F Ck K ⌬k f ,

ž / n

. p, d

The following result shows that, in fact, if the distance between an approximation polynomial and the approximated function is bounded by a K-functional then the uniform smoothness of this sequence can always be estimated by this K-functional. THEOREM 2.2. ␲n, d satisfy

Let f g L p ŽT d . and the polynomial sequence Tn4 , Tn g 5 Tn y f 5 p , d F Ck K ⌬k f ,

1

ž / n

. p, d

Then there exists CkX ) 0 such that for j s 1, . . . , k, 1 n

2j

5 ⌬ j Tn 5 p , q F CkX K ⌬j f ,

1

ž / n

, p, d

and K ⌬j Ž Tn , t . p , d F CkX K ⌬j Ž f , t . p , d . Proof. Let us first estimate 5 D ␣ Tn 5 p, d with < ␣ < s 2 j q 1. Assuming N to be such that 2 N F n - 2 Nq 1, we conclude from the assumptions in this theorem, the Bernstein inequality, Ž2.8., and Ž2.9. that 5 D ␣ Tn 5 p , d F 5 D ␣ Ž T0 y T1 . 5 p , d q 5 D ␣ Ž T2 N y Tn . 5 p , d N

q Ý 5 D ␣ Ž T2 i y T2 iy 1 . 5 p , d is1 N

F C Ý 2 iŽ2 jq1. K ⌬j f ,

ž /

is1

F CK ⌬j f ,

1

1

ž / n

2i

p, d

N

Ý 2 iŽ2 jq1.

p , d is1

n

ž / 2

i

2j

F Cn2 jq1 K ⌬j f ,

1

ž / n

. p, d

Combining the last estimate with Ž2.4., we deduce, because of < ␦ Ž t .< ; t, 1 n

1

½ ž ž //

5 ⌬ j Tn 5 p , d F C 5 I y W 2j

n

2

j

1

ž / 5

Tn 5 p , d q K ⌬j f ,

n

p, d

.

155

TRIGONOMETRIC POLYNOMIAL INEQUALITIES

To get the first assertion we have to prove that the first term of the last inequality on the right-hand side can be estimated by the second one. We therefore write this term as

ž

j

1

IyW

ž // ž ž // Ž

F

Tn

n2

p, d

IyW

j

1

n2

Tn y f .

q p, d

ž

IyW

1

j

ž // n2

f

. p, d

By the boundedness of W Ž t . Žsee Ž2.3.., the assumption, and Ž2.8. we get

ž

IyW

j

1

ž // n

2

Ž Tn y f . p, d

F C j 5 Tn y f 5 p , d F Ck K ⌬j f ,

1

ž / n

. p, d

On the other hand, Ž2.5. tells us

ž

IyW

j

1

ž // n

2

f p, d

1

F CK ⌬j f ,

ž / n

. p, d

In this way we obtain from the foregoing 1 n

2j

5 ⌬ j Tn 5 p , d F Ck K ⌬j f ,

1

ž / n

,

j s 1, . . . , k.

p, d

To show the second assertion we notice that if nt F 1 then we get, with the first assertion and Ž2.9., K ⌬j Ž Tn , t . p , d F t 2 j 5 ⌬ j Tn 5 p , d 2j

F Ck Ž tn . K ⌬j f ,

1

ž / n

p, d

F Ck K ⌬j Ž f , t . p , d .

If tn ) 1, i.e., t ) 1rn, we get from the assumption and Ž2.8. the estimate K ⌬j Ž Tn , t . p , d F K ⌬j Ž Tn y f , t . p , d q K ⌬j Ž f , t . p , d F 5 Tn y f 5 p , d q K ⌬j Ž f , t . p , d F Ck K ⌬j f ,

1

ž ž / n

p, d

q K ⌬j Ž f , t . p , d F Ck K ⌬j Ž f , t . p , d ,

/

which verifies the second assertion of this theorem.

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KNOOP AND ZHOU

3. THE LOWER ESTIMATES FOR LINEAR OPERATORS AND THE PROOF OF THEOREM 1.2 If we consider the best approximation polynomials as operators of the approximated function, then these operators are in general not linear. There are, however, only a few cases for which one can get the exact form of the best approximation polynomials. Thus, the linear method or the linear operator approach is a good alternative for approximation by polynomials; the linear positive method in particular has attracted special interest in the past. In connection with the approximation degree for the linear method one investigates, among other things, the Jackson type estimate Ži.e., the upper bound of the approximation error., the inverse theorem, and the saturation phenomenon. We shall refer to w11, 12, 14x for the detailed information concerning these subjects. In w8x Žsee also w4, 9x. we introduce a new technique. Using this technique we unify the abovementioned subjects for a large class of positive linear operators in one form. In this way we obtain for example the best lower and upper estimates for the Bernstein operator Žsee w9, 15x., which completely characterizes the approximation behavior of this operator. For the positive convolution operators the following results generalize the theorems in w8x Žsee also w17x.. To begin with let ⌽ be a positive kernel. The convolution operator L defined by this ⌽ is Lf s f ) ⌽ , which satisfies Lf s 1 for f ' 1 and Lf g L p ŽT d . whenever f g L p ŽT d ., 1 F p F ⬁. We shall denote by ⌽j the kernel of L j, which is given by L1 [ L and L j f [ LŽ L jy1 f .. THEOREM 3.1.

Let there exist some j G 1 such that ⌽j g C 1 ŽT d . and

⑀y2 [

HT

< grad ⌽j Ž t . < 2 ⌽j Ž t .

d

dt - q⬁.

Then for fixed k g ⺞ there exists a constant Ck ) 0, which does not depend on ⑀ , such that for all f g L p ŽT d . and 1 F p F ⬁, K 2 kq1 Ž f , ⑀ . p , d F Ck 5 Ž I y L . f . 5 p , d . k

Proof. In w8x we proved that for d s 1 one has 5 D i L ji N f 5 p , 1 F

1 N

i r2 i

⑀

5 f 5 p, 1 ,

i , N s 1, 2, . . . .

157

TRIGONOMETRIC POLYNOMIAL INEQUALITIES

Without any additional difficulties one can use the method of w8x to show this inequality for d ) 1. Thus, for < ␣ < s i, 5 D ␣ L ji N f 5 p , d F

1 N

i r2 i

⑀

5 f 5 p, d ,

i , N s 1, 2, . . . .

Ž 3.1.

To verify the assertion of this theorem we observe that since ⌽j g C 1 ŽT d . one has L ji f g Wpi ŽT d . whenever f g L p ŽT d .. Let us denote Lk [ I y Ž I y L j . k . Thus the above imply Lk2 kq1 f g Wp2 kq1 ŽT d . whenever f g L p ŽT d .. On the other hand, the definition of the K-functional tells us K 2 kq1 Ž f , ⑀ . p , d F 5 f y Lk2 kq1 f 5 p , d q ⑀ 2 kq1

sup < ␣
5 D ␣ Lk2 kq1 f 5 p , d .

Ž 3.2. Clearly, 5 f y Lk2 kq1 f 5 p , d F Ck , j 5 Ž I y L . k f 5 p , d .

Ž 3.3.

To estimate the second term of the right-hand side of Ž3.2., we use the fact that Lk2 kq1 is a linear combination of L jl with l s 2 k q 1, . . . , k Ž2 k q 1.. We can therefore replace L k2 kq 1 with L jŽ2 kq 1. and estimate 5 D ␣ L jŽ2 kq1. f 5 p, d as 5 D ␣ L jŽ2 kq1. f 5 p , d s

⬁

Ý

D ␣ L jl Ž I y L j . f

ls2 kq1

s

⬁

Ý

p, d

⭈⭈⭈

l 1s2 kq1

⬁

Ý

k

D ␣ L jl k Ž I y L j . f

l k sl ky1 k

F 5Ž I y L j . f 5 p , d

p, d

⬁

Ý

⭈⭈⭈

l1s1

Ž 3.4.

⬁

1

Ý

⑀ 2 kq1 l kŽ2 kq1.r2

l k sl ky1

F Ck ⑀y2 ky1 5 Ž I y L . f 5 p , d , k

where we have used Ž3.1. to get the first inequality. The assertion then follows from Ž3.2. ᎐ Ž3.4.. Let us now use this result to get the best lower and upper estimates in terms of K ⌬k Ž f, h. p, d , if the kernel ⌽ satisfies some additional conditions. THEOREM 3.2. Let the condition of Theorem 3.1 be fulfilled. If there exists some constant A ) 0 such that for all trigonometric polynomials Tn of degree F max 1, ⑀y1 4 and 1 F p F ⬁, 5 Tn y LTn y ␩ Ž ⑀ . ⌬Tn 5 p , d F A ⑀ 3

sup < ␣ q ␤
˜ ␤ Tn 5 p , d 5 D␣D

Ž 3.5.

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with ␩ Ž ⑀ . ; ⑀ 2 , then there hold for some Ck ) 0, which does not depend on ⑀ , the inequalities k k 5 5 Cy1 k K ⌬ Ž f , ⑀ . p , d F Ž I y L . f p , d F Ck K ⌬ Ž f , ⑀ . p , d , k

᭙ f g Lp Ž T d . .

Ž 3.6.

Proof. We may assume ⑀ F 1. It follows from Ž1.1. and Ž3.5. for n F ⑀y1 that 5 Tn y LTn 5 F C⑀ 2

sup < ␣ q ␤
˜ ␤ Tn 5 p , d . 5 D␣D

By using induction we conclude from the above estimate and Ž3.5. that for n F ⑀y1 , 5 Ž I y L . k Tn y ␩ Ž ⑀ . k ⌬k Tn 5 p , d F Ck ⑀ 2 kq1

sup < ␣ q ␤
˜ ␤ Tn 5 p , d , 5 D␣D

᭙Tn g ␲n , d .

Ž 3.7.

Let us verify the second inequality of Ž3.6.. On the other hand, it is clear that according to the definition of the K-functional we only need to show that for f g Wp2 k ŽT d ., 5 Ž I y L . k f 5 p , d F C⑀ 2 k 5 ⌬k f 5 p , d .

Ž 3.8.

To this end, choose n such that ⑀ F 1rn - 2 ⑀ and let Tn be a best approximation polynomial of f. Thus, Ž2.10. of Theorem 2.1 implies 5 f y Tn 5 p , d F C⑀ 2 k 5 ⌬k f 5 p , d . On the other hand, as 5 ⌬k Tn 5 p, d F C 5 ⌬k f 5 p, d Žsee Theorem 2.2., we obtain, by using Ž3.7., 5 Ž I y L . k f 5 p , d F 5 Ž I y L . k Ž f y Tn . 5 p , d q 5 Ž I y L . k Tn 5 p , d F C⑀ 2 k 5 ⌬k f 5 p , d q C⑀ 2 kq1

sup < ␣ q ␤
˜ ␤ Tn 5 p , d . 5 D␣D

The inequality Ž1.4. of Theorem 1.1 tells us that sup < ␣q ␤
ž

/

159

TRIGONOMETRIC POLYNOMIAL INEQUALITIES

where Tn is as above. Theorems 2.1 and 3.1 yield 5 f y Tn 5 p , d F CK 2 kq1 Ž f , ⑀ . p , d F C 5 Ž I y L . k f 5 p , d , while Ž3.7. and the above estimate imply k ⑀ 2 k 5 ⌬k Tn 5 p , d F C 5 Ž I y L . Tn 5 p , d q ⑀ 2 kq1

ž žŽ

k F C 5 I y L . f 5 p , d q ⑀ 2 kq1

sup < ␣ q ␤
sup < ␣ q ␤
˜ ␤ Tn 5 p , d 5 D␣D

/

˜ ␤ Tn 5 p , d . 5 D␣D

/

We then apply Theorem 1.1 to obtain from the last estimate k ⑀ 2 k 5 ⌬k Tn 5 p , d F C 5 Ž I y L . f 5 p , d q ⑀ 2 kq1

ž

sup < ␣
5 D ␣ Tn 5 p , d .

/

It remains to show that

⑀ 2 kq1 5 D ␣ Tn 5 p , d F C 5 Ž I y L . f 5 p , d . k

To this end, we use first the Bernstein inequality to get

⑀ 2 kq1 5 D ␣ Tn 5 p , d F C⑀ 2 kq1 Ž 5 D ␣ Ž Tn y Lk2 kq1 Tn . 5 p , d q 5 D ␣ Lk2 kq1 Tn 5 p , d . k F C 5 Ž I y L . Tn 5 p , d q ⑀ 2 kq1 5 D ␣ Lk2 kq1 Tn 5 p , d .

ž

/

Moreover, the approach in the proof of Theorem 3.1 Žsee Ž3.4.. tells us

⑀ 2 kq1 5 D ␣ Lk2 kq1 Tn 5 p , d F C 5 Ž I y L . Tn 5 p , d . k

It follows from the last two estimates that

⑀ 2 kq1 5 D ␣ Tn 5 p , d F C 5 Ž I y L . Tn 5 p , d k

F C 5 Ž I y L . Ž Tn y f . 5 p , d q C 5 Ž I y L . f 5 p , d k

k

F C 5 Ž Tn y f . 5 p , d q C 5 Ž I y L . f 5 p , d k

F C 5Ž I y L. f 5 p , d . k

This finishes the proof of this theorem. It is clear that Theorem 3.2 involves the Jackson inequality, the inverse theorem, and the characterization of saturation. In many cases the quantity ⑀ 2 in Theorems 3.1 and 3.2 is equivalent to the second moment of the operator L. In w3x one may find many operators which satisfy the conditions of these theorems. Let us now verify Theorem 1.2.

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Proof of Theorem 1.2. In view of Theorems 3.1 and 3.2, the only things we have to do are the calculation of ⑀ for K n, r Žsay ⑀ n . instead of ⌽j and the check of Ž3.5. for In, r . For the quantity ⑀ n we have

⑀y2 n [

HT

< grad K n , r Ž t . < 2 d

dt ; n2 .

K n, r Ž t .

Ž 3.9.

Indeed, by the definition we obtain

⑀y2 n s 2d

␲

H0

< kXn , r Ž t . < 2 kn, r Ž t .

dt.

The last integral is equivalent to n2 as shown in w8x. The check of Ž3.5. for In, r will be done in two cases, namely, r G 3 and r s 2. The case r G 3 is simple. In fact, using Taylor’s formula and straightforward calculation, we conclude that for g g Wp3 ŽT d . there holds 5 g y In , r g y Cn ⌬ g 5 p , d F Cny3 sup 5 D ␣ g 5 p , d , < ␣
Ž 3.10.

where < Cn < ; ny2 . It remains to show the case r s 2. We have the following assertion: there holds for all 1 F p F ⬁ and Tm g ␲m, d with m F ⑀y1 the estimate n 5 Tm y In , 2 Tm y Cn ⌬Tm 5 p , d F Cny3

sup < ␣ q ␤
˜ ␤ Tm 5 p , d , Ž 3.11. 5 D␣D

where < Cn < ; ny2 . The proof of Ž3.11. is involved. To begin with, we consider m F n. In w5x we show that for d s 1 one has Jn , 2 Tn s Tn q CnTnY y C2, nT˜nX q C3, nT˜nZ ,

᭙Tn g ␲n , 1 ,

where < Cn < ; ny2 , < C2, n < ; ny3 , and < C3, n < ; ny3 . Hence, in ␲n, 1 we have

˜ q C3, n D˜ 3 . Jn , 2 s I q Cn D 2 y C2, n D Therefore, in ␲n, d we may write In, 2 as d

In , 2 s

Ł ž I q Cn Di2 y C2, n D˜i q C3, n D˜i3 /

is1

d

s I q Cn ⌬ y C2, n

Ý D˜i

ž / is1

TRIGONOMETRIC POLYNOMIAL INEQUALITIES

q C2,2 n

161

˜i D˜j q R Ž D . , D

Ý i/j, 1Fi , jFd

where RŽ D. s

˜ ␤j , a j, n D ␣ j D

Ý 1FjF4

d

with < a j, n < F Cn and 3 F < ␣ j q ␤ j < F 3d. Hence, we conclude from the Bernstein inequality that for any Tn g ␲n, d , y< ␣ jq␤ j <

5 R Ž D . Tn 5 p , d F C d ny3

sup < ␣ q ␤
˜ ␤ Tn 5 p , d . 5 D␣D

Moreover, 5 C2,2 n

˜i D˜j Tn 5 p , d D

Ý i/j, 1Fi , jFd

˜i Tn 5 p , d . F Cny3 sup 5 D i

˜i Tn 5 p, d let k be such that nr2 - 2 k F n and let T2 j be a To estimate 5 D best approximation polynomial of Tn . Now the Bernstein inequality and Theorem 2.1 Žsee Ž2.6.. tell us that ˜i Tn 5 p , d F 5D

k

Ý 5 D˜iT2

j

˜i T2 jy 1 5 p , d q 5 D˜i Tn y D˜i T2 k 5 p , d yD

js1

˜i T1 y D˜i T0 5 p , d q5D k

FC

Ý 2 jK 3 js1

ž

Tn ,

1 2j

/

p, d

F C sup 5 D ␣ Tn 5 p , d . < ␣
The above calculation implies Ž3.11. for m F n. To relax the restriction m F n we notice that ⑀y1 n F cn for some c ) 0. Thus, we need only deal with n F m F cn for Ž3.11. and show that for such m Ž3.11. is still true. To this end, let TnU g ␲n, d be a best approximation polynomial of Tm g ␲m, d . We then use the fact that Ž3.11. is true for m F n and Theorem 2.1 to obtain 5 Tm y In , 2 Tm y Cn ⌬Tm 5 p , d F 5 TnU y In , 2 TnU y Cn ⌬TnU 5 p , d q Cny3 sup 5 D ␣ Tm 5 p , d < ␣
F Cny3 F Cny3

ž

sup < ␣ q ␤
sup < ␣ q ␤
˜ ␤ Ž Tm y TnU . 5 p , d q 5 D␣D

sup < ␣ q ␤
˜ ␤ Tm 5 p , d 5 D␣D

˜ ␤ Tm 5 p , d . 5 D␣D

Having Ž3.9. ᎐ Ž3.11., we apply Theorems 3.1 and 3.2 to finish the proof.

/

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