J Fourier Anal Appl (2008) 14: 16–38 DOI 10.1007/s00041-007-9004-y

On Approximation Methods Generated by Bochner-Riesz Kernels Konstantin Runovski · Hans-Jürgen Schmeisser

Received: 8 August 2006 / Published online: 23 January 2008 © Birkhäuser Boston 2008

Abstract Means and families of operators generated by Bochner-Riesz kernels are studied. Some sharp results on their convergence are achieved. The equivalence of the approximation errors of these methods to smoothness quantities related to the Laplacian is proved. Keywords Bochner-Riesz kernels and means · Families of operators · Necessary and sufficient conditions of convergence · K-functional · Realizations and moduli of smoothness related to the Laplacian Mathematics Subject Classification (2000) 42A10 · 42A15 1 Introduction and Notation In the present article we are concerned with approximation properties of families of operators defined by Sn;λ (f, x) = (2n + 1)−d · (α)

2n 

    f tnk + λ · Kn(α) x − tnk − λ ,

n ∈ N,

(1.1)

k=0

Communicated by Hans G. Feichtinger. K. Runovski Slavyansk University, Batyuka Str. 19, 84166 Slavyansk, Ukraine K. Runovski () Institute of Mathematics, National Academy of Sciences, Tereshenkovska Str. 3, 01601 Kiev, Ukraine e-mail: [email protected] H.-J. Schmeisser Mathematical Institute, University of Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany e-mail: [email protected]

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17

where Kn(α) (h) =



1−

|k|≤n

|k|2 n2

α · eikh

(1.2)

is the Bochner-Riesz kernel of order α ≥ 0, in Lp -spaces of 2π -periodic functions of d variables for 0 < p ≤ +∞. In formulas (1.1)–(1.2) x, k, λ are d-dimensional vectors, |k| = (k12 +. . .+kd2 )1/2 , kh = k1 h1 + . . . kd hd and tnk =

2πk , 2n + 1

k ∈ Zd ;

2n 

=

k=0

2n 

···

k1 =0

2n 

.

kd =0

Together with the families of type (1.1) we investigate the Riesz means which are given by  f (x + h) · Kn(α) (h) dh, n ∈ N. (1.3) Sn(α) (f ; x) = (2π)−d Td

Here, Td = [0, 2π)d denotes the d-dimensional torus. Bochner-Riesz means, being a classical method of trigonometric approximation, are intensively studied by many mathematicians. For references and more details we refer to [18, Chapter 7]. Here, we only mention that the Bochner-Riesz means converge in Lp for all 1 ≤ p ≤ +∞, provided α > (d − 1)/2. The number (d − 1)/2 is known as the critical index for the convergence of Riesz means (see [18, p. 255]). Moreover, in this case their approximation error in the norm of the space C of continuous 2π -periodic functions (as usual, it corresponds to p = +∞) turns out to be equivalent to the modulus of smoothness at the point 1/n which is given by        1 •   ω (f ; δ)∞ =  d (f (x + u) − f (x)) du , δ > 0, (1.4)  δ · μ(D)    |u|≤δ ∞

where μ(D) is the measure of the d-dimensional unit ball D[21]. Using standard arguments of duality and interpolation, one can easily transfer this result to the case 1 ≤ p < +∞. Hence, one has   f − S (α) (f )  ω• (f, 1/n)p , f ∈ Lp , n p

n ∈ N, 1

(1.5)

for 1 ≤ p ≤ +∞ and α > (d − 1)/2. Combining (1.5) and the results in [6] on the equivalences of averaged moduli of continuity and K-functionals related to the Laplacian, one can also state that   f − S (α) (f )  ω◦ (f ; 1/n)p , n p

f ∈ Lp , n ∈ N,

(1.6)

1 As usual, A (f )  B (f ), if there exist positive constants c and c such that c A (f ) ≤ B (f ) ≤ n n n 1 2 1 n c2 An (f ) for all f and n.

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for 1 ≤ p ≤ +∞ and α > (d − 1)/2. Here,      1 ◦  , ω (f ; δ)p = sup  (f (x + u) − f (x)) dS(u)  hd−1 · m(S)  |u|=h 0
δ > 0, 2

(1.7) is another spherical modulus of smoothness. In contrast to (1.4) it contains the average on the surface of the ball of radiush. Clearly, S denotes the unit sphere in Rd . Moreover, it holds   f − S (α) (f )  K(f, 1/n)p , f ∈ Lp , n ∈ N, (1.8) n p for 1 ≤ p ≤ +∞ and α > (d − 1)/2, where

K(f ; δ)p = inf f − gp + δ 2 · gp , g∈Wp2

δ > 0,

(1.9)

is a K-functional. In (1.9) Wp2 = Wp2 (Td ), 1 ≤ p < +∞ is the Sobolev space on the 2 torus Td (see, for instance, [20, pp. 54–56] or [16, Chapter 3]). If p = +∞ then W∞ 2 3 means the space C . Clearly, summation methods for Fourier series as in (1.3) are no longer relevant for approximation in the spaces Lp with 0 < p < 1, whereas the method of approximation by families of operators makes sense for all 0 < p ≤ +∞ [13, 14]. Moreover, the families of type (1.1) can be treated by the same scheme we have applied to the Fourier means. The only difference is that their norms and approximation errors must be taken in average with respect to the parameterλ. The exact definitions and details can be found in the next section (see also [2] for the one-dimensional case). We shall show that in contrast to the corresponding Bochner-Riesz means (1.3) the families given by (1.1) with α > (d − 1)/2 converge in Lp for some p < 1 as well. Moreover, the range of admissible parameters p will be determined precisely. This extension is very natural in the following sense. If both methods converge in Lp for some 0 < p ≤ +∞, then their approximation errors turns out to be equivalent in Lp . It should be noticed that in the case 0 < p < 1 the K-functional given in (1.9) can not be used for such characterizations. For, it was shown in [8] that it is equal to 0. In accordance with the concept of “Realization,” which was introduced in [12], the K-functional can be replaced by the quantity

 ; δ)p = inf f − tp + δ 2 · tp , δ > 0, (1.10) K(f t∈T1/δ

where Tσ =

⎧ ⎨ ⎩

t (x) =

 k∈Zd

  ck eikx : c−k = ck , |k| = k12 + . . . + kd2 ≤ σ 1 2

⎫ ⎬ ⎭

,

2 dS(u) denotes the integration with respect to the Lebesgue measure on the sphere. 3 C m is the space of 2π -periodic functions having continuous derivatives up to the order m.

σ > 0,

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19

stands for the space of all real valued trigonometric polynomials of order at most σ . As it follows from the results in [3, 4], and [8] the quantities given in (1.9) and (1.10) are equivalent for 1 ≤ p ≤ +∞. However, note that (1.10) makes sense for all 0 < p ≤ +∞. (α) Our aim is to show that the approximation error of {Sn;λ } with α > 0 in Lp (T 2d )  ; 1/n)p are equivalent in all cases where the family (1.1) converges. In view and K(f of the above mentioned results this is a direct extension of equivalence (1.8). We will also see that a similar result holds true for the means (1.3). In other words, equivalence (1.5) is valid for all admissable pairs (1/p, α) with 1 ≤ p ≤ +∞ and α > 0 if (α) the method Sn converges in Lp . This statement is essentially stronger than the result on equivalence (1.5) obtained in [21] and its consequences (1.6) and (1.8), because the condition α > (d − 1)/2 is required no longer. Next, we want to consider the problem of characterization of the rate of convergence of the families (1.1) in Lp -spaces with 0 < p < 1 by means of smoothness moduli. Obviously, the moduli (1.4) and (1.7) are not suitable because integrability of the given function f is supposed. As it was shown in [4] the modulus   d       ω(f ; δ)p = sup 2 df (·) − (f (· + heν ) + f (· − heν )) , δ > 0, (1.11)   |h|≤δ ν=1

p

where eν are the unit vectors in direction of the coordinates of Td , is equivalent to the K-functional given in (1.9) for 1 ≤ p ≤ +∞. On the other hand, it is welldefined for all 0 < p ≤ +∞. Taking all these facts into account one might expect the equivalence to the realization (1.10) to be valid for all 0 < p ≤ +∞ as well. As it was proved in [7], it is not so if d > 1. More presicely, the modulus (1.11) turns out to be equivalent to the realization given in (1.10) if and only if d/(d + 2) < p ≤ +∞. Combining this fact and the preceding discussion on the equivalence of the approximation error of the method (1.1) and the realization (1.10), we will conclude (α) that in case of convergence of the method {Sn;λ } in Lp and d > 1  Td

  f − S (α) (f )p dλ n;λ p

1/p   ω(f ; 1/n)p ,

f ∈ Lp , n ∈ N,

(1.12)

holds if and only if d/(d + 2) < p ≤ +∞. The article is organized as follows. In Section 2 we formulate the main results. Section 3 is devoted to estimates of the norms of the families (1.1). In Section 4 we collect some general facts on the convergence of families of operators due to the Banach-Steinhaus theorem. In Section 5 helpful inequalities for trigonometric polynomials are proved. The proofs of the main results are given in Section 6.

2 Main Results We shall deal with 2π -periodic functions in Lp , where 0 < p ≤ +∞ (as usual, we put L∞ = C with standard norm). Sometimes the functions may depend additionally

20

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Fig. 1 (d = 1)

on a parameter λ ∈ T d . We denote by  · p or  · p;x the p-norm (quasi-norm, if 0 < p < 1) with respect to x. The p- (quasi-)norm with respect to the parameter λ is denoted by the symbol  · p;λ . Moreover,  · p stands for  · p;x p;λ , that is, for the p-(quasi)norm with respect to both x and λ. For the sake of simplicity we will use the notation “norm” for all 0 < p ≤ +∞. A sequence of linear operators {Ln }n∈N , mapping Lp (1 ≤ p ≤ +∞) into the space Tn of real valued trigonometric polynomials of order at most n (”sequence of linear polynomial operators”) is said to be convergent (or converges) in Lp , if lim f − Ln (f )p = 0.

n→+∞

(2.1)

for each f ∈ Lp (T d ). On analogy, a family of linear operators {Ln;λ }n∈N, λ∈T d , mapping Lp (0 < p ≤ +∞) into Tn converges in Lp , if lim f − Ln;λ (f )p = 0.

n→+∞

(2.2)

for each f ∈ Lp (T d ). In order to formulate the main results of this article, we split the domain R2+ of pairs (1/p, α) into three parts (see Figures 1 and 2):       1 1 1 1 d −1

(d) = , α ∈ R2+ : α > max ,d − − , (2.3) p 2 p 2 2       1 1 1 1 2   , α ∈ R+ : 0 ≤ α ≤ d  −  − , (2.4) (d) = p p 2 2       1 1 1 1 d −1 2   (d) = , α ∈ R+ : 0 ≤ α ≤ , α > d −  − . (2.5) p 2 p 2 2 (α)

Theorem 1 The family {Sn;λ } converges in Lp if (1/p, α) ∈ (d) and it diverges in Lp if (1/p, α) ∈ (d).

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21

Fig. 2 (d > 1)

Theorem 1 gives an exact criterion for the convergence of the families generated by Bochner-Riesz kernels in Lp - spaces for 0 < p ≤ 1 and p = +∞. In this sense it (α) can be refomulated as follows: (1) The family {Sn;λ } converges in C, L1 or in Lp for (α)

all 1 ≤ p ≤ +∞ if and only ifα > (d − 1)/2. (2) Let 0 < p < 1. The family {Sn;λ } converges in Lp if and only if α > d(1/p − 1/2) − 1/2. As it was mentioned above, the criterion of convergence for Bochner-Riesz means (α) Sn is the same in the spaces C, L1 or in Lp for all 1 ≤ p ≤ +∞, simultaneously. The following assertion strengthens this property. Theorem 2 If 1 ≤ p ≤ +∞, α > (d − 1)/2, and f ∈ Lp (T d ) then     f − S (α) (f )  f − Sn(α) (f )  K(f ; 1/n)p n;λ p p  ; 1/n)p  ω• (f ; 1/n)p  K(f

(2.6)

◦

 ω (f ; 1/n)p   ω(f ; 1/n)p . Recently, Ditzian (cf. [5], Theorem 2) gave a direct proof of the equivalence   f − S (α) (f )  K(f ; 1/n)p n p in the nonperiodic case. Next, we consider the spaces Lp (T d ) with 0 < p < 1. Clearly, the only concepts making sense in this case are the family (1.1), the realization (1.10) and the modulus given in (1.11). Theorem 3 If 0 < p < 1 and α > d(1/p − 1/2) − 1/2, then   f − S (α) (f )  K(f  ; 1/n)p , n;λ p

f ∈ Lp , n ∈ N.

(2.7)

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Taking into account that the relations between the realization (1.10) and the modulus defined in (1.11) essentially depend on the dimension [18], we should distinguish the cases d = 1 and d > 1 when the problem of comparison of the approximation error of the method (1.1) and the quantity (1.11) is considered. We first deal with the one-dimensional case. Clearly, the modulus given in (1.11) coincides with the classical modulus of continuity of second order ω2 (f ; δ)p . Theorem 4 If d = 1, 0 < p < 1 and α > 1/p − 1, then   f − S (α) (f )  ω2 (f ; 1/n)p , n;λ p

f ∈ Lp , n ∈ N.

(2.8)

Theorem 5 Let d > 1, 0 < p < 1 and α > d(1/p − 1/2) − 1/2. If d/(d + 2) < p < 1 then   f − S (α) (f )   ω(f ; 1/n)p , f ∈ Lp , n ∈ N. (2.9) n;λ p If 0 < p ≤ d/(d + 2), the upper estimate in (2.9) is valid and the lower estimate is false. Theorems 1–5 give complete solutions to the problems concerning convergence and approximative properties of the families given in (1.1) in (d) and (d). In contrast to these cases the problem of convergence in the domain (d) is not completely studied even for the Bochner-Riesz means (1.3) if d ≥ 3. The description of those pairs (1/p; α) ∈ (d) for which the convergence problem is already solved can be found in [17, Chapter 9]. for the nonperiodic case. Taking into account the interrelations between periodic and nonperiodic multipliers given in [18, Chapter 7] one can assert that all results obtained in the nonperiodic case remain valid in the periodic case as well. The following statements contain some properties of the methods (1.1) and (1.3) for (d). Henceforth, we shall denote the subdomains of (d) where the BochnerRiesz means and the corresponding families have the convergence property with (d) and

(d), respectively. In contrast to the results for the domains (d) and (d) we are not able to give their complete description. Theorem 6 Let 1 ≤ p ≤ +∞. If (1/p, α) ∈ (d) then     f − S (α) (f ) ≤ (2π)−d/p · f − S (α) (f ) , n n;λ p p

f ∈ Lp , n ∈ N,

(2.10)

in particular,

(d) ⊂ (d). Theorem 7 Let 1 ≤ p ≤ +∞. If d > 1, α > 0 and (1/p; α) ∈ (d) then   f − Sn(α) (f )  K(f ; 1/n)p  K(f  ; 1/n)p p •

◦

 ω (f ; 1/n)p  ω (f ; 1/n)p  ω(f ; 1/n)p ,

f ∈ Lp , n ∈ N.

(2.11)

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Analysing Theorems 2 and 7 we can see that in contrast to the result on the equivalence (1.5) in [21] the condition α > (d − 1)/2 which was essentially used in the proof turns out to be not necessary. Just the convergence property of the BochnerRiesz means is required. Thus, in the case α > 0 we obtain the following criterion: (α) Equivalence (1.5) holds in Lp if and only if the means Sn have the convergence property. Theorem 8 Let 1 ≤ p ≤ +∞. If d > 1, α > 0 and (1/p; α) ∈

(d) then     f − S (α) (f )  f − Sn(α) (f )  K(f ; 1/n)p n;λ p p •

 ; 1/n)p  ω (f ; 1/n)p  K(f

(2.12)

◦

 ω (f ; 1/n)p   ω(f ; 1/n)p , for all f ∈ Lp , n ∈ N. Theorems 1–8 show that for the means as well as for the families generated by the Bochner-Riesz kernels there is an alternative, which is valid for all admissable pairs (1/p, α) ∈ R2+ with α > 0: Either the method (1.1) (or (1.3) if 1 ≤ p ≤ +∞) diverges in Lp or its approximation error is equivalent to the K-functional (or its realization if 0 < p < 1) related to the Laplacian.

3 Norms of Families of Operators As usual, the norm of a linear and bounded operator Ln mapping Lp with 1 ≤ p ≤ +∞ into Tn , is given by Ln (p) = sup Ln (f )p . f p ≤1

(3.1)

A sequence {Ln }n∈N , of operators is said to be bounded if the sequence of their norms is bounded by some positive constant independent of n. The concept of a norm is very important in Approximation Theory, because in view of the BanachSteinhaus theorem (see, for instance, [1, pp. 220, 249] the behavior of the sequence of norms determines approximative properties of the method (for further details; see [19, pp. 466, 595]). Observe, that for nontrivial linear polynomial operators in Lp with 0 < p < 1 quantity (3.1) is always equal to +∞[2, Lemma 3.2, p. 685]. In contrast to single polynomial operators the concept of the norm of a family of operators (see [2, p. 678], for the one-dimensional version) makes sense for all 0 < p ≤ +∞. Let {Ln;λ }n∈N, λ∈Td , be a family of operators mapping Lp (Td ) with 0 < p ≤ +∞ into Tn such that the function Ln;λ (f ; x) of two variables x and λ belongs to Lp (T2d ) for each f ∈ Lp . We define the (averaged) norm of such a family by {Ln;λ }(p) = (2π)−d/p · sup Ln;λ (f ; x)p . f p ≤1

(3.2)

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On the analogy to single operators a family is said to be bounded if the sequence of its norms is bounded. Each family of operators can be considered as a single linear operator mapping Lp (Td ) into Lp (T2d ). From this point of view (3.2) is (up to the normalizing constant) the norm of this operator in the usual sense. This observation will enable us to apply the standard techniques of Functional Analysis to families of operators. As is known, the Riesz means are bounded in Lp for all 1 ≤ p ≤ +∞ if and only if α > (d − 1)/2 (cf. [18, Chapter 7]). Now, we study the norms of the families (1.1). We do not consider pairs of parameters (1/p, α) in the domain (d) which is given in (2.5). The boundedness problem of the norms seems to be much more difficult in this case and is beyond the scope of this article. Recall that it is not completely solved even for the Bochner-Riesz means given in (1.3). (α) Theorem 9 The family {Sn;λ } is bounded in Lp if (1/p, α) ∈ (d) and it is unbounded in Lp if (1/p, α) ∈ (d).

Proof We split the proof into several steps. Step 1. Let us define the Fourier transform of ψ ∈ L1 (Rd ) by  (x) = ψ ψ(ξ ) · e−ixξ dξ. Rd

The generator of the Bochner-Riesz kernel (1.2) is given by (a+ = max(a, 0))  α ψα (ξ ) = 1 − |ξ |2 + , ξ ∈ Rd . (3.3) By the same arguments as in [2], Theorem 4.2, p. 687, where the one-dimensional (α) case is considered, it follows that the family {Sn;λ } is bounded in Lp if the Fourier d transform of its generator ψα belongs to Lp (R ) ( p = min(1, p)). It is well-known (see, for instance, [17, Chapter 9, Section 2.2, pp. 389–390]) that α (x) = π −α (α + 1)|x|−α−d/2 Jα+d/2 (|x|), ψ

(3.4)

where Js (x), s > −1/2, is the Bessel function of order s. Using the properties of the Bessel functions, in particular, their asymptotic formula [17, Chapter 8, Section 5, pp. 356–357], we obtain (0 < q < +∞)   ψ α 

Lq

 (Rd )



|x|≥1

|x|−q(α+d/2+1/2) dx 



+∞

r −q(α+d/2+1/2)+d−1 dr. (3.5)

1

α ∈ Lq (Rd ) if and only if α > d(1/q − 1/2) − 1/2 and the family {S } is Hence, ψ n;λ bounded in Lp for (1/p, α) ∈ (d). (α)

Step 2. Let (1/p, α) ∈ (d) and 1 ≤ p ≤ +∞. Integrating formula (1.1) with (α) (α) respect to the parameter λ we obtain a connection between {Sn;λ } and Sn for

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1 ≤ p ≤ +∞ which can be formally represented as  (α) (α) −d Sn = (2π) Sn;λ dλ.

(3.6)

Td

Using Minkovski’s generalized inequality, Hölder’s inequality, and (3.6) leads to   (α)      (α) S  = sup S (α) (f ) ≤ (2π)−d sup S (f ) dλ n n n;λ (p) p p f ≤1

≤ (2π)−d

f ≤1

 Td

1−1/p dλ

Td

  (α)   (α) sup Sn;λ (f ; x)p =  Sn;λ (p) , (3.7)

f p ≤1 (α)

for n ∈ N. As is known [11], the means Sn are unbounded in Lp with 1 ≤ p ≤ +∞ (α) if (1/p, α) ∈ (d). Thus, the unboundedness of the family {Sn;λ } in this case follows immediately from (3.7). Step 3. Now, let 0 < p < 1. We will show that  (α)    d(1/p−1)  (α)   S  · Kn p , n;λ (p)  n

n ∈ N.

(3.8)

Indeed, for any f ∈ Lp and n ∈ N we get  (α) p S (f ) n;λ p −dp

≤ (2n + 1)

 2n    k p   f t +λ · n

k=0

Td

Td

 (α)   K x − t k − λ p dx n

n

 dλ

p p  = (2n + 1)d(1−p) · f p · Kn(α) p which yields the upper estimate. In order to show the lower estimate we consider the 2π -periodic function f∗ which is defined on [−π, π)d by  (μ(Dδ ))−1/p , h ∈ Dδ π f∗ (h) = , , δ= 2n + 1 0, otherwise where Dδ = {h ∈ [−π, π)d : |h| < δ} and μ(e) is the Lebesgue measure of the set e ⊂ Rd . For each λ ∈ Dδ and for each vector k ∈ Zd \ {0} with components 0 ≤ kj ≤ 2n, j = 1, . . . , d, we have k    t + λ ≥ t k  − |λ| ≥ 2δ − δ = δ. n n Hence, by the definition of f∗ and (1.1),   f∗ tnk + λ = 0 for λ ∈ Dδ , k ∈ Zd , k = 0, 0 ≤ kj ≤ 2n, j = 1, . . . , d.

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Therefore, Sn;λ (f∗ ; x) = (2n + 1)−d f∗ (λ)Kn(α) (x − λ), (α)

x ∈ Td , λ ∈ Dδ .

(3.9)

(α)

Since Sn;λ (f∗ ; x) is 2δ-periodic as a function of the variable λ and f∗ p = 1, we obtain p   (α) p −d  (α)   S Sn;λ (f∗ ; x)p n;λ (p) ≥ (2π) 



−d

= (2δ)

[−δ,δ]d

−d



≥ (2δ)

Dδ −d

= (2π)

Td



  (α) S (f∗ ; x)p dx n;λ

  (α) S (f∗ ; x)p dx

dλ 

 (2n + 1)

d(1−p)

dλ



n;λ

Td



|f∗ (λ)| · p

Dδ

Td

  (α) K (x − λ)p dx n

 dλ

 p = (2π)−d (2n + 1)d(1−p) · Kn(α) p , from (3.9). This completes the proof of (3.8). Step 4. Suppose 0 < p ≤ 1. We claim: If there exists a positive constant C ≡ C(d; p; α) such that   nd(1/p−1) · Kn(α) p ≤ C (3.10) for all n ∈ N, then α > d(1/p − 1/2) − 1/2. Indeed, let us consider the sequence of functions {Fn }+∞ n=1 given by   x p    (α) n−dp · Kn  , x ∈ [−πn, πn]d Fn (x) = . n 0, otherwise Clearly, the functions Fn , n ∈ N, are nonnegative and measurable. Let x0 ∈ Rd . Then there exists n0 ∈ N such that x0 ∈ [−πn, πn]d for n ≥ n0 . The function ψα (ξ )eiξ x0 of the variable ξ is Riemann-integrable on [−1, 1]d . By definition of the Riemann integral we get     k −d i nk x0 α (−x0 ). ·e ψα = ψα (ξ ) · eiξ x0 dξ = ψ lim n · n→+∞ n [−1,1]d d k∈Z

Therefore, lim

n→+∞

(3.10) implies  Fn (x)dx = n−dp · Rd

  α (−x0 )p . Fn (x0 ) = ψ



[−πn,πn]d

    p  (α) x p  dx = nd(1−p) · Kn(α) p ≤ C p . Kn n

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27

Thus, we have proved that the sequence {Fn }+∞ n=1 satisfies all conditions of Fatou’s lemma and the integral of its limit can be estimated by the same constant. α L (Rd ) ≤ C. Using (3.5) we see that α > α ∈ Lp (Rd ) and ψ Consequently, ψ p d(1/p − 1/2) − 1/2. Now we are able to complete the proof of Theorem 9. Let (1/p, α) ∈ (d) and 0 < p < 1. Then 0 ≤ α ≤ d(1/p − 1/2) − 1/2. Therefore, the sequence (α) {nd(1/p−1) · Kn p }n∈N is unbounded. Using equivalence (3.8), the unboundedness (α) of the family {Sn;λ } follows in this case. 

4 Preliminaries In this section we will discuss some general convergence criteria of the approximation methods of type (1.1) and (1.3). We shall deal with the Fourier means  f (x + h) · Wn (h) dh, n ∈ N, (4.1) Ln (f ; x) = (2π)−d Td

and the families of operators −d

Ln;λ (f ; x) = (2n + 1)

2n      · f tnk + λ · Wn x − tnk − λ ,

n ∈ N,

(4.2)

k=0

which are generated by the kernel Wn (h) =

 k∈Zd

ψ

  k · eikh , n

n ∈ N,

(4.3)

where ψ(ξ ) is a real valued centrally symmetric function with compact support contained in D 1 = {ξ : |ξ | ≤ 1} satisfying ψ(0) = 1.4 In accordance with the Banach-Steinhaus principle [19, pp. 466, 595], which is a reformulation of the Banach-Steinhaus theorem for operators of type (4.1), the sequence of linear polynomial operators {Ln }n∈N converges in Lp , 1 ≤ p ≤ +∞, if and only if the sequence {Ln (eim· ; x)}n∈N converges for each m ∈ Zd and the sequence of norms {Ln (p) }n∈N is bounded. Observe, for each m ∈ Zd and n ∈ N, Ln (eim· ; x) =

 k∈Zd

ψ

   k imx e · (2π)−d eimh eikh dh d n T

m  m eimx = ψ eimx , =ψ − n n

(4.4)

4 Fourier means (4.1), families (4.2), and kernels (4.3) depend on ψ , which is omitted for the sake of

simplicity and brevity.

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and, therefore, Ln (T ; x) =



ψ

m n

m∈Zd

· cm eimx ,

T (x) =



cm eimx ∈ T ,

n ∈ N, (4.5)

m∈Zd

where T is the space of all real-valued trigonometric polynomials. Hence, we can reformulate this principle as follows. We omit the standard proofs. Theorem 10 Let ξ = 0 be a point of continuity of the function ψ. Then the means Ln given by (4.1) converge in Lp with 1 ≤ p ≤ +∞ if and only if the sequence of their norms {Ln (p) }n∈N is bounded. The convergence of the operators Ln has been considered also by Feichtinger, Weisz [9, 10] It turns out that the same statement holds true for families of operators (4.2) replacing the concept of the norm (3.1) by the concept of the averaged norm (3.2). In order to prove this assertion we notice first that for each m ∈ Zd satisfying |m| ≤ n and n ∈ N 2n    ν     eiνx · (2n + 1)−d · Ln;0 eim· ; x = ψ exp itnk (m − ν) n d k=0

ν∈Z

=



ψ

|ν|≤n

ν  n

eiνx ·

d 

δ(mj ; νj ).

(4.6)

j =1

Here, δ(mj ; νj ) ≡ (2n + 1)−1 ·

2n 

  exp itnk (mj − νj )

k=0

 =

1, mj ≡ νj (mod(2n + 1)) . 0, otherwise

Since |mj | + |νj | ≤ |m| + |ν| ≤ 2n < 2n + 1 we have δ(mj ; νj ) = 1 for all j = 1, . . . , d if and only if m = ν. Therefore, taking into account the equality Ln;λ = S−λ ◦ Ln;0 ◦ Sλ ,

(4.7)

where St (g; x) = g(x + t) is the translation operator, we conclude from (4.6) that m   eimx , λ ∈ Rd , |m| ≤ n. Ln;λ eim· ; x = ψ (4.8) n In particular, it holds (n ∈ N).  m · cm eimx , ψ Ln;λ (T ; x) = n d m∈Z

T (x) =

 m∈Zd

cm eimx ∈ Tn .

(4.9)

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29

Now, we are able to formulate and prove the counterpart of Theorem 10 for families of operators. Clearly, for all 0 < p ≤ +∞ (recall, p  = min(1, p)), p 

p 

p 

f + gp ≤ f p + gp ,

  f, g ∈ Lp Td .

(4.10)

Theorem 11 Let ξ = 0 be a point of continuity of the function ψ. Then the family {Ln;λ } given in (4.2) converges in Lp with 0 < p ≤ +∞ if and only if the sequence ({Ln;λ }(p) )n∈N is bounded.

5 Inequalities for Trigonometric Polynomials In this section we will show that the approximation error in Lp of means and families generated by the Bochner-Riesz kernels of order n with parameter α can be estimated from above and from below on the space Tn of trigonometric polynomials of order at most n by the norm of the Laplacian if (1/p, α) belongs to the range of parameters where the method is bounded. Before we formulate and prove the main results we give some general facts related to Fourier multipliers for spaces of trigonometric polynomials. Let g be a real or complex valued function defined on Rd . It generates operators {Aσ (g)}σ ≥1 on the space T given by  k  Aσ (g)t (x) = g ck eikx ∈ T . (5.1) ck eikx , t (x) = σ d d k∈Z

k∈Z

We shall deal with inequalities of type Aσ (μ)tp ≤ c(d; p; μ; ν) · Aσ (ν)tp ,

t ∈ Tσ , σ ≥ 1.

(5.2)

We say that (5.2) is valid for some 0 < p ≤ +∞ (or valid in Lp ), if it is valid in the Lp -norm for all t ∈ Tσ and σ ≥ 1 with some positive constant independent of t and σ . The set of all p for which inequality (5.2) is valid is called its range of validity. Henceforth, we suppose that ν(ξ ) = 0 for ξ = 0. Let us define X (ξ ) =

μ(ξ ) , ν(ξ )

ξ ∈ Rd \ {0}.

We assume that X is somehow defined at the point ξ = 0. Consider the inequality Aσ (X )tp ≤ c (d; p; μ; ν) · tp ,

t ∈ Tσ , σ ≥ 1,

(5.3)

which is associated with (5.2). Clearly, it is of the same type, but the operator on its right-hand side is the identity. Let (A) and (B) be inequalities of type (5.2). We say that inequality (A) implies the inequality (B) for some p if the validity of (A) for p implies the validity of (B) for p. We also say that inequality (A) implies inequality (B) if it is so for all 0 < p ≤ +∞.

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Lemma 1 Let μ(0) = ν(0) = 0. Then inequality (5.3) implies (5.2) independently of the value X (0). Proof We notice that X (ξ ) ν(ξ ) = μ(ξ ) for all ξ ∈ Rd independently of X (0). This implies that for σ > 0 Aσ (X ) ◦ Aσ (ν) = Aσ (μ).

(5.4)

For any t ∈ Tσ we put τ = Aσ (ν)t. Applying (5.3) to τ and using (5.4), the claim immediately follows.  The next result [16, pp. 150–151], [15, Theorem 3.2] gives sufficient conditions for the validity of (5.3). Recall, p  = min(1, p). Theorem 12 Let 0 < p ≤ +∞ and let X be a continuous function with compact  ∈ Lp(Rd ), then inequality (5.3) is valid in Lp . support. If X If, for instance, the function X is infinitely differentiable on Rd and has compact support, then its Fourier transform belongs to Lp (Rd ) for all p > 0. In this case, the range of validity of (5.3) is [0, +∞). We can give stronger assertions on the validity of (5.3) in terms of the smoothness of the function X . Let us denote by W1k (Rd ) the Sobolev space of all integrable functions whose derivatives up to the order k ◦

belong to L1 (Rd ). The notation W k1 (Rd ) is used for the set of functions in W1k (Rd ) having compact supports (for details and further properties; see, for instance, [20, pp. 54–56]). ◦

Corollary 1 Let 0 < p ≤ +∞, k = [ d/ p ] + 1. If X ∈ W k1 (Rd ), then inequality (5.3) is valid in Lp . Proof Using elementary properties of the Fourier transform, we find ν X (x), (x) = (−i)|ν|1 D  xν X

x ∈ Rd , |ν|1 ≤ k,

ν

where |ν|1 = ν1 + . . . + νd , x ν = x1ν1 . . . xd d and Dν X =   X (x) ≤

  |ν|1 =k

−1   ν  D X  |x | L ν

|ν|1 =k

1 (R

d)

∂ |ν|1 X . Hence, . . . ∂ξdνd

∂ξ1ν1

≤ c · |x|−k ,

|x| ≥ 1.

(5.5)

Taking into account that p k > d and using (5.5), we get   X 

 

1 + ≤ c (Rd )

Lp

|x|≥1

   pk |x|− dx ≤ c

1 +

Thus, the statement follows from Theorem 12.

+∞

 p k+d−1 r − dr < +∞.

1



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31

Now, we are able to formulate and prove the main result of this section. Recall, that Dr and D r denote the sets {ξ ∈ Rd : |ξ | < r} and {ξ ∈ Rd : |ξ | ≤ r}, respectively. Moreover, (d) ⊂ (d) is the subdomain where the Bochner-Riesz means are bounded (see Section 2).  Theorem 13 Let (1/p, α) ∈ (d) (d) and let α > 0. Then,   T − S (α) (T )  n−2 · T p , T ∈ Tn , n ∈ N. n p (α)

In (5.6) the operator Sn affecting the constants.

(5.6)

(α)

can be replaced by Sn,λ for any fixed λ ∈ Rd without

Proof We split the proof into several steps. Step 1. Comparing (4.5) and (4.9), we have for each T ∈ Tn , n ∈ N and λ ∈ Rd (α)

Sn(α) (T ; x) = Sn;λ (T ; x),

x ∈ Td .

(5.7)

This simple observation immediately implies the second  part of Theorem 13. Moreover, it enables us to verify that for (1/p, α) ∈ (d) (d) and α > 0,   (α) S (T )  T p , T ∈ Tn , n ∈ N. (5.8) n p Indeed, if (1/p, α) ∈ (d), then the means Sn , n ∈ N, are bounded by Theorem 10 and (5.8) is valid. If (1/p, α) ∈ (d), then for any T ∈ Tn and n ∈ N, (α)

     (α)

 S (T ) = (2π)−d/p · S (α) (T ) ≤  S (α)  · T p ≤ cT p n n,λ (p) n;λ p p by (5.7), (3.2), and Theorem 9. Step 2. We define  ϕ(ξ ) =

1, ξ ∈ D1 ϕ0 (ξ ) = 0, ξ ∈ / D2



1, 0,

ξ ∈ D1/2 ; ξ∈ / D3/4

(5.9)

ϕ1 (ξ ) = ϕ(ξ ) − ϕ0 (ξ ). In addition, suppose that ϕ and ϕ0 are infinitely differentiable in Rd and radial functions. We put  |ξ |−2 (1 − ψα (ξ )), ξ = 0 ; X(α) (ξ ) = α (ξ )ϕ(ξ ), α (ξ ) = (5.10) α, ξ =0 where ψα is given by (3.3). Clearly, X(α) (ξ ) = α (ξ )ϕ0 (ξ ) + α (ξ )ϕ1 (ξ ),

ξ ∈ Rd .

(5.11)

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Taking into account that the function y −2 (1 − (1 − y 2 )α+ ) (extended to the point 0 by α) is analytic in the interval (−1, 1) we see that the function α ϕ0 is infinitely differentiable in Rd . By (5.9) the function |ξ |−2 ϕ1 (ξ ) has the same property. Using Corollary 1 and (4.10), (5.1), (4.5), and (5.8) we find that p 

An (X(α) )T p

p 

p 

≤ An (α ϕ0 )T p +  An (α ϕ1 )T p p   p   =  An (α ϕ0 )T p +  An | · |−2 ϕ1 (·) ◦ An (1 − ψα (·))T p  p p   p  (α) ≤ c T p + T − Sn (T )p ≤ c · T p ,

(5.12)

for T ∈ Tn , n ∈ N. By (4.5) and (5.1) the upper estimate in (5.6) is an inequality of type (5.2) with μ(ξ ) ≡ μα (ξ ) = (1 − ψα (ξ )) ϕ(ξ ) and ν(ξ ) = |ξ |2 . Moreover, X(α) (ξ ) = μ(ξ )/ν(ξ ) for ξ = 0. Using Lemma 1 and (5.12) we obtain   T − S (α) (T )  n−2 · T p , (T ∈ Tn , n ∈ N). (5.13) n p Step 3. In order to prove the lower estimate we put α (ξ ) = (α (ξ ))−1 ;

X (α) (ξ ) = α (ξ ) ϕ(ξ ),

(5.14)

where α has the meaning of (5.10). Clearly, X (α) (ξ ) = α (ξ ) ϕ0 (ξ ) + α (ξ )ϕ1 (ξ ),

ξ ∈ Rd .

(5.15)

Taking into account that the function y 2 (1 − (1 − y 2 )α+ )−1 (extended to 0 by α −1 ) is analytic in the interval (−1, 1) we find that the function α ϕ0 is infinitely differentiable in Rd . Using Corollary 1 we obtain An (α ϕ0 )T p  T p , Setting

  k = d/ p + 1,

T ∈ Tn , n ∈ N.

(5.16)

  m = α −1 (k + 1) + 1,

(5.17)

we have k ≤ αm − 1. By (5.10) and (5.14) the second item in (5.15) can be represented as ⎫ ⎧ m−1 ⎬ ⎨  α ϕ1 (ξ ) = |ξ |2 ϕ1 (ξ ) (ψα (ξ ))j + ϕ1 (ξ )g(|ξ |) , (5.18) ⎭ ⎩ j =0

where

αm  1 − y2 +  α  , g(y) =  1 − 1 − y2 +

y = 0.

(5.19)

It is easily seen that lim

y→1(+0)

g (s) (y) = 0,

s ∈ N.

(5.20)

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33

Now we prove that lim

y→1(−0)

g (s) (y) = 0,

1 ≤ s ≤ m − 1.

(5.21)

Indeed, for l ∈ N0 (if l = 0, we suppose that j = 0 and c0 (α; 0) = 1) 

1 − xα

−1 (l)

=

l 

 −(j +1) cj (α; l) x αj −l 1 − x α ,

0 < x < 1.

(5.22)

j =1

This equality can be easily checked by induction on l. We put F (x) =

x αm , 1 − xα

x > 0.

(5.23)

By (5.22) we have for s ∈ N F (s) (x) =

s   αm (s−l)  −1 (l) 1 − xα x l=0

=

s  l 

 −(j +1) dj l (α; m; s)x αm−(s−l) x αj −l 1 − x α

l=0 j =1

⎛ ⎞ s  l    −(j +1) ⎠ = x αm−s ⎝ dj l (α; m; s)x αj 1 − x α

(5.24)

l=0 j =1

≡

x αm−s h(x),

0 < x < 1.

Taking into account that the function h(x) is continuous for 0 ≤ x < 1 it follows from (5.24) that lim F (s) (x) = 0,

x→+0

1 ≤ s ≤ m − 1.

(5.25)

Combining (5.19) and (5.23), for s ∈ N we obtain s       2 (s) g (y) = F 1 − y = F (j ) 1 − y 2 Ps;j (y), (s)

0 < y < 1,

(5.26)

j =1

where Ps;j (y) are algebraic polynomials. Thus, (5.20) is a consequence of (5.25) and (5.26). Using (5.20), (5.21), and (5.9) we conclude that the function ϕ1 g belongs to C m−1 (R). Since the point 0 does not belong to its support and the function |ξ | is infinitely differentiable in Rd \ {0}, the function ϕ1 (ξ ) g(|ξ |) belongs to C m−1 (Rd ).5 ◦

Because of (5.17) it belongs also to W k1 (Rd ), where k = [ d/ p ] + 1. 5 As usual, C m (Rd ) is the space of functions having continuous derivatives up to the order m.

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Using Corollary 1 for the functions |ξ |2 , ϕ1 and ϕ1 (ξ ) g(|ξ |) as well as (5.18), (4.10), (5.1), (4.5), and (5.8), for T ∈ Tn , n ∈ N we find that p 

An (α ϕ1 )T p

# m−1 $    p p  j An ϕ1 ψ T  + An (ϕ1 (·)g(| · |))T p ≤ c1 α

p

j =0

# m−1 $     p p  j An ψ T  + An (ϕ1 (·)g(| · |))T p ≤ c2 α

j =1

(5.27)

p

# m−1 $      (α) j p p  p  ≤ c3 T  + T p ≤ c4 T p .  Sn j =1

p

Now, as a consequence, of (5.15), (5.16), and (5.27) we get   (α)  p p  p  p  An X T p ≤ An (α ϕ0 )T p + An (α ϕ1 )T p ≤ cT p

(5.28)

for T ∈ Tn , n ∈ N. Note that by (4.5) and (5.1) the lower estimate in (5.6) is an inequality of type (5.2) with μ(ξ ) = |ξ |2 ϕ(ξ ), ν(ξ ) ≡ να (ξ ) = (1 − ψα (ξ ))−1 . Moreover, X (α) (ξ ) = μ(ξ )/ν(ξ ) for ξ = 0. Using Lemma 1 and (5.28) we see that   n−2 · T p  T − Sn(α) (T )p ,

T ∈ Tn , n ∈ N.

(5.29) 

Finally, (5.6) follows from (5.13) and (5.29), and the proof is complete.

6 Proofs of the Main Results Proof of Theorem 1 The statement follows immediately from Theorems 9 and 11.  Proof of Theorem 2 Let (1/p, α) ∈ R2+ . By the same arguments as in the proof of (3.7) we get     f − S (α) (f ) ≤ (2π)−d/p f − S (α) (f ) , n n;λ p p

f ∈ Lp , n ∈ N,

(6.1)

for any (1/p, α) ∈ R2+ . Comparing formulas (4.5) and (4.9) it is easily seen that (α)

Sn;λ ◦ Sn(α) = Sn(α) ◦ Sn(α) ,

λ ∈ Rd , n ∈ N.

(6.2)

J Fourier Anal Appl (2008) 14: 16–38

35 (α)

(α)

Taking into account that the norms of the methods Sn and {Sn;λ } are bounded in Lp if 1 ≤ p ≤ +∞ and α > (d − 1)/2 and using (6.2), we get     f − S (α) (f ) ≤ f − Sn(α) (f ) n;λ p p  (α)  (α)  (α)    (α)  (α) + Sn (f ) − Sn Snα (f ) p + Sn;λ Sn (f ) − Sn;λ (f )p ≤ (2π)d/p

  (α)   (α)     (α) · 1 + Sn (p) +  Sn;λ (p) · f − Sn (f )p

(6.3)

  (α) ≤ cf − Sn (f )p , for f ∈ Lp and n ∈ N. Thus, the first equivalence in (2.6) follows from (6.1) and (6.3). As already mentioned in Section 1, under the conditions of Theorem 2 the equivalence   f − S (α) (f )  ω• (f ; 1/n)p , f ∈ Lp , n ∈ N, n p •

was proved in [21]. A proof of the equivalence of K(f ; δ)p to ω (f ; δ)p and to ◦ ω (f ; δ)p for all 1 ≤ p ≤ +∞ can be found in [6]. The relation  ω(f ; δ)p  K(f ; δ)p ,

f ∈ Lp , δ ≥ 0,

for 1 ≤ p ≤ +∞ was shown in [4]. As pointed out in the Introduction of [7] the equivalence  ; 1/n)p , f ∈ Lp ,  ω(f ; 1/n)p  K(f

n ∈ N,

for 1 ≤ p ≤ +∞ can be obtained by combining results from [3] with results from [4].  Proof of Theorem 3 Under the hypothesis of Theorem 3 (1/p, α) belongs to the set (d) [see also (2.3)]. Therefore, Theorem 13 (the upper estimate in (5.6) for families), Theorem 9, (3.2), and (4.10) imply       f − S (α) (f )p ≤ f − T p + T − S (α) (T )p + S (α) (f − T )p p n;λ n;λ n;λ p p p   (α) p  p p ≤ (2π)d 1 +  Sn;λ (p) · f − T p + c n−2p · T p  p p ≤ c f − T p + n−2p · T p  p ≤ c

f − T p + n−2 · T p . for all f ∈ Lp , n ∈ N and T ∈ Tn . Hence,   f − S (α) (f )  K(f  ; 1/n)p , n;λ p

f ∈ Lp , n ∈ N.

(6.4)

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In order to prove the lower estimate in (2.7) we note first that for f ∈ Lp , n ∈ N       ; 1/n)p ≤ 2(2π)−d/p · max f − S (α) (f ) , n−2 · S (α) (f ) . (6.5) K(f n;λ n;λ p p By Theorem 13 (the lower estimate in (5.6) for families) we have for all β, γ ∈ Rd  (α)   (α)  (α)  (α) n−2 · Sn;γ (f )p ≤ c · Sn;γ (f ) − Sn;β Sn;γ (f ) p ,

f ∈ Lp , n ∈ N, (6.6)

where the positive constant c does not depend on β, γ and n. Furthermore,  (α)   (α) (α) (α) (α)  (α)  (α)  Sn;γ − Sn;β ◦ Sn;γ = Sn;γ − I + I − Sn;β + Sn;β ◦ I − Sn;γ ,

n ∈ N, (6.7)

where I is the identity. By virtue of (6.6) and (6.7), for f ∈ Lp , n ∈ N and for all β, γ ∈ Rd   (α) p p  p (α) (α) n−2p · Sn;γ (f )p ≤ c f − Sn;γ (f )p + f − Sn,β (f )p  (α)  p  (α) + Sn;β f − Sn,γ (f ) p .

(6.8)

Integrating with respect to β over Td , with help of (3.2) we obtain   (α) p    (α) p p (α)  · f − Sn;γ (f )p ≤ c 1 +  Sn;λ (f )p n−2p · Sn;γ (p) +



(2π)−d/p f

p (α) − Sn;λ (f )p

(6.9)

 ,

for each α ∈ Rd . Next, we integrate on both sides of (6.9) with respect to γ over Td . Then, by Theorem 9 we get  (α) p p  (α) n−2p · Sn;λ (f )p ≤ c

f − Sn;λ (f )p ,

f ∈ Lp , n ∈ N.

(6.10)

Using (6.5) and (6.10) we see that    ; 1/n)p  f − S (α)  , K(f n;λ p

f ∈ Lp , n ∈ N.

Now, equivalence (2.7) follows from (6.4) and (6.11).

(6.11) 

Proof of Theorem 4 It was proved in [8] that for d = 1 and 0 < p ≤ +∞  ; 1/n)p , ω2 (f ; 1/n)p  K(f

f ∈ Lp , n ∈ N.

Hence, (2.8) follows immediately from (6.12) and Theorem 3.

(6.12) 

Proof of Theorem 5 It was shown in [7] that for 0 < p ≤ +∞  ; 1/n)p   K(f ω(f ; 1/n)p ,

f ∈ Lp , n ∈ N,

(6.13)

J Fourier Anal Appl (2008) 14: 16–38

37

and, moreover,  ; 1/n)p   K(f ω(f ; 1/n)p ,

f ∈ Lp , n ∈ N,

(6.14)

if and only if d/(d + 2) < p ≤ +∞. Thus, all statements in Theorem 5 follow from (6.13), (6.14), and Theorem 3.  Proof of Theorem 6 The proof is contained in the proof of Theorem 2 [see (6.1)].  Proof of Theorem 7 As already mentioned in the proof of Theorem 2 the K-functional (1.9), its realization (1.10) as well as the moduli (1.4), (1.7), and (1.11) taken at the point 1/n are equivalent to each other for 1 ≤ p ≤ +∞. It remains to prove that under the conditions of Theorem 7   f − S (α) (f )  K(f  ; 1/n)p , f ∈ Lp , n ∈ N. (6.15) n p (α)

First, we note that in view of Theorem 10 the sequence (Sn )n∈N , is bounded if (1/p, α) ∈ (d). By virtue of Theorem 13 [the upper estimate in (5.6)] we obtain for f ∈ Lp , n ∈ N and T ∈ Tn       f − Sn(α) (f ) ≤ f − T p + T − Sn(α) (T ) + Sn(α) (f − T ) p p p   (α)   ≤ 1 + Sn (p) · f − T p + c n−2 · T p   ≤ c f − T p + n−2 · T p . Hence,

  f − S (α) (f )  K(f  ; 1/n)p , n p

f ∈ Lp , n ∈ N.

(6.16)

Theorem 13 [the lower estimate in (5.6)] implies      ; 1/n)p ≤ f − Sn(α) (f ) + n−2 · Sn(α) (f ) K(f p p    (α)   (α) (α) ≤ f − Sn (f )p + cSn f − Sn (f ) p

(6.17)

  (α) ≤ c f − Sn (f )p , for f ∈ Lp and n ∈ N. Now (6.15) follows from (6.16) and (6.17).



Proof of Theorem 8 The proof of the first equivalence ain (2.12) coincides with the proof of the first equivalence in (2.6), because it was based only on the boundedness of the Bochner-Riesz means and the corresponding families. In our case this follows immediately from the conditions of Theorem 8 because of Theorems 10 and 11. Now, all other equivalences in (2.12) follow from Theorem 7.  Acknowledgements

We would like to thank the referees for useful suggestions and comments.

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