3 Families generated by kernels of type (GR)

Volume 1, Number 3 (2010), 112 133

ON CONVERGENCE OF FAMILIES OF LINEAR POLYNOMIAL OPERATORS GENERATED BY MATRICES OF MULTIPLIERS

K. Runovski, H.-J. Schmeisser Communicated by D.D. Haroske

Key words: trigonometric approximation, convergence, Fourier multipliers, Jackson, Cesaro and Fejer-Korovkin kernels.

AMS Mathematics Subject Classication: 42A10, 42A45, 42B99.

Abstract. The convergence of families of linear polynomial operators with kernels generated by matrices of multipliers is studied in the scale of the L_p-spaces with 0<

p ≤+∞. An element a_{n, k} of generating matrix is represented as a sum of the value of the generator ϕ(k/n) and a certain "small" remainder rn, k. It is shown that under some conditions with respect to the remainder the convergence depends only on the properties of the Fourier transform of the generator ϕ. The results enable us to nd explicit ranges for convergence of approximation methods generated by some classical kernels.

1 Introduction

In this paper we continue the systematic study of methods of trigonometric approxi- mation started in [1] and [5] - [10]. We consider Fourier means, interpolation means and families of linear polynomial operators, which are dened as follows. Let

A ={an, k ∈C: an,−k =an, k, |k| ≤rn, n ∈N0} , (1) where r≡r(A) is a real positive number, be a matrix of multipliers (We put a_n,k = 0 if |k|> rn.). It generates the kernels Wn(A) given by

W_n(A)(h) = X

k∈Z^d

a_{n, k}e^ikh, n∈N₀, h∈R^d . (2) If f ∈ L_p(T^d),1 ≤ p ≤ ∞ (T^d stands for the d-dimensional torus), then the Fourier means are given by

F_n^(A)(f;x) = (2π)^−d Z

T^d

f(h)W_n(A)(x−h)dh , n∈N₀, x∈T^d. (3) If f belongs to the space C(T^d) of continuous 2π - periodic (with respect to each variable) functions then the interpolation means are dened as

I_n^(A)(f;x) = (2N+ 1)^−d· X2N

ν=0

f(t^ν_N)·W_n(A) (x−t^ν_N) , n∈N₀, x∈T^d. (4)

Here

N = [ρ n], ρ≥r(A); t^ν_N = 2πν

2N + 1, ν ∈Z^d; X2N

ν=0

≡ X2N

ν1=0

. . . X2N

νd=0

.

The functions dened in (1) - (4) are trigonometric polynomials of spherical order not exceeding r(A)n. If f ∈ L_p(T^d),0 < p < ∞, or if f ∈ C(T^d) (p = +∞) then we consider the functions given by

L^(A)_n;λ(f;x) = (2N + 1)^−d· X2N

ν=0

f(t^ν_N +λ)·W_n(A) (x−t^ν_N −λ) . (5) In the case that f ∈ Lp(T^d),0 < p < ∞, formula (5) makes sense for almost all λ∈R^dandx∈T^d. We understandλ as a parameter and call{L^(A)_n;λ}a family of linear (trigonometric) polynomial operators. In contrast to classical methods of trigonometric approximation the method of approximation by families is comparatively new (see e.g. [6], [7]). Its systematic study was continued in [1], [5] and other works. For applications of the method, in particular for an algorithm of stochastic approximation (SA-algorithm), we refer to [8]. We are interested in the approximation process

L^(A)_n;λ(f;x)→f(x) (n →+∞)

The convergence of the family{L^(A)_n;λ}must be understood in the sense of an averaging with respect to the parameterλ. More precisely, we consider the limit process

°°L^(A)_n;λ(f;x)−f(x)°

°p =

³ 1 (2π)^d

Z

T^d

³ Z

T^d

¯¯L^(A)_n;λ(f;x)−f(x)¯

¯^pdx

´ dλ

´¹

p →0

if n→+∞ for admissiblep, 0< p≤+∞(see Section 2 for detailed explanation).

In [5] the problem of convergence has been investigated for kernels of type (1) - (2) with

A(ϕ) = {a_{n, k}}: a_0,0 = 1; a_{n, k} =ϕ µ k

σ(n)

¶

, k ∈Z^d, n∈N , (6) where σ(n) is a certain strongly increasing sequence of positive real numbers of order n and ϕis a complex-valued continuous function on R^d, d∈N,with compact support satisfying ϕ(0) = 1 and ϕ(−ξ) = ϕ(ξ) for each ξ ∈ R^d. In this case it is r(A) = r(ϕ) := sup{ |ξ| : ϕ(ξ) 6= 0}. We call ϕ generator of the kernels W_n and we use the notations W_n(ϕ) and { L^(ϕ)_n;λ} in place of W_n(A) and { L^(A)_n;λ}, respectively. More precisely, it was shown in [5]

• that the family {L^(ϕ)_σ;λ} converges in L_p for all 1 ≤ p ≤ +∞ if the Fourier transformϕbof the generatorϕbelongs toL₁(R^d)and that the convergence inL_p for some 0< p < 1 is equivalent to the condition: ϕb ∈ Lp(R^d). For this reason the method of approximation by a family of linear trigonometric polynomial operators is relevant both for p ≥ 1 and p < 1 with the range of admissible parameters depending on the properties of its generatorϕ;

• that the approximation error for families is equivalent to the approximation error of the corresponding Fourier means in the case of L_p(T^d),1 ≤ p ≤ +∞

and to the approximation error of the corresponding interpolation means in the case of C(T^d) (p= +∞);

• that the ranges of convergence (the values of p such that the family converges in Lp) can be determined precisely for appropriate families with classical kernels (Fejer, de la Vallee-Poussin, Rogosinski, Bochner-Riesz) .

In this paper we elaborate an approach to the study of the convergence problem for families which are not generated by some function ϕ. In particular, we show

• that in many situations the matrix of multipliers can be represented in the form:

A=A(ϕ) +R , R={r_{n, k}}; lim

n→+∞r_{n, k} = 0, k∈Z^d, (7) whereA(ϕ)is of type (6) and the matrix of remainders R satises some relevant conditions of "smallness" (see Section 1 for exact denitions) which enable us to reduce the convergence problem for {L^(A)_n;λ}to the study of the associated family {L^(ϕ)_σ;λ};

• that for powers of kernels of type (6), that is, for kernels (q is a natural number) W_n(ϕ, q)(h) = (γ_n(ϕ, q))⁻¹(W_n(ϕ))^q(h), (8) where

γ_n(ϕ, q) = (2π)^−d Z

T^d

(W_n(ϕ)(h))^qdh (9)

is a normalizing factor, the ranges of convergence of corresponding families can be explicitly expressed in terms of the range of convergence of the family generated by the initial kernels W_n(ϕ);

• that for families with classical kernels of types (7) and (8)-(9) (Fejer-Korovkin, Cesaro, (generalized) Jackson) the ranges of convergence can be determined precisely by applying general results.

The paper is organized as follows. Section 2 is devoted to preliminaries. The gen- eral results are formulated and proved in Sections 3 and 4. Sharp ranges of convergence of the families generated by (generalized) Jackson, Fejer-Korovkin and Cesaro kernels are determined in Sections 5, 6 and 7-9, respectively. In view of the comparison prin- ciple stated in [5], Lemma 2.2, the results of this paper formulated for families (5) immediately imply the classical results on convergence of the corresponding Fourier means (3) and the interpolation means (4) in the spaces L_p with 1≤p≤+∞ and C, respectively, in both general and special cases.

2 Denitions, notations and preliminary remarks

L_p-spaces. As usual, L_p ≡ L_p(T^d), where 0 < p < +∞, T^d = [0,2π)^d, is the space of measurable real-valued functions f(x) which are 2π-periodic with respect to each variable such that

kfk_p =

³ Z

T^d

|f(x)|^p dx

´_1/p

< +∞ .

C ≡ C(T^d) (p = +∞) is the space of real-valued 2π-periodic continuous functions equipped with the Chebyshev norm

kfk∞ = max

x∈T^d |f(x)| .

For L_p-spaces of non-periodic functions dened on a measurable set Ω ⊆R^d we will use the notation L_p(Ω).

Often we deal with functions in Lp(T^2d)which depend on both the main variable x∈T^dand the parameter λ∈T^d. Let us denote by k · k_p or k · k_p;x the L_p(T^d)-norm with respect to x. For the L_p(T^d)-norm with respect to the parameter λ we use the symbol k · kp;λ. For shortness, Lp stands for the space Lp(T^2d) equipped with the norm

k · k_p = (2π)^−d/p k k · k_p;xk_p;λ . (10) Analogously, we use the symbol k · k_∞ for the norm in the space C(T^2d). Clearly, Lp with 0 < p < ∞ and C(T^d) can be considered as subspaces of Lp and C(T^2d), respectively. In this case,

kfk_p = kfk_p , f ∈L_p (f ∈C if p=∞) .

The functional k · k_p is a norm if and only if 1 ≤ p ≤ +∞. For 0 < p < 1 it is a quasi-norm, and the "triangle" inequality is valid for its p-th power. If we put e

p= min(1, p)then the inequality

kf+gk^p_p^e ≤ kfk^p_p^e + kgk^p_p^e, f, g ∈L_p , (11) will be valid for all 0 < p ≤ +∞. This inequality is very convenient, because both cases can be treated simultaneously. Moreover, for the sake of convenience we shall use the notation "norm" also in the case 0< p <1.

Spaces of trigonometric polynomials. Let σ be a real non-negative number. Let us denote by T_σ the space of all real-valued trigonometric polynomials of (spherical) order σ. It means

T_σ = n

T(x) = X

k∈Z^d

c_ke^ikx :c_−k =c_k, |k| ≡(k₁²+. . .+k²_d)^1/2 ≤σ o

,

wherekx=k₁x₁+. . .+k_dx_d and c is a complex conjugate to c. Further, T stands for the space of all real-valued trigonometric polynomials of arbitrary order. We denote by Tσ, p, where 0< p ≤+∞, the space Tσ, if it is equipped with the Lp-norm and we

use the symbol Tσ, p to denote the subspace of Lp which consists of functions g(x, λ) such that g(x, λ) as a function of x belongs to T_σ for almost all λ. Clearly, T_{σ, p} can be considered as a subspace of T_{σ, p} with identity of the norms. As we can see, in our notation the line over the index p indicates that we are dealing with functions of 2d variables.

Types of kernels and generators. Recall that the kernels and conditions with respect to the generator ϕ of the approximation method have been given in the Introduction (see, in particular, (1), (2), and (6)). Henceforth, the class of all admissible generators will be denoted by K. The dierent types of kernels under consideration can be classied as follows.

Type (G). These are kernels (2) corresponding to matrices A dened by (6). The main characteristics of the generator ϕ is the set Pϕ = {p ∈ (0,+∞] : ϕb∈ Lp(R^d)}, where ϕb is the Fourier transform of ϕ (see below). The corresponding approximation methods were studied in [5].

Types (GR^α) and (GRα) (0 < α < 1). Let A, Wn(A) be as in (1) and (2). Let 0< q <+∞. We put

M_q;A(n) = (n+ 1)^d(1/q−1)kW_n(A)k_q, n∈N₀; M_q;A = sup

n

M_q;A(n) ; (12) M_q;A(n) = (n+ 1)^d(1/q−1)kW_n(A)k₂, n∈N₀; M_q;A= sup

n

M_q;A(n). (13) Clearly,

M_q;A(n) = (2π)^d/2(n+ 1)^d(1/q−1)



 X

|k|≤rn

|a_{n, k}|²





1/2

, n∈N₀, (14) by Parseval's equation.

A kernel is said to be of type (GR^α) or (GR_α) if it satises (7) with M_α;R<+∞

orM_α;R<+∞ for their matrix R of remainders, respectively. Since

M_α;R(n) ≤ (2π)^{d(1/α−1/2)}M_α;R(n), n ∈N₀, (15) by Holder's inequality, each kernel of type (GR_α) is also of type (GR^α), that is,

GR_α ⊂ GR^α . (16)

Notations (12) will be used also for kernels of type (G). In this case, that is, if A = A(ϕ), we use the symbols M_q;ϕ(n) and M_q;ϕ instead of M_q;A(n) and M_q;A, respectively.

Type (G^q) (q ∈ N). These are powers of kernels of type (G) given by (8)-(9).

It will be shown in Section 4 that γ_n(ϕ, q) 6= 0 if n ≥ n₀, where n₀ ≡ n₀(ϕ, q) is a certain integer, and, consequently, the functions W_n(ϕ, q)(h) are well-dened for n≥n0. Clearly, they belong to T_qr(ϕ)n.

Fourier transform and convolution. The Fourier transform and its inverse are given by

b g(ξ) =

Z

R^d

g(x)e^−ixξdx , g^∨(x) = (2π)^−d Z

R^d

g(ξ)e^ixξdξ , g∈L1(R^d). Letg₁ and g₂ be functions in L₁(R^d). The function

\ g1∗g2(ξ) =

Z

R^d

g1(ξ−η)g2(η)dη (17) is called convolution ofg₁ and g₂. Its main property reads as

\

g1? g2(x) = gb1(x)·gb2(x), x∈R^d. (18) ForϕinKwe denote by(ϕ)^q_∗ itsqth convolution power, that is, ϕ∗. . .∗ϕ. Applying (17), (18) we get

(ϕ)^q_∗(0) = Z

R^d(q−1)

ϕ(ξ₁). . . ϕ(ξ_q−1)ϕ(−(ξ₁+. . .+ξ_q−1))dξ₁. . . dξ_q−1

= ((ϕ(·))b ^q)^∨(0) = (2π)^−d Z

R^d

(ϕ(x))b ^qdx . (19)

General linear trigonometric polynomial operators. As it was shown in [5]

approximation methods (3) - (5) are examples of linear bounded operators of general type

Lσ : Lp −→ Trσ, p ⊂ Lp , σ ≥0, (20)

where0< p ≤+∞ and r >0. The operator norm of L_σ is given by k L_σk_(p) = sup

kfkp≤1

k L_σ(f)k_p . (21) In particular, k { L^(A)_n;λ} k_(p) stands for the operator norm (quasinorm) of the family { L^(A)_n;λ}in the sense of (21). The sequence (L_σ) is called uniformly bounded on L_p if their operator norms are bounded by a constant independent of σ, that is,

sup

σ≥0

k L_σk_(p) < +∞, (22) and it is said to be convergent in L_p if for each f ∈L_p (f ∈C if p=∞)

σ→+∞lim kf − L_σ(f)k_p = 0 . (23)

Obviously, for the approximation processes dened in (3) and (4) these concepts coin- cide with the norm, the boundedness and the convergence in L_p in usual sense.

Taking into account that both the Uniform Boundedness Principle of functional analysis and the Marcinkiewicz Interpolation Theorem can be extended to the case of quasi-normed spaces (see, for example, [3], Theorem 1.3.2 and Appendix G ) we obtain the following statements for the operators (20).

Theorem A (Banach-Steinhaus Convergence Principle). Let0< p ≤+∞. A sequence (L_σ) of linear bounded operators of type (20) converges in L_p (in C, if p=∞) if and only if the following conditions are satised:

a) lim

σ→+∞ke^ik·− L_σ(e^ik·)k_p = 0 for each k∈Z^d; b) (L_σ) is uniformly bounded on L_p.

Theorem B (Marcinkiewicz Interpolation Principle). Let a sequence (L_σ) of linear operators of type (20) be bounded on Lp0 and on Lp1, where 0< p0 < p1 ≤+∞. Then it is bounded onL_p for all p₀ ≤p≤p₁ and there exists a constant c(p, p₀, p₁) such that

k L_σk_(p) ≤ c(p, p₀, p₁)k L_σk^1−θ_(p0)· k L_σk^θ_(p1), σ ≥0, µ1

p = 1−θ p₀ + θ

p₁

¶

. (24) Relations up to constants. By ⁰⁰A . B⁰⁰ we denote the relation A(f, σ) ≤ cB(f, σ), where c is a positive constant independent of f ∈ L_p (or f ∈ C) and σ ≥ 0. The symbol ⁰⁰ ³ ⁰⁰ indicates equivalence. It means that A . B and B . A simultaneously.

3 Families generated by kernels of type (GR)

In this section we formulate and prove some results on explicit ranges of convergence of families (5) generated by kernels of types (GR^α) and (GRα). Our approach is mainly based on the following lemmas which have been proved in [5].

Lemma 1. Let A be of type (1) , 0 < p ≤+∞, pe= min(1, p), pb=p for 0 < p <

+∞ and pb= 1 for p= +∞. Then

M_bp;A(n) . k { L^(A)_n;λ} k_(p) . M_p;eA(n), n∈N₀. (25) Lemma 2. Let 0 < p ≤1, ϕ∈ K. Then M_p;ϕ < +∞ if and only if p∈ P_ϕ, where Pϕ ={p∈(0,+∞] : ϕb∈Lp(R^d)}.

Lemma 3. For any T ∈ T_ρn, n ∈N₀, ρ≥r(A), λ∈R^d L^(A)_n;λ(T;x) = X

k∈Z^d

an, kcke^ikx , (26)

where c_k are the Fourier coecients of T.

Our result on convergence of families generated by kernels of type (GR^α) can be formulated as follows.

Theorem 1. Let A be of type (GR^α) for some 0 < α < 1, that is, A = A(ϕ) +R, where ϕ ∈ K and R = {r_{n, k}} satises M_α;R < +∞. Assume that 1 ∈ P_ϕ and α /∈ P_ϕ. Then the family { L^(A)_{n, λ}} converges in L_p if and only if p∈ P_ϕ.

Proof. Let 0< p≤+∞. Suppose k ∈Z^d,A ∈R^d and n ≥ |k|/ρ. Using (26) we get ke^ik·− L^(A)_n;λ(e^ik·)k_p = |1−a_{n, k}| · ke^ik·k_p = (2π)^d/p

¯¯

¯1−ϕ

³ k σ(n)

´

−r_{n, k}

¯¯

¯ .

Taking into account that ϕ is continuous,σ(n)³n, lim

n→+∞r_{n, k} = 0 by (7) we obtain

n→+∞lim ke^ik·− L^A_n;λ(e^ik·)k_p = 0, k∈Z^d. (27) In view of Theorem A we conclude therefrom that the family { L^(A)_{n, λ}} converges in L_p if and only if it is uniformly bounded on L_p.

Let rst p ∈ P_ϕ. Since 1 ∈ P_ϕ, pealso belongs to P_ϕ and M_p;eϕ <+∞ by Lemma 2. In view of (12) and p > αe we get by Nikol'skij's inequality for dierent norms for trigonometric polynomials (see, e.g., [12], Proposition 3.3.2)

M_p;eR(n) ≤ c M_α;R(n), n ∈N₀. (28) This impliesMp;Re <+∞. Hence, we obtain

sup

n

k { L^(A)_n;λ} k^p_(p)^e . M_p;e^pe_A≤M_p;ϕe^pe +M_p;e^pe_R<+∞. by Lemma 1 and { L^(A)_n;λ} converges in L_p.

Let now p≥α and p /∈ Pϕ. In this case pb=pe=p. By Lemma 1 we get

(M_p;ϕ(n))^p−(M_p;R(n))^p ≤(M_p;A(n))^p .k { L^(A)_n;λ} k^ep_(p) (29) for n ∈ N₀. Since M_p;ϕ = +∞ by Lemma 2 and M_p;R < +∞ in view of (28), the supremum at the left-hand side of (29) is equal to +∞. Hence, in this case { L^(A)_n;λ} is not uniformly bounded and divergent in L_p.

Finally we consider the case 0< p < α. Assume that the family { L^(A)_n;λ} converges in L_p. Then { L^(A)_n;λ} being uniformly bounded in L_p and L₁ should be uniformly bounded inL_αby the interpolation argument (Theorem B). This contradicts the above

result. ¤

We give a few remarks. Theorem 1 contains the General Convergence Theorem for kernels of type (G), which we have proved in [5], Theorem 4.1, as a special case corresponding to R = 0. To nd an estimate for the value M_α;R can be a rather sophisticated problem. The replacement of the requirement M_α;R < +∞ by the stronger condition M_α;R < +∞ is one possible approache to its solution. Another method is based on the concept of majorant.

Lemma 4. Suppose ψ ∈ K and ψ(x) =b O(|x|^−δ), (x→+∞) for some δ > d. Then

M_p;ψ(n) .







1 , d/δ < p≤+∞

(ln(n+ 1))^1/p , p=d/δ n^d/p−δ , 0< p < d/δ

. (30)

Proof. By Poisson's summation formula and properties of the Fourier transform we get

W_n(ψ)(x) = X

ν∈Z^d

ψ\(·/n)(2πν−x) = n^d X

ν∈Z^d

ψb(n(2πν −x)) for x∈[−π, π]^d. It follows therefrom the estimate

|W_n(ψ)(x)| ≤ n^d

³

|ψ(−nx)b | + cX

ν6=0

|nν|^−δ

´

n^d|ψ(−nx)b | + c₁n^−(δ−d) ,

(31)

where c and c₁ do not depend on x and n. Inequality (31) immediately implies (30) for p = +∞. Taking into account that n^−(δ−d) is bounded and integrating (31) over [−π, π]^d we conclude

kW_n(ψ)k^p_p . n^dp Z

[−π,π]^d

|ψ(nx)b |^pdx . n^d(p−1) Z

1≤|x|≤2πn

|x|^−δpdx .

n^d(p−1) Zn

1

r^−δp+d−1dr

(32)

for 0< p <+∞. Now (30) follows from (2.8) and (1.3). ¤ In view of (1.6) and (2.6) we obtain by Theorem 1 the following statement. In particular, it contains the result on convergence of families generated by kernels of type (GR_α) as a special case.

Theorem 2. Let A=A(ϕ) +R⁰+R⁰⁰, where ϕ∈ K, R⁰ ={r_{n, k}} satises M_α;R⁰ <

+∞ for some 0< α <1 and R⁰⁰={r⁰⁰_{n, k}} has the representation r⁰⁰_{n, k} = λ_nψ

µk n

¶

, n∈N; ψ ∈ K, where

ψ ∈ K; ψ(x) =b O(|x|^−δ), (x→+∞) ; λn=O(n^δ−d/α) ; d < δ < d/α .

Assume that 1∈ Pϕ and α /∈ Pϕ. Then the family{ L^(A)_{n, λ}} converges in Lp if and only if p∈ P_ϕ.

Proof. By (15) and Lemma 4 we obtain

M_α;^α_R0+R⁰⁰ ≤ M_α;^α_R0+M_α;^α_R00 =M_α;^α_R0+ µ

sup

n |λn|Mα;ψ(n)

¶_α

≤

≤ (2π)^d(1−α/2)M^α_α;R0 +c µ

sup

n |λ_n|n^d/α−δ

¶_α

<+∞.

Now the statement follows from Theorem 1. ¤

4 Families generated by powers of kernels of type (G)

In this Section we consider kernels of type (G^q). Let ϕ∈ Kand q ∈N such that Z

R^d

(ϕ(x))b ^qdx 6= 0. (33)

For the kernels

W₀(h) ≡ 1 ; W_n(h) = X

k∈Z^d

ϕ µk

n

¶

e^ikh, n∈N, (34) of type (G) we introduce theirqth power by (8)-(9). It is easy to see that

γ_n≡γ_n(ϕ, q) = (2π)^−d Z

T^d

(W_n(ϕ))^qdh = X

k1+...+kq=0

Yq

j=1

ϕ µk_j

n

¶

. (35) Up to the factor n^d(1−q) these quantities are the Riemannian sums of the integral at the left-hand side of (19). In view of (33) they are not equal to 0 for n ≥ n₀, where n0 ≡n0(ϕ, q) is a certain integer. By this reason the kernels Wn(ϕ, q)are well dened by (8)-(9) at least forn ≥n₀.

Theorem 3. Suppose that ϕ ∈ K satises (33) and let 1 ∈ P_ϕ. Then the family generated by the kernels W_n(ϕ, q) converges in L_p if and only if p∈q⁻¹P_ϕ.

Proof. By (8) we get

W_n(ϕ, q)(h) = X

k∈Z^d

a_{n, k}e^ikh, n≥n₀, (36) where

a_{n, k} ≡ a_{n, k}(ϕ, q) = γ_n⁻¹ X

k1+...+kq=k

Yq

j=1

ϕ µk_j

n

¶

, k ∈Z^d. (37) First we prove that

n→lim+∞a_{n, k} = 1 (38)

for each k ∈Z^d. If k ∈Z^d and n≥n₀ then a_{n, k}−1 = γ_n⁻¹



 X

k1+...+kq=k

Yq

j=1

ϕ µk_j

n

¶

− X

k1+...+kq=0

Yq

j=1

ϕ µk_j

n

¶

=

= γ_n⁻¹



 X

k1,..., kq−1

Yq−1

j=1

ϕ µk1

n

¶ . . . ϕ

µkq−1

n

¶

·

· µ

ϕ µk

n − k₁+. . .+k_q−1 n

¶

−ϕ µ

−k₁+. . .+k_q−1 n

¶¶¶

(39)

by (35) and (37).

In order to estimate the right-hand side of (39) we observe rst that

n→lim+∞n^d(1−q)γ_n = (2π)^−d Z

R^d

(ϕ(x))b ^q dx 6= 0 (40) by (33) and (35). Moreover, by uniform continuity of ϕfor each ε >0 there exists an integer n₁ such that

sup

ξ∈R^d

¯¯

¯¯ϕ µk

n −ξ

¶

− ϕ(−ξ)

¯¯

¯¯ < ε , n≥n₁ . (41) Taking into account that the number of nonvanishing items in the sum over k₁, . . . , k_q−1 in (39) does not exceed cn^d(q−1), where c is a positive constant inde- pendent of n and using (40) and (41) we get from (39)

|a_{n, k}−1| ≤ c⁰n^d(1−q)

¯¯

¯¯

¯¯ X

k1,..., kq−1

sup

ξ∈R^d

¯¯

¯¯ϕ µk

n −ξ

¶

− ϕ(−ξ)

¯¯

¯¯×

× Ã

sup

ξ∈R^d

|ϕ(ξ)|

!_q−1¯

¯¯

¯¯

¯ ≤ c⁰⁰ε

for suciently large n (k is xed). This completes the proof of (38).

Let 0< s <+∞. Combining (8), (9) and (40) we see that

n^d(1/s−1)kW_n(ϕ, q)k_s = n^d(1/s−1)|γ_n⁻¹| kW_n(ϕ)k^q_qs ³

³ ¡

n^d(1/qs−1)kWn(ϕ)kqs

¢_q

It follows the equivalence

M_s;A(n) ³ M_qs;ϕ(n), n≥n₀ , (42) using the notations in (12), whereA ={a_{n, k}} is given by (37).

Letp∈q⁻¹P_ϕ, that is,qp ∈ P_ϕ. Then qep∈ P_ϕ as well. Indeed, for 0< p≤1 it is obvious. If1≤p≤+∞, thenqpe=q ∈ P_ϕ becauseq ≥1and 1∈ P_ϕ. By the General Convergence Theorem [5], Theorem 4.1, (see also Theorem 1 with R= 0)

sup

n k { L^(ϕ)_n;λ} k_(qep) < +∞. (43) Applying Lemma 1 and (42) withs =pewe nd

k { L^(A)_n;λ} k_(p) . Mep, A(n) ³ Mqep;ϕ(n) . k { L^(ϕ)_n;λ} k_(qep) (44) for n ≥ n₀. By (43), (44) the family { L^(A)_n;λ} is uniformly bounded on L_p. Thus, its convergence follows from (38) and Theorem A.

Let now p /∈q⁻¹P_ϕ, that is, qp /∈ P_ϕ. Then M_qp;ϕ = +∞ by Lemma 2. In view of (42) with s = p it implies M_{p, A} = +∞. Hence, { L^(A)_n;λ} is not uniformly bounded on

Lp by Lemma 1. The proof is complete. ¤

5 Families generated by (generalized) Jackson kernels

The (generalized) Jackson kernels are dened by

Jn;q(h) = γ_{n, q}⁻¹ (Fn(h))^q, n∈N0, (2π)⁻¹ Z2π

0

(Fn(h))^q)dh = γn, q, (45) where q ∈ N and F_n(h) are the Fejer kernels generated by ϕ(ξ) = ( 1− |ξ|)₊ (a₊ = max(a,0)). The classical Fejer and Jackson kernels correspond to q = 1 and q = 2, respectively. In the general case J_n;q(h) is of type G^q. They have been introduced by S. Stechkin [13] in order to prove the Jackson type estimate (direct theorem of approximation theory) for moduli of smoothness of higher orders. Taking into account thatP_ϕ = (1/2,+∞](see, e.g. [5], Section 5) and applying Theorem 3 we immediately obtain the following statement.

Theorem 4. The family{J_n;λ^(q)}, q∈N, generated by the generalized Jackson kernels (45) converges in L_p if and only if 1/(2q)< p≤+∞.

Combining this result with the comparison principle [5], Lemma 2.2, we obtain the classical statements on the convergence of the Fourier means and the interpolation means generated byJ_n;q(h)in the spacesL_p for 1≤p≤+∞ andC, respectively (see, e.g. [2], Chapter 7, Ÿ2, [13]).

6 Families generated by Fejer-Korovkin kernels

The Fejer-Korovkin kernelsK_n(h)are dened by (1)-(2), where A={a_{n, k} : |k| ≤ n−2, n≥2} and

an, k =

(n− |k|+ 1 ) sin|k|+ 1

n π −(n− |k|+ 1 ) sin|k| −1

n π

2nsin(π/n) . (46)

They were introduced by P. Korovkin [4]. The corresponding Fourier means of type (3) turn out to solve the problem of the existence of positive methods of summation which would be relevant for the proof of the Jackson type estimate. In this connection we mention that the Fejer kernels do not have this property. By elementary calculations we get

a_{n, k} = µ

1− |k| n

¶

cosπ|k|

n + 1

πsinπ|k|

n + λ_n sinπ|k|

n , |k| ≤ n−2, (47) where

λn = 1−(n/π) sin(π/n)−2(sin(π/(2n)))²

nsin(π/n) ≥0, n≥2, (48)

Clearly,

λn . n⁻². (49)

By (47) A=A(ϕ) +R⁰+R⁰⁰, where

ϕ(ξ) =

( (1− |ξ|) cosπξ + (1/π) sinπ|ξ| , |ξ| ≤1

0 , |ξ|>1 (50)

R⁰ ={r_{n, k}⁰ }: r_{n, k}⁰ =











−λ_n−1sinπ(n−1) n −2ϕ

µn−1 n

¶

, |k|=n−1

0 , otherwise (51)

R⁰⁰ = {r⁰_{n, k}}: r⁰⁰_{n, k} = λ_nsinπ|k|

n , |k| ≤n . (52)

By straightforward calculation one has b

ϕ(x) = 4π² (cos(x/2))²

(x−π)²(x+π)² , P_ϕ = (1/4,+∞] . (53) Using (49) - (51) we obtain

sup

|k|=n−1

r_{n, k}⁰ . n⁻³. In view of (14) this leads to

M_1/4;R⁰ < +∞ . (54)

To estimate the contribution of R⁰⁰ we rewrite formula (5.7) as r⁰⁰_{n, k} = λnψ

µk n

¶

, |k| ≤n , (55)

where ψ(ξ) = sinπ|ξ|. By straightforward calculation ψ(x) =b 4π²cos²(x/2)

π²−x² ; ψ(x) =b O(|x|⁻²), x→+∞. (56) By means of (49), (53), (54) - (56) we see that all conditions of Theorem 2 are satised with d = 1, δ = 2 and α = 1/4. Applying Theorem 2 to the Fejer-Korovkin kernels we obtain the following result.

Theorem 5. The family {Kn;λ} generated by the Fejer-Korovkin kernels converges in L_p if and only if 1/4< p ≤+∞.

Combining this result with the comparison principle [5], Lemma 2.2, we obtain, in particular, the statement of P. Korovkin on the convergence of the Fourier means generated byKn(h)in C (cf. [4]).

7 Families generated by Cesaro kernels I: structure of the ma- trix of multipliers

As known (see, e.g. [15], Vol. 1), the Cesaro kernels Cn^(α)(h) with index α > 0 are dened by (1)-(2), where A≡A_α ={a_{n, k} :|k| ≤n−1, n∈N} and

a^(α)_{n, k} = Γ(n− |k|+α) Γ(n)

Γ(n− |k|) Γ(n+α), |k| ≤n−1. (57) Obviously, the case α = 1 corresponds to the Fejer kernels. Applying the asymptotic formula for theΓ-function

Γ(z) = (2π)^1/2e^−zz^z−1/2 ¡

1 + (12z)⁻¹+O(z⁻²)¢

, z 6= 0, |argz|< π , we get

a^(α)_{n, k} = µ

1− |k| n

¶_αµ

1 + α n− |k|

¶_{n−|k|+α−1/2}µ n n+α

¶_n+α−1/2

1 + ( 12(n− |k|+α) )⁻¹+O( (n− |k|)⁻²) 1 + ( 12(n− |k|) )⁻¹+O( (n− |k|)⁻²)

1 +O(n⁻¹) 1 +O(n⁻¹) ≡

≡ ϕ_α µk

n

¶

·J₁·J₂·J₃·J₄

(58)

for n ∈N,|k| ≤n−1from (57). Here ϕ_α(ξ) = (1− |ξ|)^α₊ and b_{n, k} =O(a_{n, k}) means that |b_{n, k}| ≤ca_{n, k}, wherec does not depend on k and n. Using Taylor expansions for expz, lnz and (1−z)⁻¹ we obtain

J₁ = e^α µ

1 + α(α−1)

2(n− |k|) + O¡

(n− |k|)⁻²¢¶

, (59)

J2 = e^−α¡

1 + O¡ n⁻¹¢¢

, (60)

J₃ = 1 + O¡

(n− |k|)⁻²¢

, (61)

J₄ = 1 + O¡ n⁻¹¢

. (62)

By (58) - (62)

a^(α)_{n, k} =ϕ_α µk

n

¶ µ

1 + α(α−1)

2(n− |k|) +O¡

(n− |k|)⁻²¢¶

(1 +λ_n) , (63) whereλ_n=O(n⁻¹).

We introduce the matrix A_α = {b_{n, k}}, where b_{n, k} are dened by the right-hand side of (63) withλ_n = 0. By (63) we get

Aα = A(ϕα) + R⁰_α + R⁰⁰_α, (64)

where

R⁰_α = ¡

r_{n, k}⁰ (α)¢

, r⁰_{n, k}(α) = α(α−1)

2 n⁻¹ϕ_α−1 µk

n

¶

, (65)

R⁰⁰_α = ¡

r⁰⁰_{n, k}(α)¢

, r⁰⁰_{n, k}(α) = O µ

n⁻²ϕα−2

µk n

¶ ¶

. (66)

Taking into account that

Wn(Aα)(h) = (1 +λn)Wn(Aα)(h)

it follows from Lemma 1 and Theorem A that the ranges of convergence of the Cesaro families { C_n;^(α)_λ} and the families generated by A_α coincide. For this reason we deal with the matrix A_α in the following.

8 Families generated by Cesaro kernels II: Fourier transform of the generator

As it was shown in the previous section, the matrix Aα is of type (7) with ϕ(ξ) ≡ ϕ_α(ξ) = (1− |ξ|)^α₊. Following our approach we have to study the properties of the Fourier transform of the generator ϕ_α and to dtermine the set P_α = {p ∈ (0,+∞] :

b

ϕα ∈Lp(R)}.

Let ψ, ψ₀ and ψ₁ be real valued even innitely dierentiable functions dened on R and satisfying

ψ(ξ) =

( 1, ξ ∈D₁ 0, ξ /∈D5/4

; ψ0(ξ) =

( 1, ξ ∈D_1/2 0, ξ /∈D3/4

; ψ1(ξ) = ψ(ξ)−ψ0(ξ), (67)

where D_ρ={ξ : |ξ| ≤ρ}. It is clear that

ϕ_α(ξ) = (ϕ_αψ₀)(ξ) + (ϕ_αψ₁)(ξ), ξ ∈R. (68) In view of

( 1−y)^α = X3

ν=0

(−1)^ν[α]_ν

ν! y^ν + y⁴g(y), −1< y <1,

where [α]ν =α(α−1). . .(α−ν+ 1) and g(y) is analytic on (−1,1), we obtain (ϕ_αψ₀)(ξ) =

X3

ν=0

(−1)^ν[α]_ν

ν! |ξ|^νψ₀(ξ) + |ξ|⁴g(|ξ|)ψ₀(ξ). (69) As it was shown in [11], Lemma 4.1 and Theorem 4.1, for any δ >0, δ 6= 2k for each k ∈None has ¯

¯¯¡

| · |^δψ₀(·)¢_∧ (x)

¯¯

¯ ³ |x|^−(δ+1), |x| → +∞, (70)