PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 91(105) (2012), 163–175 DOI: 10.2298/PIM1205163M

ON AN INTERPOLATION PROCESS OF LAGRANGE–HERMITE TYPE

Giuseppe Mastroianni, Gradimir V. Milovanović, and Incoronata Notarangelo

Abstract. We consider a Lagrange–Hermite polynomial, interpolating a function at the Jacobi zeros and, with its first (r − 1) derivatives, at the points ±1. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator in certain suitable weighted L^{p}-spaces, 1 < p < ∞, proving a Marcinkiewicz inequality involving the derivative of the polynomial at ±1. Moreover, we give optimal estimates for the error of this process also in the weighted uniform metric.

1. Introduction

Let us denote by L_{m,r} (v^{α} , f) the polynomial of Lagrange–Hermite type based on the Jacobi zeros x_{k} = x_{m,k} (v^{α}) related to the weight v^{α} (x) = (1 − x^{2})^{α} and whose jth order derivatives at ±1 are equal to f^{(j)} (±1), j = 0, 1, …, r − 1, i.e.,

L_{m,r} (v^{α} , f, x_{k}) = f(x_{k}), k = 1, …, m,
L_{m,r} (v^{α} , f)^{(j)} (±1) = f^{(j)} (±1), j = 0, 1, …, r − 1,

where f^{(0)} ≡ f.

This interpolation process is sometimes useful in the numerical treatment of differential equations with boundary conditions. The authors had already took into consideration a similar procedure obtaining some results that the reader can find in [3, pp. 260, 272]. In the present paper we are going to study the behaviour

2010 Mathematics Subject Classification: Primary 41A05; Secondary 41A10. Key words and phrases: Hermite–Lagrange interpolation, approximation by polynomials,

orthogonal polynomials, Jacobi weights.

The first author was partially supported by GNCS (project 2011: Tecniche numeriche per problemi di propagazione di onde elastiche in multidomini) and by University of Basilicata (local funds).

The second author was supported in part by the Serbian Ministry of Education and Science (Project: Approximation of integral and differential operators and applications, grant number #174015).

The third author was supported by University of Basilicata (local funds). This paper is dedicated to Academician Anton Bilimović.

of the sequence {L_{m,r} (v^{α} , f)}_{m} in certain suitable weighted L^{p}-spaces and give necessary and sufficient conditions on the weights for the uniform boundedness of {L_{m}(v^{α})}_{m}. Optimal estimates of the error will be given and a Marcinkiewicz inequality involving the derivatives of the polynomial at ±1 will be proved. The results of this paper cover the ones available in literature.

In Section 2 we will state our main results and in Section 3 we will prove them.

2. Main Results

In the following C denotes a positive constant which may have different values in different formulas. We will write C ≠ C(a, b, …) to say that C is independent of the parameters a, b, … If A, B > 0 are quantities depending on some parameters, we write A ∼ B, if there exists a positive constant C independent of the parameters of A and B, such that B/C ⩽ A ⩽ CB.

Now we introduce some function spaces, related to a Jacobi weight of the form

(2.1)  v^{γ}(x) = (1 − x^{2})^{γ}, γ ⩾ 0, x ∈ (−1, 1).

Letting L^{p}, 1 ⩽ p < ∞, denote the space of all measurable functions f with ∥f∥_{p}^{p} = ∫_{−1}^{1} |f|^{p}, we say f ∈ L_{v^{γ}}^{p} if fv^{γ} ∈ L^{p}, i.e., ∥f∥_{L_{v^{γ}}^{p}}^{p} = ∫_{−1}^{1} |fv^{γ}|^{p} < ∞. For

p = ∞ and γ > 0, we set L_{v^{γ}}^{∞} = C_{v^{γ}} = {f ∈ C^{0}(−1, 1) : lim_{|x|→1} (fv^{γ})(x) = 0} and C_{v^{0}} ≡ C^{0}[−1, 1]. Moreover, we set

C_{r}^{0} = {f ∈ C^{0}(−1, 1) : f is (r − 1) − times differentiable at ± 1} ,

where r ⩾ 1 is an integer number. Of course, C_{r}^{0} ⊂ C_{v^{γ}} , γ ⩾ 0, and C_{r}^{0} ⊃ C^{r−1} [−1, 1], where C^{r−1} [−1, 1] is the collection of all functions whose (r − 1)th derivative is continuous on [−1, 1].

The Sobolev type spaces are defined as follows

W_{p}^{s} = W_{p}^{s}(v^{γ}) = {f ∈ L_{v^{γ}}^{p} : f^{(s−1)} ∈ AC(−1, 1) and ∥f^{(s)} φ^{s}v^{γ}∥_{p} < ∞},

where φ(x) = \sqrt{1 − x^{2}} , AC(−1, 1) is the set of the absolutely continuous functions in every compact of (−1, 1), 1 ⩽ p ⩽ ∞ and s ⩾ 1 is an integer.

Let v^{α} (x) = (1 − x^{2})^{α}, α > −1, and {p_{m}(v^{α})}_{m} be the corresponding sequence of orthonormal polynomials with positive leading coefficients. For every function f ∈ C_{r}^{0}, r ⩾ 1, an expression of the polynomial L_{m,r} (v^{α} , f, x), α > −1, is given by

L_{m,r} (v^{α} , f, x) = ∑_{k=1}^{m} v^{r}(x)\frac{l_{k}(x)}{v^{r}(x_{k})}f(x_{k})

+ (1 − x)^{r}p_{m}(v^{α} , x) ∑_{i=0}^{r−1} \frac{(1 + x)^{i}}{i!} ( \frac{f}{(1 − ·)^{r}p_{m}(v^{α})}) ^{(i)} (−1)

+ (1 + x)^{r}p_{m}(v^{α} , x) ∑_{i=0}^{r−1} \frac{(1 − x)^{i}}{i!} ( \frac{f}{(1 + ·)^{r}p_{m}(v^{α})}) ^{(i)} (1)

=: A(x) + B_{1}(x) + B_{2}(x), (2.2) where x_{k}, k = 1, …, m, are zeros of p_{m}(v^{α}) and

l_{k}(x) = \frac{p_{m}(v^{α} , x)}{p′_{m}(v^{α} , x_{k})(x − x_{k})} , k = 1, …, m.

We complete the definition of L_{m,r} (v^{α} , f) setting L_{m,0} (v^{α} , f) = L_{m}(v^{α} , f).

Finally, letting ℙ_{m} be the space of all polynomials of degree at most m, we denote by E_{m}(f)_{v^{γ} ,p} = inf_{P_{m} ∈ℙ_{m}} ∥(f − P_{m})v^{γ}∥_{p}, 1 ⩽ p ⩽ ∞, the error of best polynomial approximation in L_{v^{γ}}^{p} .

Now we are able to study the behaviour of the sequence {L_{m,r} (v^{α})}_{m}, r ⩾ 1, in the above introduced function spaces.

Theorem 2.1. Let v^{α} and v^{γ} be two Jacobi weight functions defined in (2.1), with γ ⩾ 0 and α > −1. Then, for every f ∈ C_{r}^{0}, r ⩾ 1, we have

(2.3) ∥L_{m,r} (v^{α} , f)v^{γ}∥_{∞} ⩽ C\{ ∥fv^{γ}∥_{∞} logm + \frac{1}{m^{2γ}} ∑_{j=0}^{r−1} \frac{|f^{(j)} (−1)| + |f^{(j)} (1)|}{j! m^{2j}} \} ,

where C ≠ C(m, f), if and only if

(2.4)  \frac{α}{2} + \frac{1}{4} ⩽ γ + r ⩽ \frac{α}{2} + \frac{5}{4}.

Moreover, if f ∈ C^{r−1} [−1, 1], the condition (2.4) implies

(2.5)  ∥[f − L_{m,r} (v^{α} , f)]v^{γ}∥_{∞} ⩽ CE_{m+2r−1} (f)_{v^{γ} ,∞} logm, C ≠ C(m, f).

The above theorem includes some special cases that are well-known in literature. For example, for γ = 0 and r ⩾ 1 we get Theorem 4.2.5 in [3, p. 260]. In the case r = 0 and γ > 0 we obtain Theorem 2.2 in [4] (see also [3, p. 272]), and for γ = r = 0 we get Theorem 14.4 in [9, p. 335].

Theorem 2.2. Let 1 ⩽ p < ∞, γ ⩾ 0, and α > −1. Then, for every function f ∈ C_{r}^{0}, r ⩾ 1, there exists a constant C ≠ C(m, f) such that

(2.6) ∥L_{m,r} (v^{α} , f)v^{γ}∥_{p} ⩽ C\{ ∥fv^{γ}∥_{∞} + \frac{1}{m^{2γ+2/p}} ∑_{j=0}^{r−1} \frac{|f^{(j)} (−1)| + |f^{(j)} (1)|}{j! m^{2j}} \}

if and only if

(2.7)  \frac{v^{γ+r}}{\sqrt{v^{α} φ}} ∈ L^{p} and \frac{\sqrt{v^{α} φ}}{v^{γ+r}} ∈ L^{1}

i.e.,

−\frac{1}{p} < γ + r − \frac{α}{2} − \frac{1}{4} < 1.

Letting

(2.8)  σ_{m}(f) = ∑_{j=0}^{r−1} \frac{|f^{(j)} (−1)| + |f^{(j)} (1)|}{j!m^{2j}} ,

it is easy to deduce from the proof of the above theorem that the only condition v^{γ+r} /\sqrt{v^{α} φ} ∈ L^{p} is necessary and sufficient to obtain the bound

∥L_{m,r} (v^{α} , f)v^{γ}∥_{p} ⩽ C∥f∥_{∞} + \frac{σ_{m}(f)}{m^{2γ+2/p}} , 1 ⩽ p < ∞,

which, for r = 0, follows from a well-know theorem of P. Nevai [7, p. 680].

The following theorem is a refinement of the previous one and implies some interesting consequences.

Theorem 2.3. Let 1 < p < ∞, γ ⩾ 0, and α > −1. Then, for every function f ∈ C_{r}^{0}, r ⩾ 1, there exists a constant C ≠ C(m, f) such that

(2.9)  ∥L_{m,r} (v^{α} , f)v^{γ}∥_{p} ⩽ C{ ( ∑_{k=1}^{m} Δx_{k}|fv^{γ}|^{p}(x_{k})) ^{1/p}

+ \frac{1}{m^{2γ+2/p}} ∑_{j=0}^{r−1} \frac{|f^{(j)} (−1)| + |f^{(j)} (1)|}{j!m^{2j}} \} (2.10)

if and only if

(2.11)  \frac{v^{γ+r}}{\sqrt{v^{α} φ}} ∈ L^{p} and \frac{\sqrt{v^{α} φ}}{v^{γ+r}} ∈ L^{q}, q^{−1} + p^{−1} = 1,

i.e.,

−\frac{1}{p} < γ + r − \frac{α}{2} − \frac{1}{4} < \frac{1}{q}.

Now we state some estimates of the error f − L_{m,r} (v^{α} , f) for f varying in the previously introduced spaces.

Proposition 2.1. For any f ∈ C^{r−1} [−1, 1], r ⩾ 1, under the assumptions (2.7) we have

(2.12)  ∥[f − L_{m,r} (v^{α} , f)]v^{γ} ∥_{p} ⩽ CE_{m+2r−1} (f)_{v^{γ} ,∞} ,

where 1 ⩽ p < ∞ and C ≠ C(m, f). Moreover, if f ∈ W_{p}^{r}, r ⩾ 1, and 1 < p < ∞, then the conditions (2.11) imply

(2.13)  ∥[f − L_{m,r} (v^{α} , f)]v^{γ} ∥_{p} ⩽ \frac{C}{m^{r}} ∥f^{(r)} φ^{r}v^{γ} ∥_{p}, C ≠ C(m, f).

Note that (2.13) shows that, if f ∈ W_{p}^{r}, with r ⩾ 1 and 1 < p < ∞, the polynomial L_{m,r} (v^{α} , f) converges with the order of the best polynomial approximation in L_{v^{γ}}^{p} . Therefore, in the usual way, we can establish the next corollary that shows the uniform boundedness (with respect to m) of the operator L_{m,r} (v^{α}) in Sobolev spaces (under the assumptions (2.11)).

Corollary 2.1. Under the conditions (2.11), for every f ∈ W_{p}^{r} with r ⩾ 1 and 1 < p < ∞, we have

(2.14)  sup_{m} ∥L_{m,r} (v^{α} , f)∥_{W_{p}^{r}} ⩽ C∥f∥_{W_{p}^{r}} , C ≠ C(f).

Coming back to Theorem 2.3, the estimate (2.9) with the notation (2.8) can be written as

∥L_{m,r} (v^{α} , f)v^{γ} ∥_{p} ⩽ C\{ ( ∑_{k=1}^{m} Δx_{k}|fv^{γ}|^{p}(x_{k})) ^{1/p} + \frac{σ_{m}(f)}{m^{2γ+2/p}} \} =: Γ_{m}(f).

Of course, if f is a polynomial P of degree m + 2r − 1, the inequality

Γ_{m}(P) ⩾ C∥L_{m,r} (v^{α} , P)v^{γ}∥_{p} = C∥P v^{γ}∥_{p}, 1 < p < ∞,

is equivalent to the conditions (2.11).

Moreover, it is easy to prove that for arbitrary α > −1, γ ⩾ 0, and r ⩾ 1, the inverse inequality Γ_{m}(P) ⩽ C∥P v^{γ}∥_{p} holds true for 1 ⩽ p < ∞. In fact, the bound

(2.15)  ( ∑_{k=1}^{m} Δx_{k}|P v^{γ}|^{p}(x_{k})) ^{1/p} ⩽ C∥P v^{γ}∥_{p}

is well-known (see, for example, [7, p. 675]). In order to obtain

\frac{σ_{m}(P)}{m^{2γ+2/p}} ⩽ C∥P v^{γ}∥_{p}

it suffices to apply the inequalities of Markov, Schur and Nikol’skiĭ.

Therefore, we can state a new Marcinkiewicz inequality involving the derivatives of a polynomial at ±1.

Corollary 2.2. Let x_{k}, k = 1, …, m, be the zeros of the mth Jacobi polynomial p_{m}(v^{α}) and let 1 < p < ∞. Then, for every polynomial P ∈ ℙ_{m+2r−1} , the following equivalence

(2.16)  ∥P v^{γ}∥_{p} ∼ ( ∑_{k=1}^{m} Δx_{k}|P v^{γ}|^{p}(x_{k})) ^{1/p} + \frac{σ_{m}(P)}{m^{2γ+2/p}}

holds true, with the constants in “∼" independent of m and P, if and only if

−\frac{1}{p} < r + γ − \frac{α}{2} − \frac{1}{4} < \frac{1}{q} (p^{−1} + q^{−1} = 1).

Finally, we want to observe that if we introduce the mth Christoffel function of the weight v^{γp},

λ_{m}(v^{γp}, x) = [ ∑_{k=0}^{m−1} p_{k}^{2}(v^{γp}, x)]^{−1} ∼ v^{γp}(x)\frac{\sqrt{1 − x^{2}}}{m} ,

then the sum in (2.16) can be replaced by

( ∑_{k=1}^{m} λ_{m}(v^{γp}, x_{k})|P (x_{k})|^{p}) ^{1/p}.

3. Proofs

In this section we will frequently use the Remez-type inequality in the following form

(3.1)  (∀P_{m} ∈ ℙ_{m}) ∥P_{m}v^{γ}∥_{p} ⩽ C∥P_{m}v^{γ}∥_{L^{p} (A_{m} )} ,

where A_{m} = [−1 + am^{−2} , 1 − am^{−2} ] , with a > 0 fixed, and C ≠ C(m, P_{m}). If I is a subinterval of (−1, 1), the Hilbert transform H(f, t) is defined as follows

H(f, t) = ∫_{I} \frac{f(x)}{x − t} dx, t ∈ I,

where the integral is understood in the Cauchy principal value sense. For 1 < p < ∞, the following property is well-known:

∫_{I} gH(f) = − ∫_{I} fH(g), f ∈ L^{p} and g ∈ L^{q}, \frac{1}{p} + \frac{1}{q} = 1.

Moreover, with v^{σ}(x) = (1 − x^{2})^{σ} and 1 < p < ∞, one has

∥(Hf)v^{σ}∥_{p} ⩽ C∥fv^{σ}∥_{p} if and only if − \frac{1}{p} < σ < \frac{1}{q}.

Now, recalling (2.2) with the notation (2.8), we can state the following lemma:

Lemma 3.1. With the notation (2.2), we have

∥(B_{1} + B_{2})v^{γ}∥_{∞} ⩽ \frac{C}{m^{2γ}} σ_{m}(f), C ≠ C(f, m),

if and only if \frac{α}{2} + \frac{1}{4} ⩽ γ + r ⩽ \frac{α}{2} + \frac{5}{4}. Moreover, for p ∈ [1, ∞), we get

∥(B_{1} + B_{2})v^{γ}∥_{p} ⩽ \frac{C}{m^{2γ+2/p}} σ_{m}(f), C ≠ C(f, m),

if and only if −\frac{1}{p} < γ + r − \frac{α}{2} − \frac{1}{4} < \frac{1}{q}.

Proof. We estimate only the norm ∥B_{1}v^{γ}∥_{p}, 1 ⩽ p ⩽ ∞, since the estimate of ∥B_{2}v^{γ}∥_{p} is similar. Using the Remez inequality (3.1) and letting

Ā_{i} = \frac{1}{i!} ( \frac{f}{(1 − ·)^{r}p_{m}(v^{α})}) ^{(i)} (−1),

we can write

∥B_{1}v^{γ}∥_{p} ⩽ C∥B_{1}v^{γ}∥_{L^{p} (A_{m} )}

⩽ ∑_{i=0}^{r−1} |Ā_{i}|∥(1 − x)^{r}(1 + x)^{i}(1 − x^{2})^{γ}p_{m}(v^{α} , x)∥_{L^{p} (A_{m} )} =: ∑_{i=0}^{r−1} |Ā_{i}|b_{i}. (3.2)

Of course, we have

b_{i} ⩽ ∥(1 − x)^{r}(1 + x)^{i}(1 − x^{2})^{γ}p_{m}(v^{α} , x)∥_{L^{p} (−1+a/m^{2}, 0)}
+ ∥(1 − x)^{r}(1 + x)^{i}(1 − x^{2})^{γ}p_{m}(v^{α} , x)∥_{L^{p} (0,1−a/m^{2} )}
:= I_{1} + I_{2}.

Moreover, by virtue of the estimate |p_{m}(v^{α} , x)| ⩽ Cv^{− \frac{α}{2} − \frac{1}{4}} (x), |x| ⩽ 1 − a/m^{2}, we have

I_{1} + I_{2} ⩽ C\{ ∥(1 + x)^{γ+i− \frac{α}{2} − \frac{1}{4}}  ∥_{L^{p} (−1+a/m^{2}, 0)} + ∥(1 − x)^{γ+r− \frac{α}{2} − \frac{1}{4}}  ∥_{L^{p} (0,1−a/m^{2} )} \}.

Now, under the assumptions on the parameters α, γ and r (and only in this case), the first summand is dominated by C (m^{−2} )^{γ+i− \frac{α}{2} − \frac{1}{4} + \frac{1}{p}} , with γ +i− \frac{α}{2} − \frac{1}{4} + \frac{1}{p} ⩽ 0, 1 ⩽ p ⩽ ∞, while the second summand is bounded. In particular, for p < ∞ we have γ + i − \frac{α}{2} − \frac{1}{4} + \frac{1}{p} < 0, while for p = ∞ we have γ + i − \frac{α}{2} − \frac{1}{4} = 0 only in the case i = r − 1. In any case, since |p_{m}(v^{α} , ±1)| ∼ m^{α+1/2} (see for instance [3, p. 251, formula (4.2.10)]) we conclude that

I_{1} + I_{2} ⩽ \frac{C}{m^{2γ+2/p}} \frac{|p_{m}(v^{α} , −1)|}{m^{2i}} , 1 ⩽ p ⩽ ∞,

taking into account that 2γ + 2i = α + 1/2 for i = r − 1 and p = ∞. Therefore, recalling (3.2), we have

∥B_{1}v^{γ}∥_{p} ⩽ C \frac{|p_{m}(v^{α} , −1)|}{m^{2γ+2/p}} ∑_{i=0}^{r−1} \frac{Ā_{i}}{m^{2i}} .

It remains to estimate Ā_{i}, i = 0, 1, …, r − 1. We have

|Ā_{i} | ⩽ \frac{1}{i!} ∑_{j=0}^{i} \binom{i}{j} |f^{(j)} (−1)||( \frac{1}{(1 − x)^{r}} \frac{1}{p_{m}(v^{α} , x)})^{(i−j)} (−1)|

and, taking into account that [7, p. 674, formula (23)]

( \frac{1}{p_{m}(v^{α} , x)})^{(k)}(−1) ⩽ C \frac{m^{2k}}{|p_{m}(v^{α} , −1)|} ,

we obtain

|( \frac{1}{(1 − x)^{r}} \frac{1}{p_{m}(v^{α} , x)})^{(i−j)} (−1)| ⩽ C \frac{m^{2i−2j}}{|p_{m}(v^{α} , −1)|} .

Hence we get

|Ā_{i}| ⩽ \frac{C}{i! |p_{m}(v^{α} , −1)|} ∑_{j=0}^{i} \binom{i}{j} |f^{(j)} (−1)|m^{2i−2j}

and, for 1 ⩽ p ⩽ ∞,

∥B_{1}v^{γ}∥_{p} ⩽ \frac{C}{m^{2γ+2/p}} ∑_{i=0}^{r−1} \frac{1}{i!} ∑_{j=0}^{i} \binom{i}{j} \frac{|f^{(j)} (−1)|}{m^{2j}}

= \frac{C}{m^{2γ+2/p}} ∑_{j=0}^{r−1} \frac{|f^{(j)} (−1)|}{m^{2j}} ∑_{i=j}^{r−1} \frac{1}{i!} \binom{i}{j}

⩽ \frac{C}{m^{2γ+2/p}} ∑_{j=0}^{r−1} \frac{|f^{(j)} (−1)|}{j!m^{2j}} ,

which completes the proof. □

Proof of Theorem 2.1. In Lemma 3.1, we proved that

∥(B_{1} + B_{2})v^{γ}∥_{∞} ⩽ \frac{C}{m^{2γ}} σ_{m}(f).

Then, it remains to prove that ∥Av^{γ}∥_{∞} ⩽ C∥fv^{γ}∥_{∞} logm. But, the latter inequality can be found in [3, p. 262] with a minor change. So, the proof of (2.3) is complete.

Concerning the estimate of the error (2.5), we refer to the proof of Proposition 2.1. □

We are going to prove Theorem 2.3 before Theorem 2.2.

Proof of Theorem 2.3. Taking into account Lemma 3.1, to prove the theorem, it suffices to show that the inequality

(3.3)  ∥Av^{γ}∥_{p}^{p} ⩽ C ∑_{k=1}^{m} Δx_{k}|fv^{γ}|^{p}(x_{k}), 1 < p < ∞,

is equivalent to the conditions (2.11).

We first prove that (2.11) implies (3.3). To this end, using (3.1) and, letting g = v^{γ(p−1)} |A|^{p−1} sgnA in the interval A_{m}, we can write

∥Av^{γ}∥_{L^{p} (A_{m} )}^{p} = ∫_{A_{m}} ∑_{k=1}^{m} v^{γ+r} (x)\frac{l_{k}(x)f(x_{k})}{v^{r}(x_{k})} g(x) dx,

where

l_{k}(x) = \frac{p_{m}(v^{α} , x)}{p′_{m}(v^{α} , x_{k})(x − x_{k})} ,

whence we deduce

∥Av^{γ}∥_{L^{p} (A_{m} )}^{p} = ∑_{k=1}^{m} \frac{f(x_{k})v^{γ}(x_{k})}{p′_{m}(v^{α} , x_{k})v^{γ+r} (x_{k})} ∫_{A_{m}} \frac{v^{γ+r} (x)p_{m}(v^{α} , x)}{x − x_{k}}g(x) dx

⩽ C ∑_{k=1}^{m} \frac{Δx_{k}|fv^{γ}|(x_{k})}{v^{γ+r− \frac{α}{2} − \frac{1}{4}} (x_{k})} |∫_{A_{m}} v^{γ+r} (x)p_{m}(v^{α} , x)\frac{g(x)}{x − x_{k}} dx|,

since, with Δx_{k} = x_{k+1} − x_{k}, 1/|p′_{m}(v^{α} , x_{k})| ∼ Δx_{k}v^{\frac{α}{2} + \frac{1}{4}} (x_{k}). Denoting by G(x_{k}) the absolute value of the integral at the right-hand side and using the Hölder inequality, we get

∥Av^{γ}∥_{L^{p} (A_{m} )}^{p} ⩽ ( ∑_{k=1}^{m} Δx_{k}|fv^{γ}|^{p}(x_{k})) ^{1/p}( ∑_{k=1}^{m} Δx_{k} [ \frac{\sqrt{v^{α} φ}}{v^{γ+r}} (x_{k})G(x_{k})]^{q}) ^{1/q}.

It remains to prove that the L^{q} norm is bounded by C∥Av^{γ}∥_{L^{p} (A_{m} )}^{p−1} . We note that for an arbitrary polynomial Q ∈ ℙ_{m}, we can write

G(x_{k}) = |∫_{A_{m}} \frac{p_{m}(v^{α} , x)Q(x) − p_{m}(v^{α} , x_{k})Q(x_{k})}{x − x_{k}} v^{γ+r} (x)\frac{g(x)}{Q(x)}dx| .

Therefore, G(t) is a polynomial of degree 2m − 1. Then, using the Marcinkiewicz inequality (2.15), the L^{q} norm is dominated by a positive constant C times the norm ∥ \frac{\sqrt{v^{α} φ}}{v^{γ+r}} G∥_{L^{q}(A_{m} )} that is bounded under the assumption \frac{\sqrt{v^{α} φ}}{v^{γ+r}} ∈ L^{q}. Moreover, denoting by H the Hilbert transform defined on the interval A_{m}, we can write

∥ \frac{\sqrt{v^{α} φ}}{v^{γ+r}} G∥_{L^{q}(A_{m} )} ⩽ ∥ \frac{\sqrt{v^{α} φ}}{v^{γ+r}} H(p_{m}(v^{α})v^{γ+r} g)∥_{L^{q}(A_{m} )}

+ ∥ \frac{\sqrt{v^{α} φ}}{v^{γ+r}} Qp_{m}(v^{α})H( \frac{gv^{γ+r}}{Q} ) ∥_{L^{q}(A_{m} )} .

Taking also into account the assumption \frac{v^{γ+r}}{\sqrt{v^{α} φ}} ∈ L^{p}, the Hilbert transform is a bounded operator and the first norm is dominated by

∥ \frac{\sqrt{v^{α} φ}}{v^{γ+r}} p_{m}(v^{α})v^{γ+r} g∥_{L^{q}(A_{m} )} ⩽ C∥g∥_{L^{q}(A_{m} )} = ∥Av^{γ}∥_{L^{p} (A_{m} )}^{p−1} ,

since |p_{m}(v^{α} , x)\sqrt{(v^{α} φ)(x)}| ⩽ C.

In order to prove the estimate of the second norm at the right-hand side, we choose a polynomial Q such that Q(x) ∼ v^{γ+r} (x) for x ∈ A_{m} (see [2]). Consequently, the second norm is dominated by

∥H ( \frac{gv^{γ+r}}{Q} ) ∥_{q} ⩽ C ∥ \frac{gv^{γ+r}}{Q} ∥_{q} ⩽ C∥g∥_{q} = ∥Av^{γ}∥_{L^{p} (A_{m} )}^{p−1} .

Then (2.11) implies (3.3).

Now, we prove that (2.11) is a consequence of (3.3). To this end, for any f ∈ C_{r}^{0}, we consider a piecewise linear function F_{m} such that

F_{m}^{(i)} (±1) = 0,  i = 0, 1, …, r − 1,
F_{m}(x_{k}) = 0,  for x_{k} ∈ [−a, a], with a < \frac{1}{4} fixed,
F_{m}(x_{k}) = |f(x_{k})| sgn{−x_{k}p′_{m}(v^{α} , x_{k})} , for x_{k} ∉ [−a, a].

Taking into account that sgn(−x_{k}) = sgn(x − x_{k}) for x ∈ [−a, a] and x_{k} ∉ [−a, a], we get

|L_{m,r} (v^{α} , F_{m}, x)v^{γ}(x)| = |v^{γ+r} (x)p_{m}(v^{α} , x)| ∑_{k=1}^{m} \frac{|F_{m}v^{γ}|(x_{k})}{|p′_{m}(v^{α} , x_{k})v^{γ+r} (x_{k})||x − x_{k}|}

⩾ C | \frac{v^{γ+r} (x)}{2}p_{m}(v^{α} , x)| ∑_{k=1}^{m} Δx_{k}v^{α+ \frac{1}{4} −γ−r} (x_{k})|F_{m}v^{γ}|(x_{k}),

since |p′_{m}(v^{α} , x_{k})|^{−1} ∼ Δx_{k}v^{\frac{α}{2} + \frac{1}{4}} (x_{k}) and |x − x_{k}| ⩽ 2. Moreover, by virtue of a result in [6], we have

lim inf_{m→∞} ∥v^{γ+r} χ_{a}p_{m}(v^{α})∥_{p} ⩾ C ∥ \frac{v^{γ+r} χ_{a}}{\sqrt{v^{α} φ}} ∥_{p} ⩾ C,

being χ_{a} the characteristic function of [−a, a].

Then, collecting the previous inequalities, by (3.3), we obtain

∑_{k=1}^{m} Δx_{k} \frac{\sqrt{v^{α} φ}}{v^{γ+r}} (x_{k})|F_{m}v^{γ}|(x_{k}) ⩽ ∥L_{m,r} (v^{α} , F_{m})χ_{a}v^{γ}∥_{L^{p} (A_{m} )}

⩽ C( ∑_{k=1}^{m} Δx_{k}|F_{m}v^{γ}|^{p}(x_{k})) ^{1/p}. (3.4)

Now, letting

a_{k} = (Δx_{k})^{1/q} \frac{\sqrt{v^{α} φ}}{v^{γ+r}} (x_{k}), c_{k} = (Δx_{k})^{1/p}|F_{m}v^{γ}|(x_{k}), ∥c̄∥_{p}^{∗} = ( ∑_{k=1}^{m} |c_{k}|^{p}) ^{1/p},

where c̄ = (c_{1}, c_{2}, …, c_{m}), we can write (3.4) as ∑ _{k=1}^{m} a_{k}c_{k} ⩽ C∥c̄∥_{p}^{∗} and, since C ≠ C(m, F_{m}),

sup_{m} sup_{c̄} ∑_{k=1}^{m} a_{k} \frac{c_{k}}{∥c̄∥_{p}^{∗}} ⩽ C.

Hence, we get sup_{m} (∑ _{k=1}^{m} |a_{k}|^{q})^{1/q} ⩽ C, i.e.,

sup_{m}( ∑_{k=1}^{m} Δx_{k} ( \frac{\sqrt{v^{α} φ}}{v^{γ+r}} (x_{k}))^{q} ) ^{1/q} ⩽ C.

The latter inequality is equivalent to ∥ \frac{\sqrt{v^{α} φ}}{v^{γ+r}} ∥_{q} < ∞ which is, therefore, a consequence of (3.3).

Finally, we prove that (3.3) implies also ∥ \frac{v^{γ+r}}{\sqrt{v^{α} φ}} ∥_{p} < ∞. To this end, since (3.3) holds true for every f ∈ C^{0}(−1, 1), letting g(x) = f(x)v^{r}(x), we have

∥Av^{γ}∥_{L^{p} (A_{m} )} = ∥L_{m}(v^{α} , f)v^{γ+r} ∥_{L^{p} (A_{m} )} ⩽ C∥fv^{γ+r} ∥_{∞} ,

i.e.,

sup_{m} sup_{∥fv^{γ+r} ∥_{∞} =1} ∥L_{m}(v^{α} , f)v^{γ+r} ∥_{L^{p} (A_{m} )} ⩽ C, C ≠ C(m, f).

Therefore, using [5], we get sup_{m} ∥p_{m}(v^{α})v^{γ+r} ∥_{p} ⩽ C, i.e., \frac{v^{γ+r}}{\sqrt{v^{α} φ}} ∈ L^{p}, and the proof is complete. □

Proof of Theorem 2.2. We first show that (2.7) implies (2.6). Taking into account Lemma 3.1, it remains to estimate only the quantity ∥Av^{γ}∥_{L^{p} (A_{m} )}^{p} , where A is given by (2.2). Using the same argument of the previous proof, we have

∥Av^{γ}∥_{L^{p} (A_{m} )}^{p} ⩽ C∥fv^{γ}∥_{∞} ∑_{k=1}^{m} Δx_{k}v^{−γ−r+ \frac{α}{2} + \frac{1}{4}} (x_{k})|G(x_{k})|

and

G(x_{k}) = ∫_{A_{m}} v^{γ+r} (x)\frac{p_{m}(v^{α} , x)}{x − x_{k}}g(x)dx

= ∫_{A_{m}} \frac{p_{m}(v^{α} , x)Q(x) − p_{m}(v^{α} , x_{k})Q(x_{k})}{x − x_{k}} v^{γ+r} (x)\frac{g(x)}{Q(x)}dx,

where, as in the proof of Theorem 2.3, Q ∈ ℙ_{m} is equivalent to the weight v^{γ+r} in the interval A_{m}. Then G(t) is a polynomial of degree 2m − 1 and, using a Marcinkiewicz inequality, we get

∥Av^{γ}∥_{L^{p} (A_{m} )}^{p} ⩽ C∥fv^{γ}∥_{∞} ∥v^{−γ−r+ \frac{α}{2} + \frac{1}{4}} G∥_{L^{1} (A_{m} )}

and the L^{1} norm is bounded under our hypotheses. In fact, expressing G by means of the Hilbert transform, we have

∥v^{−γ−r+ \frac{α}{2} + \frac{1}{4}} G∥_{L^{1} (A_{m} )} ⩽ ∥v^{−γ−r+ \frac{α}{2} + \frac{1}{4}} H(p_{m}(v^{α})v^{γ+r} g)∥_{L^{1} (A_{m} )}

+ ∥v^{−γ−r+ \frac{α}{2} + \frac{1}{4}} p_{m}(v^{α})QH( \frac{v^{γ+r} g}{Q}) ∥_{L^{1} (A_{m} )} .

Concerning the second summand at the right-hand side, using the Hölder inequality, the boundedness of H and Q ∼ v^{γ+r} , we get

∥v^{−γ−r+ \frac{α}{2} + \frac{1}{4}} p_{m}(v^{α})QH( \frac{v^{γ+r} g}{Q}) ∥_{L^{1} (A_{m} )} ⩽ ∥H( \frac{v^{γ+r} g}{Q}) ∥_{1} ⩽ C∥g∥_{q}.

In order to estimate the first summand, we note that the function under the sign of the Hilbert transform is bounded and the one outside is L(log^{+} L) (see [8]). Therefore, with Γ = sgnH(p_{m}(v^{α})v^{γ+r} g) and ϱ = γ + r − α/2 − 1/4, we can write

∥v^{−ϱ} H(p_{m}(v^{α})v^{γ+r} g)∥_{L^{1} (A_{m} )} ⩽ ∥p_{m}(v^{α})v^{γ+r} gH(v^{−ϱ} Γ)∥_{L^{1} (A_{m} )}
⩽ C∥v^{ϱ}gH(v^{−ϱ} Γ)∥_{L^{1} (A_{m} )}
⩽ C∥g∥_{q}∥v^{ϱ}H(v^{−ϱ} Γ)∥_{L^{p} (A_{m} )}
⩽ C∥g∥_{q},

since the L^{p}-norm is bounded (see, for example, [7, p. 676]). Therefore, (2.7) implies (2.6).

In order to prove that (2.6) is a consequence of (2.7) it suffices use the same arguments of the proof of Theorem 2.3 (the part dealing with the necessary condition (2.9) ⇒ (2.11)) replacing p by ∞ and q by 1. So, the theorem is completely proved. □

Proof of Proposition 2.1. The proof is based on the following result due to Gopengauz [1]: “For every function f ∈ C^{s}, s ⩾ 0, there exists a polynomial q ∈ ℙ_{m+2s−1} such that, for i = 0, 1, …, s, one has q^{(i)} (±1) = f^{(i)} (±1) and

|(f^{(i)} (x) − q^{(i)} (x))| ⩽ C( \frac{\sqrt{1 − x^{2}}}{m} ) ^{s−i} ω(f^{(i)} , \frac{\sqrt{1 − x^{2}}}{m} ) _{∞} , |x| ⩽ 1,

where C ≠ C(m, f, x) and ω(·, ·)_{∞} is the ordinary modulus of smoothness” (in the uniform norm).

Then, if f ∈ C^{r−1} , we have

v^{γ}(x)|f(x) − L_{m,r} (v^{α} , f, x)| = |f(x) − q(x)|v^{γ}(x) + |v^{γ+r} (x)L_{m,r} ( v^{α} , \frac{f − q}{v^{r}} , x) |,

whence, using (2.6),

∥[f − L_{m,r} (v^{α} , f)]v^{γ} ∥_{p} ⩽ C∥(f − q)v^{γ} ∥_{∞} .

Therefore, for every polynomial P of degree m + 2r − 1, we get

∥[f − L_{m,r} (v^{α} , f)]v^{γ} ∥_{p} ⩽ C∥[(f − q) − P]v^{γ} ∥_{∞}

and, assuming the infimum on P, the estimate (2.12) follows.

Now we prove (2.13). Using the polynomial q of Gopengauz and (2.9), we get

(3.5) ∥[f − L_{m,r} (v^{α} , f)]v^{γ} ∥_{p} ⩽ C∥(f − q)v^{γ} ∥_{p}

+ C( ∑_{k=1}^{m} Δx_{k}[ω( f, \frac{φ(x_{k})}{m} )_{∞} v^{γ}(x_{k})]^{p}) ^{1/p}.

Now, we have

|f(x) − q(x)|v^{γ}(x) ⩽ Cv^{γ}(x) ω( f, \frac{φ(x)}{m} )_{∞}

⩽ Cv^{γ}(x) ∫_{x− \frac{φ(x)}{m}}^{x+ \frac{φ(x)}{m}} |f′(t)| dt ⩽ \frac{C}{m} \frac{m}{φ(x)} ∫_{x− \frac{φ(x)}{m}}^{x+ \frac{φ(x)}{m}} |f′φv^{γ}|(t) dt,

since 1 ± x ∼ 1 ± t if |x − t| ⩽ C \frac{φ(x)}{m} for x, t ∈ [x_{1}, x_{m}].

Then, using the maximal function of f′φv^{γ}, the first summand in (3.5) is dominated by

\frac{C}{m}( ∫_{−1}^{1} ( \frac{m}{φ(x)} ∫_{x− \frac{φ(x)}{m}}^{x+ \frac{φ(x)}{m}} |f′φv^{γ}|(t) dt) ^{p}dx) ^{1/p} ⩽ \frac{C}{m}∥f′φv^{γ}∥_{p}.

Concerning the sum in (3.5), for a sufficiently large s, we have

ω( f, \frac{φ(x_{k})}{m} )_{∞} v^{γ}(x_{k}) ⩽ s ω( f, \frac{φ(x_{k})}{sm} )_{∞} v^{γ}(x_{k})
⩽ C s ∫_{x_{k} − \frac{φ(x_{k} )}{sm}}^{x_{+} \frac{φ(x_{k} )}{sm}} |f′(t)|v^{γ}(t) dt ⩽ C ∫_{x_{k−1}}^{x_{k+1}} |f′v^{γ}|(t) dt.

Then, using the Hölder inequality in the latter integral, we get

Δx_{k}[ω( f, \frac{φ(x_{k})}{m} )_{∞} ]^{p} ⩽ C(Δx_{k})^{p} ∫_{x_{k−1}}^{x_{k+1}} |f′v^{γ}|^{p}(t) dt ⩽ \frac{C}{m^{p}} ∫_{x_{k−1}}^{x_{k+1}} |f′φv^{γ}|^{p}(t) dt,

for k = 1, …, m and x_{0} = −1. Adding up on k, we obtain that also the second term in (3.5) is dominated by \frac{C}{m} ∥f′φv^{γ}∥_{p}. Consequently, in a usual way, we deduce

∥[f − L_{m,r} (v^{α} , f)]v^{γ}∥_{p} ⩽ \frac{C}{m}E_{m+2r−2} (f′)_{v^{γ} φ,p} .

Iterating the latter relation, (2.13) follows. □

Proof of Corollary 2.1. The bound (2.14) is a consequance of (2.13) and the well-known estimate

∥(f − L_{m,r} (v^{α} , f))^{(r)} v^{γ}φ^{r}∥_{p} ⩽ Cm^{r}∥(f − L_{m,r} (v^{α} , f))v^{γ}∥_{p} + ∥f^{(r)} φ^{r}v^{γ}∥_{p}

which holds for any f ∈ W_{p}^{r}, 1 < p < ∞, and r ⩾ 1. We omit the details. □

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Department of Mathematics and Computer Sciences (Received 24 11 2011) University of Basilicata (Revised 06 03 2012) Via dell’Ateneo Lucano 10, 85100 Potenza Italy

giuseppe.mastroiannni@unibas.it

Mathematical Institute of Serbian Academy of Sciences and Arts Kneza Mihaila 36, 11000 Beograd Serbia

gvm@mi.sanu.ac.rs

Department of Mathematics and Computer Sciences University of Basilicata Via dell’Ateneo Lucano 10, 85100 Potenza Italy

incoronata.notarangelo@unibas.it