Lagrange–Hermite interpolation on the real semiaxis

Abstract

In order to approximate continuous functions on $[0,+\infty )$, we consider a Lagrange–Hermite polynomial, interpolating a finite section of the function at the zeros of some orthogonal polynomials and, with its first $(r-1)$ derivatives, at the point 0. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator. Moreover, we prove optimal estimates for the error of this process in the weighted $L^p$ and uniform metric.

1 Introduction

In this paper we discuss the weighted polynomial interpolation of continuous functions on $[0,+\infty )$, which are $(r-1)$-times differentiable at 0 and can increase with order $\mathcal {O}\left( \mathrm {e}^{x^\beta /2}\right) , \beta >1/2$, for $x\rightarrow \infty$.

The case $\beta =1$ has been treated in [1, 5, 15, 20, 22, 24–26], where the Authors considered Lagrange polynomials based on Laguerre zeros (see also [8]). Here we choose as a tool the orthonormal system $\{p_m(w)\}_m$ in $(0,+\infty )$ related to a generalized Laguerre weight of the form $w(x)=x^\alpha \mathrm {e}^{-x^\beta }$. From the numerical point of view, we observe that the weight w is nonclassical in general and, for the construction of the orthonormal system in the case $\beta \ne 1$, we can use the procedure introduced in [2] (see also [16]).

The presence of the derivatives of the function in 0 leads in a natural way to the construction of Lagrange–Hermite polynomial $\mathcal {L}_{m,r}(w,f)$ based at the zeros of the polynomial $p_m(w), 0$ as a multiple node and another additional node. Applying the operator $\mathcal {L}_{m,r}(w)$ to a suitable finite section of the function f, we obtain a new interpolation process, that we will denote by $\mathcal {L}^*_{m,r}(w)$. This new operator is not a projector into the set of all polynomials of degree at most $m+r$, $\mathbb {P}_{m+r}$, but on a special subset $\mathcal {P}_{m,r}^*\subset \mathbb {P}_{m+r}$ that can replace the space $\mathbb {P}_{m+r}$, in the sense we are going to show. Thus the projector $\mathcal {L}^*_{m,r}(w)$ can be profitably used in quadrature rules and in the numerical treatment of Boundary Value Problems on $(0,\infty )$ (see, e.g., [6, 27]).

We estimate the error of the process in weighted $L^p$ and uniform metric and compare it with the order of convergence of the best weighted polynomial approximation (that can be found in [21]). The error estimates are sharp for the considered classes of functions. All the results of this paper are new and cover the ones in the literature.

The paper is structured as follows. In Sect. 2 we recall some basic facts and give some preliminary results, in Sect. 3 we will state our main results and in Sect. 4 we will prove them.

2 Preliminary results

In the sequel $\mathcal {C}$ will stand for a positive constant that can assume different values in each formula and we shall write $\mathcal {C}\ne \mathcal {C}(a,b,\ldots )$ when $\mathcal {C}$ is independent of $a,b,\ldots$. Furthermore $A\sim B$ will mean that if A and B are positive quantities depending on some parameters, then there exists a positive constant $\mathcal {C}$ independent of these parameters such that $(A/B)^{\pm 1}\le \mathcal {C}$. Finally, we denote by $\mathbb {P}_m$ the set of all algebraic polynomials of degree at most m.

In order to introduce some interpolation processes we consider the weight

$$\begin{aligned} w(x)=x^\alpha \mathrm {e}^{-x^\beta }, \quad x\in (0,+\infty ), \end{aligned}$$

(2.1)

with $\alpha > -1, \, \beta > \frac{1}{2}$, and the corresponding sequence of orthonormal polynomials $\{p_m(w)\}_m$, with positive leading coefficient $\gamma _m$. The zeros $x_k=x_{m,k}(w)$ of $p_m(w)$, $m\ge 1$, are located as follows

$$\begin{aligned} \left( \frac{\sqrt{a_m}}{m}\right) ^2<x_1<x_2<\cdots <x_m< a_m\left( 1-\frac{c}{m^{2/3}}\right) , \end{aligned}$$

where

$$\begin{aligned} a_m=a_m(\sqrt{w})=\left[ \frac{2^{2\beta -2}{\varGamma }(\beta )^2}{{\varGamma }(2\beta )}\right] ^{1/\beta }\left( 4m+2\alpha +1\right) ^{1/\beta }\sim m^{1/\beta } \end{aligned}$$

is the Mhaskar–Rakhmanov–Saff number related to the weight $\sqrt{w}$ (see also [18, 19]). If $\beta =1$ the nodes $x_k$ are the Laguerre zeros. Nevertheless, in general w is a nonstandard weight and, for the computation of the zeros and the Christoffel numbers, we may use the Mathematica package “OrthogonalPolynomials” introduced in [2].

For any continuous function in $[0,+\infty )$, $f \in C^0([0,+\infty ))$, and $(r-1)$-times differentiable at 0, briefly $f \in C_r^0$, $r\ge 1$, we define the Lagrange–Hermite polynomial $\mathcal {L}_{m,r}(w,f)$ as follows

$$\begin{aligned} \mathcal {L}_{m,r}(w,f,x_k)=f(x_k),\quad k=1,\ldots ,m+1, \end{aligned}$$

with $x_{m+1}:=a_m$, and

$$\begin{aligned} \mathcal {L}_{m,r}(w,f)^{(j)}(0)=f^{(j)} (0), \quad j=0,1,\ldots ,r-1, \end{aligned}$$

where $f^{(0)}\equiv f$. Here we used also an idea introduced by Szabados for Lagrange and Hermite–Fejér interpolation, adding the additional node $a_m$ (see [28]).

The polynomial $\mathcal {L}_{m,r}(w,f)$ can be written as

$$\begin{aligned} \mathcal {L}_{m,r}(w,f,x)= & {} \sum _{k=1}^{m+1}\frac{x^r}{x_k^r}\,\ell _k(w,x)f(x_k) \nonumber \\&+\, (a_m-x) p_m(w,x)\sum _{i=0}^{r-1}\frac{x^i}{i!}\left( \frac{f}{(a_m-\cdot ) p_m(w)}\right) ^{(i)}(0),\quad \end{aligned}$$

(2.2)

where

$$\begin{aligned} \ell _k(w,x)=\frac{p_m(w,x)}{(x-x_k)p^{\prime }_m(w,x_k)}\frac{(a_m-x)}{(a_m-x_k)}, \quad k=1,2,\ldots , m, \end{aligned}$$

and

$$\begin{aligned} \ell _{m+1}(w,x)=\frac{p_m(w,x)}{p_m(w,a_m)} \end{aligned}$$

are the fundamental Lagrange polynomials. It is easily seen that

$$\begin{aligned} \mathcal {L}_{m,r}(w,P) = P, \quad P\in \mathbb {P}_{m+r}, \end{aligned}$$

and $\mathcal {L}_{m,r}(w)$ is a projector from $C^0_r$ into $\mathbb {P}_{m+r}$.

We are now going to introduce another Lagrange–Hermite type operator $\mathcal {L}_{m,r}^*(w)$, modifying the operator $\mathcal {L}_{m,r}(w)$. To this end, for a fixed $\theta \in (0,1)$ and for a sufficiently large m we define an index $j=j(m,\theta )$ such that

$$\begin{aligned} x_j=\min _k\{x_k:\ x_k\ge \theta a_m\}. \end{aligned}$$

(2.3)

Of course if m is small we let $j=m$. Hence we define the new Lagrange–Hermite type polynomial as follows

$$\begin{aligned} \mathcal {L}_{m,r}^*(w,f,x)= & {} \sum _{k=1}^{j}\frac{x^r}{x_k^r}\ell _k(w,x)f(x_k) \nonumber \\&+ (a_m-x) p_m(w,x)\sum _{i=0}^{r-1}\frac{x^i}{i!}\left( \frac{f}{(a_m-\cdot ) p_m(w)}\right) ^{(i)}(0). \end{aligned}$$

(2.4)

By this definition,

$$\begin{aligned} \mathcal {L}^*_{m,r}(w,f,x_k)= & {} f(x_k),\quad k=1,\ldots ,j, \\ \mathcal {L}^*_{m,r}(w,f,x_k)= & {} 0,\quad k=j+1,\ldots ,m+1, \end{aligned}$$

and

$$\begin{aligned} \mathcal {L}^*_{m,r}(w,f)^{(i)}(0)=f^{(i)}(0), \quad i=0,1,\ldots ,r-1. \end{aligned}$$

Moreover, we observe that $\mathcal {L}_{m,r}^*(w)$ does not preserve all the polynomials of degree at most $m+r$, for example $\mathcal {L}_{m,r}^*(w,1) \ne 1$.

Nevertheless, if we introduce the following set of polynomials

$$\begin{aligned} \mathcal {P}_{m,r}^* =\left\{ Q \in \mathbb {P}_{m+r} : Q(x_i)=0,\, i>j \right\} . \end{aligned}$$

then it is easy to show that for any $f \in C^0_r$, $\mathcal {L}^*_{m,r}(w,f) \in \mathcal {P}_{m,r}^*$ and for any $Q \in \mathcal {P}_{m,r}^*, \mathcal {L}^*_{m,r}(w,Q)=Q$, therefore $\mathcal {L}^*_{m,r}(w,f)$ is a projector from $C^0_r$ into $\mathcal {P}_{m,r}^*$.

In order to show some approximation properties of the projectors defined above, we are going to define some function spaces.

2.1 Function spaces

Let us consider a weight of the form

$$\begin{aligned} u(x)=x^\gamma \mathrm {e}^{-x^\beta /2},\quad x\in (0,+\infty ), \end{aligned}$$

(2.5)

with $\beta > \frac{1}{2}$ and the following weighted function spaces associated to u.

For $1 \le p < \infty$ and $\gamma >-1/p$, by $L_u^p$ we denote the set of all measurable functions f such that

$$\begin{aligned} \Vert f\Vert _{L_u^p}:=\Vert f u\Vert _p=\left( \int _{0}^{+\infty }|f u|^p(x) \,\mathrm {d}x\right) ^{1/p}<\infty . \end{aligned}$$

For $p=\infty$ and $\gamma \ge 0$, by a slight abuse of notation, we set

$$\begin{aligned} L^\infty _u:=C_u= \left\{ f\in C^0(0,+\infty ): \lim _{\begin{array}{c} x\rightarrow 0^+ \\ x\rightarrow +\infty \,\, \end{array}}f(x)u(x)=0\right\} , \end{aligned}$$

and we equip this space with the norm

$$\begin{aligned} \Vert f\Vert _{L^\infty _u}:=\Vert f u\Vert _\infty = \sup _{x\in (0,+\infty )}\left| f(x)u(x)\right| . \end{aligned}$$

Note that the Weierstrass theorem implies the limit conditions in the definition of $C_u$.

Subspaces of $L_u^p, 1\le p\le \infty$, are the Sobolev spaces, given by

$$\begin{aligned} W_s^p(u)=\left\{ f\in L^p_u:\ f^{(s-1)}\in AC(0,+\infty ), \ \Vert f^{(s)}\varphi ^s u\Vert _p<\infty \right\} , \quad 1\le s\in \mathbb {Z}, \end{aligned}$$

where $AC(0,+\infty )$ denotes the set of all functions which are absolutely continuous on every closed subset of $(0,+\infty )$ and $\varphi (x)=\sqrt{x}$. We equip these spaces with the norm

$$\begin{aligned} \Vert f\Vert _{W_s^p(u)}=\Vert f u\Vert _p+\Vert f^{(s)}\varphi ^s u\Vert _p. \end{aligned}$$

In order to define some further function spaces, we consider the s-th modulus of smoothness of $f \in L^p_u, 1\le p \le \infty , s\ge 1$,

$$\begin{aligned} {\varOmega }_{\varphi }^s(f,t)_{u,p} = \sup _{0<h\le t}\left\| {\varDelta }_{h\varphi }^s \left( f\right) u\right\| _{L^p(I_{rh})}, \end{aligned}$$

where

$$\begin{aligned} {\varDelta }_{h\varphi }^s f(x) = \sum _{i=0}^s (-1)^i \left( {\begin{array}{c}s\\ i\end{array}}\right) f\left( x+(s-i) h\varphi (x)\right) \end{aligned}$$

is the forward finite difference of order s with variable step $h\varphi (x)$, $\varphi (x)=\sqrt{x}$ and $I_{s h}=[8s^2h^2,\mathcal {C}h^{-\frac{1}{(\beta -1/2)}}]$, $h>0$.

Let $f\in L_u^p$, $1\le p\le \infty$. Then the following estimate

$$\begin{aligned} {\varOmega }_{\varphi }^s(f,t)_{u,p} \le \mathcal {C}\sup _{0<h\le t} h^s \Vert f^{(s)}\varphi ^s u\Vert _{L^p(I_{s h})}, \end{aligned}$$

(2.6)

holds with $\mathcal {C}\ne \mathcal {C}(f,t)$, provided the norm on the right-hand side is finite (see [21]).

Using these moduli of smoothness we can define the Zygmund spaces as

$$\begin{aligned} Z^p_\lambda (u)=\left\{ f\in L^p_u : \sup _{t>0} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t^\lambda } < \infty ,\, s>\lambda \right\} , \quad 0< \lambda \in \mathbb {R}, \end{aligned}$$

with the norm

$$\begin{aligned} \Vert f\Vert _{Z^p_\lambda (u)} = \Vert fu\Vert _p + \sup _{t>0} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t^\lambda }, \quad s>\lambda . \end{aligned}$$

Let us denote by

$$\begin{aligned} E_m(f)_{u,p}=\inf _{P_{m}\in \mathbb {P}_m} \left\| \left( f-P_{m}\right) u\right\| _p \end{aligned}$$

the error of best weighted polynomial approximation in $L^p_u$, $1\le p\le \infty$. Estimates for the error of best weighted approximation have been proved in [21]. In particular it has been shown that

$$\begin{aligned} \lim _{m\rightarrow \infty }E_m(f)_{u,p} = 0 \quad \forall f\in L_u^p. \end{aligned}$$

(2.7)

For our aims it is sufficient to recall the following weak Jackson inequality

$$\begin{aligned} E_m(f)_{u,p} \le \mathcal {C}\int _0^{\frac{\sqrt{a_m}}{m}} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t} \, \mathrm {d}t \end{aligned}$$

(2.8)

that holds for any function $f \in L^p_u, 1\le p\le \infty$.

In particular, if $f\in Z^p_\lambda$, by (2.8) we deduce

$$\begin{aligned} E_m(f)_{u,p}\le \mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^\lambda \sup _{t>0} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t^\lambda }, \end{aligned}$$

(2.9)

and by (2.6) we have

$$\begin{aligned} E_m(f)_{u,p}\le \mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^s \Vert f^{(s)}\varphi ^s u\Vert _p, \end{aligned}$$

(2.10)

for any $f\in W^p_s, s\ge 1$ and $1\le p \le \infty$.

We are now able to show an important property of the polynomials in $\mathcal {P}_{m,r}^*$. To this end we need some further notation. We say that $P_M$ is a polynomial of quasi best approximation for $f\in L^p_u$, $1\le p \le \infty$, if, for some $\mathcal {C}\ge 1$,

$$\begin{aligned} \left\| \left( f-P_M\right) u\right\| _p \le \mathcal {C}E_M(f)_{u,p}. \end{aligned}$$

Moreover, we denote by

$$\begin{aligned} E_{m,r}^*(f)_{u,p}=\inf _{Q \in \mathcal {P}_{m,r}^*} \Vert (f-Q)u\Vert _p, \end{aligned}$$

the error of best approximation by means of polynomials of $\mathcal {P}_{m,r}^*$. The relation between $E_m(f)_{u,p}$ and $E_{m}^*(f)_{u,p}$ is established by the following Lemma.

Lemma 1

Let w, u be the weights defined in (2.1) and (2.5) with $\alpha > -1, \beta > \frac{1}{2}$ and $u\in L^p$, $1\le p \le \infty$. Let $f\in L^p_u$ and $P_M\in \mathbb {P}_M$ one of its polynomial of quasi best approximation, with

$$\begin{aligned} M=\left\lfloor \left( \frac{\theta }{\theta +1}\right) m\right\rfloor \end{aligned}$$

(2.11)

for a fixed $\theta \in (0,1)$. Then we have

$$\begin{aligned} E_{m,r}^*(f)_{u,p} \le \Vert [f-\mathcal {L}_{m,r}^*(w,P_M)]u\Vert _p \le \mathcal {C}\{E_M(f)_{u,p} + \mathrm {e}^{-cm} \Vert fu\Vert _p\}, \end{aligned}$$

(2.12)

where $\mathcal {C}, c$ are independent of m, r, f. Moreover, for $s\ge 1$, we get

$$\begin{aligned} \left( \frac{\sqrt{a_m}}{m}\right) ^s\Vert \mathcal {L}_{m,r}^*(w,P_M)^{(s)}\varphi ^s u\Vert _{p} \le \mathcal {C}\left\{ \mathrm {e}^{-cm} \Vert fu\Vert _p + \int _0^{\frac{\sqrt{a_m}}{m}} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t} \, \mathrm {d}t\right\} . \end{aligned}$$

(2.13)

Using Lemma 1 and (2.7) it follows that the order of convergence of $E_{m,r}^*(f)_{u,p}$ is the same as that of $E_{m,r}(f)_{u,p}$. Therefore, $\bigcup _m \mathcal {P}_{m,r}^*$ is dense in $L^p_u$. In the next Section we will show the behaviour of the operators $\mathcal {L}_{m,r}(w)$ and $\mathcal {L}_{m,r}^*(w)$ in different function spaces.

3 Main results

The following results hold true in weighted uniform norm. In order to state it we need a further definition. We say that a function is quasi increasing on I if there exists a constant $\mathcal {C}$ such that $f(x)\le \mathcal {C}f(y)$ for $x<y$, $x,y\in I$.

Theorem 2

Let w and u be the previously defined weights, with $\gamma \ge 0$ and $\alpha >-1$. For every $f\in C_r^0 \cap C_u$, $r\ge 1$, we have

$$\begin{aligned} \left\| \mathcal {L}_{m,r}\left( w,f\right) u\right\| _\infty \le \mathcal {C}\left\{ (\log m)\Vert f u\Vert _\infty +\left( \frac{\sqrt{a_m}}{m}\right) ^{2\gamma } \sum _{i=0}^{r-1}\frac{|f^{(i)}(0)|}{i!}\left( \frac{\sqrt{a_m}}{m}\right) ^{2i}\right\} , \end{aligned}$$

(3.1)

where $\mathcal {C}\ne \mathcal {C}(m,f)$, if and only if

$$\begin{aligned} \frac{\alpha }{2}+\frac{1}{4}\le \gamma +r\le \frac{\alpha }{2}+\frac{5}{4}. \end{aligned}$$

(3.2)

Moreover, under the assumptions (3.2), if $\left| f^{(i)}\right|$ is quasi increasing on $[0,\alpha _m]$, with $\alpha _m=a_m/m^2$ and $i=0,1,\ldots , r-1$, we get

$$\begin{aligned} \left\| \left[ f-\mathcal {L}_{m,r}\left( w,f\right) \right] u\right\| _\infty \le \mathcal {C}(\log m) E_{m+r}(f)_{u,\infty } + \mathcal {O}\left( \left( \frac{\sqrt{a_m}}{m}\right) ^{r-1}\right) \end{aligned}$$

(3.3)

where $\mathcal {C}\ne \mathcal {C}(m,f)$ and the constant in “$\mathcal {O}$” is independent of m.

Therefore, in weighted spaces of continuous functions, the behaviour of the operators $\left\{ \mathcal {L}_{m,r}\left( w\right) \right\} _m$ is comparable with similar interpolation processes based on Jacobi zeros on bounded intervals (see, e.g., [15, 17]).

We emphasize that if r and $\gamma$ are given, then, in order to approximate a function f, we can always choose some $\alpha$ such that the Lagrange–Hermite polynomial converges with the order of the best polynomial approximation times the extra factor $\log m$. In fact, we can rewrite (3.2) as follows

$$\begin{aligned} 2r+2\gamma -\frac{5}{2}\le \alpha \le 2r+2\gamma -\frac{1}{2}. \end{aligned}$$

Moreover, since the Lebesgue constants related to the processes $\{\mathcal {L}_{m,r}^*(w)\}_m$ are bounded by the ones related to $\{\mathcal {L}_{m,r}(w)\}_m$, by Theorem 2 and Lemma 1, we deduce the following error estimate for the “truncated” process.

Corollary 3

Let $\theta \in (0,1)$. Under the assumptions of Theorem 2, for every $f\in C_r^0, r\ge 1$, for m sufficiently large (say $m>m_0)$ we have

$$\begin{aligned} \left\| \left[ f-\mathcal {L}_{m,r}^*\left( w,f\right) \right] u\right\| _\infty \le \mathcal {C}(\log m) E_{M}(f)_{u,\infty }+ \mathcal {O}\left( \left( \frac{\sqrt{a_m}}{m}\right) ^{r-1}\right) , \end{aligned}$$

where M given by (2.11), $\mathcal {C}\ne \mathcal {C}(m,f)$ and the constant in “$\mathcal {O}$” is independent of m.

In weighted $L^p$-spaces the behaviour of $\left\{ \mathcal {L}_{m,r}\left( w\right) \right\} _m$ is not optimal, while for the new process $\left\{ \mathcal {L}_{m,r}^*\left( w\right) \right\} _m$ we can state the following

Theorem 4

Let $u\in L^p, 1< p<\infty$ and $\theta \in (0,1)$ be fixed. Then, for every function $f\in C_r^0$, $r\ge 1$, we have

$$\begin{aligned} \left\| \mathcal {L}_{m,r}^*\left( w,f\right) u\right\| _p\sim & {} \left( \sum _{k=1}^j{\varDelta } x_k |f u|^p(x_k)\right) ^{1/p} \nonumber \\&+\left( \frac{\sqrt{a_m}}{m}\right) ^{2(\gamma + 1/p)} \sum _{i=0}^{r-1}\frac{|f^{(i)}(0)|}{i!}\left( \frac{\sqrt{a_m}}{m} \right) ^{2i} \end{aligned}$$

(3.4)

if and only if

$$\begin{aligned} \frac{\alpha }{2}+\frac{1}{4}-\frac{1}{p}< \gamma +r < \frac{\alpha }{2}+\frac{5}{4}-\frac{1}{p}, \end{aligned}$$

(3.5)

where the constants in “$\sim$” depend on $\theta$ and are independent of m and f.

If the function $f\in L^p_u$ fulfills the additional assumption

$$\begin{aligned} \dfrac{{\varOmega }^s_{\varphi }(f,t)_{u,\,p}}{t^{1+1/p}} \in \hbox {L}^1 [0, 1] \end{aligned}$$

(3.6)

for some $s\ge 1$, then f is continuous in $(0,+\infty )$ (see [21]). Moreover, using the same arguments as in [22], we obtain

$$\begin{aligned} \left( \sum _{k=1}^j{\varDelta } x_k|f u|^p(x_k)\right) ^{1/p} \le \mathcal {C}\left[ \Vert fu\Vert _p + \left( \frac{\sqrt{a_m}}{m} \right) ^{1/p} \int _0^{\frac{\sqrt{a_m}}{m}} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t^{1+1/p}}\,\mathrm {d}t\right] . \end{aligned}$$

Hence, if $f\in L^p_u$ fulfills (3.6) and is $(r-1)$-times differentiable at 0, the bound (3.4) becomes

$$\begin{aligned} \left\| \mathcal {L}_{m,r}^*\left( w,f\right) u\right\| _p\le & {} \mathcal {C}\left[ \Vert fu\Vert _p + \left( \frac{\sqrt{a_m}}{m} \right) ^{1/p} \int _0^{\frac{\sqrt{a_m}}{m}} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t^{1+1/p}}\,\mathrm {d}t \right. \nonumber \\&+\left. \left( \frac{\sqrt{a_m}}{m}\right) ^{2(\gamma + 1/p)} \sum _{i=0}^{r-1}\frac{|f^{(i)}(0)|}{i!}\left( \frac{\sqrt{a_m}}{m} \right) ^{2i}\right] , \end{aligned}$$

(3.7)

where $\mathcal {C}$ depends on $\theta \in (0,1)$ (fixed) and is independent of m and f. Note that if $f \in \mathcal {P}_{m,r}^*$, (3.7) becomes an equivalence.

Theorem 5

Under the assumptions (3.5), let $f\in L^p_u\, \cap \, C^0_r$ satisfy (3.6) for some $s\ge r$ and $\left| f^{(i)}\right|$ be quasi increasing on $[0,\alpha _m]$, with $\alpha _m=a_m/m^2$ and $i=0,1,\ldots , r-1$. Then, for $m>m_0$, we have

$$\begin{aligned} \Vert [f-\mathcal {L}^*_{m,r}(w,f)]u\Vert _p \le \mathcal {C}\left( \frac{\sqrt{a_m}}{m} \right) ^{1/p} \int _0^{\frac{\sqrt{a_m}}{m}} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t^{1+1/p}}\,\mathrm {d}t + \mathcal {O}\left( \left( \frac{\sqrt{a_m}}{m}\right) ^{r-1}\right) \end{aligned}$$

(3.8)

where $\mathcal {C}$ depends on $\theta$ and is independent of m and f, the constant in “$\mathcal {O}$” is independent of m.

In particular, under the assumptions of Theorem 5, if $f\in Z^p_\lambda (u), 1/p < \lambda \le r-1$, the error estimate (3.8) becomes

$$\begin{aligned} \left\| \left[ f-\mathcal {L}^*_{m,r}\left( w,f\right) \right] u\right\| _p = \mathcal {O}\left( \left( \dfrac{\sqrt{a_m}}{m}\right) ^\lambda \right) , \end{aligned}$$

where the constant in $\mathcal {O}$ is independent of m and f, that is the order of the best approximation in $Z^p_\lambda (u)$ [see (2.9)].

Analogously, if $f\in W^p_s(u), s\le r-1$, from Theorem 5 we get

$$\begin{aligned} \Vert [f-\mathcal {L}^*_{m,r}(w,f)]u\Vert _p = \mathcal {O}\left( \left( \dfrac{\sqrt{a_m}}{m}\right) ^s\right) , \end{aligned}$$

where the constant in $\mathcal {O}$ is independent of m and f [see (2.10)]. While, for $f\in W^p_r(u)$, the following corollary holds.

Corollary 6

Under the assumptions (3.5), let $f\in W^p_r(u)\cap C^0_r, 1<p<\infty$, such that $\left| f^{(i)}\right|$ is quasi increasing on $[0,\alpha _m], \alpha _m=a_m/m^2$, for $i=0,1,\ldots , r-1$ and m sufficiently large (say $m>m_0$). Then we have

$$\begin{aligned} \left\| \mathcal {L}^*_{m,r}\left( w,f\right) u\right\| _p \le \mathcal {C}\left\{ \Vert f u\Vert _p+\left( \frac{\sqrt{a_m}}{m}\right) ^r\Vert f^{(r)}\varphi ^r u\Vert _p\right\} \end{aligned}$$

(3.9)

and

$$\begin{aligned} \left\| \left[ f-\mathcal {L}^*_{m,r}\left( w,f\right) \right] u\right\| _p \le \mathcal {C}\left\{ \left( \frac{\sqrt{a_m}}{m}\right) ^r\Vert f^{(r)}\varphi ^r u\Vert _p +\mathrm {e}^{-cm}\Vert f u\Vert _p\right\} \end{aligned}$$

(3.10)

where $\mathcal {C},c$ depend on $\theta$ and are independent of m and f.

Finally, the following remark is of some interest.

Remark 7

The behaviours of the two interpolation processes $\{\mathcal {L}_{m,r}(w)\}_m$ and $\{\mathcal {L}_{m,r}^*(w)\}_m$ are essentially equivalent in weighted uniform norm. Nevertheless, the “truncated” process $\{\mathcal {L}_{m,r}^*(w)\}_m$ has the advantage of dropping $c\,m$ terms, $c<1$. This turns out to be useful in the numerical treatment of functional equations, since the dimension of the matrices obtained from the discretization of operators are strongly reduced with an evident computational saving.

On the other hand, the two operators behave in a completely different way in weighted $L^p$-norm. In fact, the equivalence (3.4) of Theorem 4 is not true for the “nontruncated” operator $\{\mathcal {L}_{m,r}(w)\}_m$.

Moreover, all the constants in Corollary 3, Theorems 4 and 5 are independent of m and f, but depend on $\theta$. For instance, the constant in the upper bound of (3.4) is $\mathcal {O}\left( (1-\theta )^{-3/4}\right)$ for $\theta \rightarrow 1$ and so $\theta$ cannot assume value 1.

We refer to Sect. 4 for more details.

4 Proofs

First of all we recall some weighted polynomial inequalities with the weight u defined in (2.5) which will be used in the sequel (see [21]).

Let $1\le p \le \infty$ and $a_m=a_m(u)\sim m^{1/\beta }$. For any $P_m\in \mathbb {P}_m$, the restricted range inequalities

$$\begin{aligned} \Vert P_m u \Vert _p \le \mathcal {C}\Vert P_m u\Vert _{L^p(I_m)},\quad I_m=\left[ c\,\frac{a_m}{m^2}, a_m\right] , \end{aligned}$$

(4.1)

and

$$\begin{aligned} \Vert P_m u \Vert _{L^p[a_m(1+\delta ),+\infty )} \le \mathcal {C}\mathrm {e}^{-c m} \Vert P_m u\Vert _p, \quad \delta >0, \end{aligned}$$

(4.2)

hold with $\mathcal {C}\ne \mathcal {C}(m,P_m)$ and $c \ne c(m,P_m)$. Moreover, we recall the Bernstein and Markov inequalities

$$\begin{aligned} \Vert P_m^{\prime }\varphi u\Vert _p \le \mathcal {C}\frac{m}{\sqrt{a_m}} \,\Vert P_m u\Vert _p, \quad \varphi (x)=\sqrt{x}, \end{aligned}$$

(4.3)

and

$$\begin{aligned} \Vert P_m^{\prime } u\Vert _p \le \mathcal {C}\frac{m^2}{a_m} \Vert P_m u\Vert _p, \end{aligned}$$

(4.4)

where $\mathcal {C}\ne \mathcal {C}(m,P_m)$ in both cases. Finally, for $1\le p<\infty$, we will need the following Nikolskii inequality

$$\begin{aligned} \Vert P_m u\Vert _\infty \le \mathcal {C}\left( \frac{m^2}{a_m}\right) ^{\frac{1}{p}} \Vert P_m u\Vert _p, \quad \mathcal {C}\ne \mathcal {C}(m,P_m). \end{aligned}$$

(4.5)

We also need some estimates for the polynomials of the orthonormal system $\{p_m(w)\}_{m\in \mathbb {N}}$, where w is the weight defined in (2.1). The generalized Laguerre weight w is a nonstandard weight and the estimates for the related orthogonal polynomials can be deduced by [7, 11, 12, 21, 22].

The estimate (which can be deduced from [9], see also [19, 21])

$$\begin{aligned} |p_m(w,x)|\left( x+\frac{a_m}{m^2}\right) ^{\frac{\alpha }{2}+\frac{1}{4}}\mathrm {e}^{-\frac{x^\beta }{2}} \root 4 \of {|a_m-x|+\frac{a_m}{m^{2/3}}} \sim \frac{|x-x_d|}{{\varDelta } x_{d\pm 1}} \end{aligned}$$

(4.6)

holds with $x \in [0, +\infty )$ and $x_d$ a zero closest to x. From (4.6) it follows that

$$\begin{aligned} |p_m(w,x)|\sqrt{w(x)\sqrt{x(a_m-x)}}\le \mathcal {C},\quad x\in I_m. \end{aligned}$$

(4.7)

Moreover, the following proposition will be useful.

Proposition 8

Let w be the weight defined by (2.1) and $\{p_m(w)\}_m$ its related orthonormal system. Then we have

$$\begin{aligned}&|p_m(w,0)| \sim \left( \frac{m}{\sqrt{a_m}}\right) ^{\alpha +\frac{1}{2}} \frac{1}{\root 4 \of {a_m}}, \end{aligned}$$

(4.8)

$$\begin{aligned}&\quad |p_m^{(k)}(w,0)| \le \mathcal {C}\left( \frac{m}{\sqrt{a_m}}\right) ^{2k}|p_m(w,0)|, \end{aligned}$$

(4.9)

and

$$\begin{aligned} \left| \left( \frac{1}{p_m(w)}\right) ^{(k)}(0)\right| \le \mathcal {C}\left( \frac{m}{\sqrt{a_m}}\right) ^{2k}\frac{1}{|p_m(w,0)|}, \end{aligned}$$

(4.10)

where $\mathcal {C}$ and the constant in $``\sim ''$ are independent of m.

Proof

Equivalence (4.8) easily follows from (4.6) for $x=0$.

In order to prove (4.9) we set

$$\begin{aligned} g(x)=\left( x+\frac{a_m}{m^2}\right) ^{\frac{\alpha }{2}+\frac{1}{4}}\mathrm {e}^{-\frac{x^\beta }{2}} \root 4 \of {|a_m-x|+\frac{a_m}{m^{2/3}}}. \end{aligned}$$

Then, using (4.4) with u replaced by g, by (4.6), we obtain

$$\begin{aligned} |p_m^{(k)}(w,x)| \le \mathcal {C}\left( \frac{m}{\sqrt{a_m}}\right) ^{2k}\frac{\Vert p_m(w)g\Vert _\infty }{\left| g(x)\right| } \le \mathcal {C}\left( \frac{m}{\sqrt{a_m}}\right) ^{2k}\frac{1}{|g(x)|}. \end{aligned}$$

Hence, for $x=0$, by (4.8), we get

$$\begin{aligned} |p_m^{(k)}(w,0)| \le \mathcal {C}\left( \frac{m}{\sqrt{a_m}}\right) ^{2k} \frac{1}{\left( \frac{\sqrt{a_m}}{m}\right) ^{\alpha +\frac{1}{2}} \root 4 \of {a_m}} \le \mathcal {C}\left( \frac{m}{\sqrt{a_m}}\right) ^{2k}|p_m(w,0)|. \end{aligned}$$

Finally, let us show (4.10) by induction. For $k=1$, by (4.9) we have

$$\begin{aligned} \left| \left( \frac{1}{p_m(w)}\right) ^{\prime }(0)\right| = \frac{\left| p_m^{\prime }(w,0)\right| }{\left| p_m(w,0)\right| ^2} \le \mathcal {C}\left( \frac{m}{\sqrt{a_m}}\right) ^{2}\frac{1}{|p_m(w,0)|}. \end{aligned}$$

For $k>1$, from the identity

$$\begin{aligned} \left( \frac{1}{p_m(w)}p_m(w)\right) ^{(k)}(0)=\sum _{i=0}^{k} \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( \frac{1}{p_m(w)}\right) ^{(i)}(0)p_m^{(k-i)}(w,0)=0 \end{aligned}$$

it follows that

$$\begin{aligned} -\left( \frac{1}{p_m(w)}\right) ^{(k)}(0)p_m(w,0)=\sum _{i=0}^{k-1} \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( \frac{1}{p_m(w)}\right) ^{(i)}(0)p_m^{(k-i)}(w,0) \end{aligned}$$

Then, using the induction hypothesis and (4.9), we obtain

$$\begin{aligned} \left| p_m(w,0)\right| \left| \left( \frac{1}{p_m(w)}\right) ^{(k)}(0)\right|= & {} \left| \sum _{i=0}^{k-1} \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( \frac{1}{p_m(w)}\right) ^{(i)}(0)p_m^{(k-i)}(w,0)\right| \\\le & {} \mathcal {C}\left( \frac{m}{\sqrt{a_m}}\right) ^{2 k}, \end{aligned}$$

which completes the proof. $\square$

Furthermore, the following relation will be useful (see [10, 21])

$$\begin{aligned} \frac{1}{|p^{\prime }_m(w,x_k)|\sqrt{w(x_k)}} \sim {\varDelta } x_k \root 4 \of {x_k} \root 4 \of {|a_m-x_k|+\frac{a_m}{m^{2/3}}}. \end{aligned}$$

(4.11)

By (4.6) and (4.11) if $x\in \left[ \left( \frac{\sqrt{a_m}}{m}\right) ^2, a_m\right]$ we have

$$\begin{aligned} u(x)\frac{|\ell _k(w,x)|}{u(x_k)} \le \mathcal {C}\left( \frac{x}{x_k}\right) ^{\gamma -\frac{\alpha }{2} - \frac{1}{4}}\left( \frac{a_m-x}{a_m-x_k}\right) ^{3/4} \frac{{\varDelta } x_k}{|x-x_k|}, \quad (x\ne x_k). \end{aligned}$$

(4.12)

Moreover, if $x_d$ is a node closest to x, we get (see [4])

$$\begin{aligned} u(x)\frac{|\ell _d(w,x)|}{u(x_d)} \sim 1. \end{aligned}$$

(4.13)

Proof of Lemma 1

We are going to prove (2.12) only for $p<\infty$, since the case $p=\infty$ is simpler.

Let $P_M \in \mathbb {P}_M$, with $M=\left\lfloor \frac{\theta m}{\theta +1}\right\rfloor$, be a polynomial of quasi best approximation for $f\in L^p_u$. So we can write

$$\begin{aligned} E_{m,r}^*(f)_{u,p} \le \left\| \left[ f-\mathcal {L}_{m,r}^*(w,P_M)\right] u\right\| _p. \end{aligned}$$

Since

$$\begin{aligned} \mathcal {L}_{m,r}^*(w,P_M,x)= PM(x) - \sum _{k=j+1}^{m}\dfrac{\ell _k(w,x)x^r}{x_k^r}P_M (x_k) \end{aligned}$$

(4.14)

then, letting $v^r(x)=x^r$, we get

$$\begin{aligned} E_{m,r}^*(f)_{u,p} \le \mathcal {C} E_M(f)_{u,p} + \left\| \sum _{k=j+1}^{m}\dfrac{\ell _k(w)v^r u}{v^r(x_k) u(x_k)}(P_M u)(x_k)\right\| _p. \end{aligned}$$

By (4.1) the second summand on the right-hand side is dominated by

$$\begin{aligned} \mathcal {C}a_m^{1/p}\,\Vert P_M u\Vert _{L^\infty [\theta a_m, +\infty )}\sup _{x\in I_m}\sum _{k=j+1}^{m}\dfrac{|\ell _k(w,x)|v^r (x) u(x)}{v^r(x_k) u(x_k)}, \end{aligned}$$

and, using (4.12) and (4.13), we have

$$\begin{aligned} \sup _{x\in I_m}\sum _{k=j+1}^{m}\dfrac{|\ell _k(w,x)|v^r(x) u(x)}{v^r(x_k) u(x_k)} \le \mathcal {C}m^\tau , \end{aligned}$$

(4.15)

for some $\tau >0$. Moreover, by (4.2) and (4.5), we obtain

$$\begin{aligned} \left\| \sum _{k=j+1}^{m}\dfrac{\ell _k(w)v^r u}{v^r(x_k) u(x_k)}(P_M u)(x_k)\right\| _p\le & {} \mathcal {C}m^\tau a_m^{1/p}\Vert P_M u\Vert _{L^\infty [\theta a_m, +\infty )} \nonumber \\\le & {} \mathcal {C}m^\tau a_m^{1/p} \mathrm {e}^{-c m} \Vert P_M u\Vert _\infty \nonumber \\\le & {} \mathcal {C}m^{\tau + \frac{2}{p}} \mathrm {e}^{-cm}\Vert P_M u\Vert _p \nonumber \\\le & {} \mathcal {C}\mathrm {e}^{-cm}\Vert f u\Vert _p. \end{aligned}$$

(4.16)

Let us now prove (2.13). By (4.14) we have

$$\begin{aligned}&\left( \dfrac{\sqrt{a_m}}{m}\right) ^s\Vert \mathcal {L}_{m,r}^*(w,P_M)^{(s)}\varphi ^s u\Vert _{p}\\&\quad \le \left( \dfrac{\sqrt{a_m}}{m}\right) ^s \Vert P_M^{(s)}\varphi ^s u\Vert _p + \left( \dfrac{\sqrt{a_m}}{m}\right) ^s \left\| \left( \sum _{k=j+1}^{m}\dfrac{\ell _k(w)v^r}{v^r(x_k)}P_M(x_k)\right) ^{(s)}\varphi ^s u\right\| _{p} \end{aligned}$$

For the first term on the right-hand side we use the following estimate (see [21])

$$\begin{aligned} \left( \dfrac{\sqrt{a_m}}{m}\right) ^s \Vert P_M^{(s)}\varphi ^s u\Vert _p \le \mathcal {C}\int _{0}^{\sqrt{a_m}/m}\frac{{\varOmega }^s_{\varphi }\left( f,t\right) _{u,p}}{t}\,\mathrm {d}t. \end{aligned}$$

While, for the second term on the right-hand side, using s times the Bernstein inequality (4.3) and (4.16), we obtain

$$\begin{aligned} \left( \dfrac{\sqrt{a_m}}{m}\right) ^s \left\| \left( \sum _{k=j+1}^{m}\dfrac{\ell _k(w)v^r}{v^r(x_k)}P_M(x_k)\right) ^{(s)}\varphi ^s u\right\| _{p}\le & {} \mathcal {C}\left\| \sum _{k=j+1}^{m}\dfrac{\ell _k(w)v^r}{v^r(x_k)}P_M(x_k)u\right\| _{p}\\\le & {} \mathcal {C}\mathrm {e}^{-cm}\Vert f u\Vert _p. \end{aligned}$$

Hence we get (2.13). $\square$

We recall the following estimate, concerning the distance ${\varDelta } x_k$ between two consecutive zeros (see [10, 21]):

$$\begin{aligned} {\varDelta } x_k = x_{k+1}-x_k \sim \dfrac{a_m}{m}\sqrt{x_k}\dfrac{1}{\sqrt{a_m-x_k+a_m m^{-2/3}}}, \end{aligned}$$

(4.17)

where the constants in “$\sim$” are independent of m and k. We remark that from (4.17), with j given by (2.3), it follows that

$$\begin{aligned} {\varDelta } x_k \sim \frac{\sqrt{a_m}}{m}\sqrt{x_k} \qquad k=1,2,\ldots j. \end{aligned}$$

Now, letting

$$\begin{aligned} \sigma _m(f)= \sum _{i=0}^{r-1} \dfrac{f^{(i)}(0)}{i!}\left( \frac{\sqrt{a_m}}{m}\right) ^{2i} \end{aligned}$$

(4.18)

and

$$\begin{aligned} A(x)= (a_m-x) p_m(w,x)\sum _{i=0}^{r-1}\frac{x^i}{i!}\left( \frac{f}{(a_m-\cdot ) p_m(w)}\right) ^{(i)}(0), \end{aligned}$$

(4.19)

we can state the following proposition.

Proposition 9

With the notation (4.18) and (4.19), we have

$$\begin{aligned} \Vert A u\Vert _\infty \le \mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^{2\gamma }\,\sigma _m(f), \quad \mathcal {C}\ne \mathcal {C}(m,f), \end{aligned}$$

(4.20)

if and only if

$$\begin{aligned} \frac{\alpha }{2}+\frac{1}{4}\le \gamma +r\le \frac{\alpha }{2}+\frac{5}{4}. \end{aligned}$$

(4.21)

Moreover, for $p\ge 1$, we get

$$\begin{aligned} \Vert A u\Vert _p \le \mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^{2(\gamma + 1/p)}\,\sigma _m(f), \quad \mathcal {C}\ne \mathcal {C}(m,f), \end{aligned}$$

(4.22)

if and only if

$$\begin{aligned} \frac{\alpha }{2}+\frac{1}{4} -\frac{1}{p} \le \gamma +r\le \frac{\alpha }{2}+\frac{5}{4} -\frac{1}{p}. \end{aligned}$$

(4.23)

Proof

We first prove (4.20). Recalling (4.19), we may write

$$\begin{aligned} |A(x)u(x)| \le \sum _{i=0}^{r-1} b_i \dfrac{|D_i|}{i!}, \end{aligned}$$

(4.24)

where

$$\begin{aligned} b_i=\Vert (a_m-\cdot )p_m(w,\cdot )u v^i\Vert _\infty , \quad v^i(x)=x^i \end{aligned}$$

and

$$\begin{aligned} |D_i|= & {} \left| \left( \frac{f}{(a_m-\cdot ) p_m(w)}\right) ^{(i)}(0)\right| \\= & {} \left| \sum _{\eta =0}^i \left( {\begin{array}{c}i\\ \eta \end{array}}\right) f^{(\eta )}(0) \left( \dfrac{1}{(a_m-\cdot )p_m(w,\cdot )}\right) ^{(i-\eta )}(0)\right| . \end{aligned}$$

By (4.10) and (4.11) we get

$$\begin{aligned} \left| \left( \dfrac{1}{(a_m-\cdot )p_m(w,\cdot )}\right) ^{(i-\eta )}(0)\right|= & {} \left| \sum _{j=0}^{i-\eta } \left( {\begin{array}{c}i\!-\!\eta \\ j\end{array}}\right) \dfrac{-1^{j}}{(a_m)^{j+1}} \left( \dfrac{1}{p_m(w,\cdot )}\right) ^{(i-\eta -j)}(0)\right| \\\le & {} \, \frac{\mathcal {C}}{|p_m(w,0)|} \sum _{j=0}^{i\!-\!\eta } \left( {\begin{array}{c}i\!-\!\eta \\ j\end{array}}\right) \dfrac{1}{(a_m)^{j+1}}\left( \frac{m}{\sqrt{a_m}}\right) ^{2(i-\eta -j)}\\= & {} \, \frac{\mathcal {C}}{a_m|p_m(w,0)|}\left( \frac{1}{a_m}+\frac{m^2}{a_m}\right) ^{i-\eta } \\\le & {} \, \mathcal {C}\left( \frac{m}{\sqrt{a_m}}\right) ^{2i-2\eta }\frac{1}{a_m|p_m(w,0)|}. \end{aligned}$$

Then we have

$$\begin{aligned} |D_i| \le \frac{\mathcal {C}}{a_m|p_m(w,0)|} \sum _{\eta =0}^i \left( {\begin{array}{c}i\\ \eta \end{array}}\right) |f^{(\eta )}(0)| \left( \frac{m}{\sqrt{a_m}}\right) ^{2i-2\eta }. \end{aligned}$$

(4.25)

On the other hand, by (4.1), (3.2) and (4.8), we get

$$\begin{aligned} b_i\le & {} \mathcal {C}\max _{x\in I_m}\left| \sqrt{w(x)}p_m(w,x)(a_m-x)x^{i+\gamma -\frac{\alpha }{2}}\right| \nonumber \\\le & {} \mathcal {C} \max _{x\in I_m}\left| (a_m-x)^{\frac{3}{4}}x^{i +\gamma - \frac{\alpha }{2} - \frac{1}{4}}\right| \nonumber \\\le & {} \mathcal {C} (a_m)^{\frac{3}{4}} \left( \frac{\sqrt{a_m}}{m}\right) ^{2i+2\gamma -\alpha - \frac{1}{2}}\sim a_m\left( \frac{\sqrt{a_m}}{m}\right) ^{2i+2\gamma }|p_m(w,0)|. \end{aligned}$$

(4.26)

Combining (4.25) and (4.26) in (4.24), we obtain

$$\begin{aligned} |A(x)u(x)|\le & {} \mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^{2\gamma }\sum _{i=0}^{r-1}\frac{1}{i!}\sum _{\eta =0}^i \left( {\begin{array}{c}i\\ \eta \end{array}}\right) |f^{(\eta )}(0)|\left( \frac{\sqrt{a_m}}{m}\right) ^{2\eta }\\= & {} \mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^{2\gamma }\sum _{i=0}^{r-1}\sum _{\eta =0}^{r-1}(i-\eta +1)_+^0|f^{(\eta )}(0)|\left( \frac{\sqrt{a_m}}{m}\right) ^{2\eta }\left( {\begin{array}{c}i\\ \eta \end{array}}\right) \frac{1}{i!}\\= & {} \mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^{2\gamma } \sum _{\eta =0}^{r-1}\left( \frac{\sqrt{a_m}}{m}\right) ^{2\eta }|f^{(\eta )}(0)|\sum _{i=0}^{r-1}(i-\eta +1)_+^0\left( {\begin{array}{c}i\\ \eta \end{array}}\right) \frac{1}{i!}\\\sim & {} \left( \frac{\sqrt{a_m}}{m}\right) ^{2\gamma }\sum _{\nu =0}^{r-1}\frac{|f^{(\nu )}(0)|}{\nu !}\left( \frac{\sqrt{a_m}}{m}\right) ^{2\nu }, \end{aligned}$$

since $\displaystyle \sum \nolimits _{i=\eta +1}^{r-1}\left( {\begin{array}{c}i\\ \eta \end{array}}\right) \frac{1}{i!} \sim 1$.

Let us now show (4.22). Recalling (4.24), we have

$$\begin{aligned} \Vert (a_m-\cdot )p_m(w)u v^i\Vert _p\le & {} \mathcal {C}\Vert (a_m-\cdot )p_m(w)u v^i\Vert _{L^p(I_m)} \\= & {} \mathcal {C}\Vert \sqrt{w}p_m(w)(a_m-\cdot )v^{i+\gamma -\frac{\alpha }{2}}\Vert _{L^p(I_m)}\\\le & {} \mathcal {C}\Vert (a_m-\cdot )^{\frac{3}{4}}v^{i+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\Vert _{L^p(I_m)}\\\le & {} \mathcal {C}(a_m)^{\frac{3}{4}}\left( \frac{\sqrt{a_m}}{m}\right) ^{2i+2\gamma -\alpha -\frac{1}{2}+\frac{2}{p}}\\= & {} \mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^{2i+2\gamma +\frac{2}{p}}a_m|p_m(w,0)| \end{aligned}$$

where we have applied (4.1), (4.6), (4.8), taking also into account the assumption (4.23). Then recalling (4.25), we deduce (4.22).

The necessity of the conditions (4.21) and (4.23) can be proved using standard arguments (see [22]). We omit the details. $\square$

Proof of Theorem 2

Let us first prove that conditions (3.2) imply inequality (3.1). Recalling (2.2) and Proposition 9, it remains to prove that

$$\begin{aligned} \left| \sum _{k=1}^{m+1}u(x)\frac{x^r}{x_k^r}\,\frac{\ell _k(w,x)}{u(x_k)}f(x_k)u(x_k) \right| \le \mathcal {C}(\log m) \max _{1\le i \le m+1}|fu|(x_i). \end{aligned}$$

Let $x \in I_m=\left[ c\frac{a_m}{m^2}, a_m\right]$ and $x_d$ be a zero closest to x. By using (4.7), (4.11) and (4.13), we get

$$\begin{aligned}&\sum _{k=1}^{m+1}u(x)\frac{x^r}{x_k^r}\,\frac{|\ell _k(w,x)|}{u(x_k)}f(x_k)u(x_k) \\&\quad \le \mathcal {C}\left[ |p_m(w,x)|\sqrt{w(x)\sqrt{x(a_m-x)}} \sum _{k\ne d, m+1} \left( \frac{x}{x_k}\right) ^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}} \left( \frac{a_m-x}{a_m-x_k}\right) ^{\frac{3}{4}}\right. \\&\qquad \quad \left. \frac{{\varDelta } x_k}{|x-x_k|} +1+\left| \frac{p_m(w,x)}{p_m(w,a_m)}\right| \frac{x^r}{a_m^r} \frac{u(x)}{u(a_m)}\right] \max _{1\le i \le m+1}|fu|(x_i). \end{aligned}$$

By (4.6) the last summand on the right-hand side is bounded by $\left( \frac{x}{a_m}\right) ^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}$ and so the whole right-hand side is dominated by

$$\begin{aligned}&\mathcal {C}\left( \sum _{k\ne d, m+1} \left( \frac{x}{x_k}\right) ^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}} \left( \frac{a_m-x}{a_m-x_k}\right) ^{\frac{3}{4}} \frac{{\varDelta } x_k}{|x-x_k|}\right. \\&\qquad \left. + 1+\left( \frac{x}{a_m}\right) ^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}} \right) \max _{1\le i \le m+1}|fu|(x_i). \end{aligned}$$

Therefore, if the parameters $r, \gamma$ and $\alpha$ fulfill the assumptions (3.2), the sum on the right-hand side is bounded by $\mathcal {C}(\log m)$ and the last term is bounded by 1.

Let us now prove that inequality (3.1) implies conditions (3.2). To this aim, for any $f\in C_u$, consider the piecewise linear function $f_1$ defined as follows

$$\begin{aligned} \left\{ \begin{array}{ll} f_1^{(i)}(0)=0, &{}\quad i=0,1,\ldots ,r-1, \\ f_1(x_k)= -|f(x_k)|\,\text {sgn}\{p^{\prime }_m(w,x_k)\}\,\ne 0, &{}\quad \text {for } 1\le x_k \le 2,\\ f_1(x_k)=0, &{}\quad \text {otherwise}. \end{array} \right. \end{aligned}$$

(4.27)

By (3.1) we have

$$\begin{aligned} \mathcal {C}(\log m) \max _{1\le x_i\le 2}|f_1 u|(x_i) \ge \Vert \mathcal {L}_{m,r}(w,f_1)u\Vert _\infty \ge \Vert \mathcal {L}_{m,r}(w,f_1)u\Vert _{L^\infty [0,1]}, \end{aligned}$$

(4.28)

where, for $x\in [0,1]$ and $x_k\in [1,2]$,

$$\begin{aligned} \mathcal {L}_{m,r}(w,f_1,x)u(x)= & {} -\sum _{1\le x_k\le 2}\frac{x^r}{x_k^r}\frac{p_m(w,x)|f_1 u|(x_k)}{|p^{\prime }_m(w,x_k)|(x-x_k)}\left( \frac{a_m-x}{a_m-x_k}\right) \frac{u(x)}{u(x_k)}\\= & {} \sum _{1\le x_k\le 2}\frac{x^r}{x_k^r}\frac{p_m(w,x)|f u|(x_k)}{|p^{\prime }_m(w,x_k)||x-x_k|}\left( \frac{a_m-x}{a_m-x_k}\right) \frac{u(x)}{u(x_k)}, \end{aligned}$$

since $x-x_k<0$. Now, if $x\in [0,1]$ and $x_k\in [1,2]$, then $|x-x_k|\le 2$ and $\frac{a_m-x}{a_m-x_k} \sim 1$. Hence, setting

$$\begin{aligned} \widetilde{A}(x)=p_m(w,x)\sqrt{w(x)\sqrt{x\left( a_m-x\right) }} \end{aligned}$$

and using (4.11), we get

$$\begin{aligned} \Vert \mathcal {L}_{m,r}(w,f_1)u\Vert _{L^\infty [0,1]} \ge \mathcal {C}\Vert \widetilde{A} v^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\Vert _{L^\infty [0,1]}\sum _{1\le x_k\le 2}\frac{{\varDelta } x_k |fu|(x_k)}{x_k^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}}. \end{aligned}$$

(4.29)

Combining (4.28) and (4.29), we obtain

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\Vert _{L^\infty [0,1]}\sum _{1\le x_k\le 2}\frac{{\varDelta } x_k |fu|(x_k)}{x_k^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}} \le \mathcal {C}(\log m)\max _{1\le x_i\le 2}|fu|(x_i) \end{aligned}$$

with $\mathcal {C}\ne \mathcal {C}(m,f)$, whence letting $b_i=|fu|(x_i)$ and $\Vert \underline{b}\,\Vert ^*_\infty =\displaystyle \max _{1\le x_i\le 2}b_i$, we have

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\Vert _{L^\infty [0,1]}\sum _{1\le x_k\le 2}\frac{{\varDelta } x_k }{x_k^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}}\frac{b_k}{\Vert \underline{b}\,\Vert ^*_\infty } \le \mathcal {C}(\log m), \quad \mathcal {C}\ne \mathcal {C}(m,f) \end{aligned}$$

and then

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\Vert _{L^\infty [0,1]}\sup _{\Vert \underline{d}\,\Vert ^*_\infty =1}\sum _{1\le x_k\le 2}\frac{{\varDelta } x_k }{x_k^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}}d_k \le \mathcal {C}(\log m), \quad \mathcal {C}\ne \mathcal {C}(m,f). \end{aligned}$$

Hence we get

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\Vert _{L^\infty [0,1]}\sum _{1\le x_k\le 2}\frac{{\varDelta } x_k}{x_k^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}} \le \mathcal {C}(\log m). \end{aligned}$$

Obviously the sum on the left-hand side is $\sim \! 1$. Moreover, by (4.6), we have

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\Vert _{L^\infty [0,1]}\ge \left| \widetilde{A}\left( \frac{a_m}{m^2}\right) v^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\left( \frac{a_m}{m^2}\right) \right| \sim \left( \frac{a_m}{m^2}\right) ^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}. \end{aligned}$$

So, from

$$\begin{aligned} \left( \frac{a_m}{m^2}\right) ^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\le \mathcal {C}(\log m) \end{aligned}$$

it follows that $r+\gamma \ge \frac{\alpha }{2}+\frac{1}{4}$.

In order to prove the necessity of the other inequality in (3.2), we introduce the piecewise linear function $f_2$ defined as

$$\begin{aligned} \left\{ \begin{array}{ll} f_2^{(i)}(0)=0, &{}\quad i=0,1,\ldots ,r-1, \\ f_2(x_k)= |f(x_k)|\,\text {sgn}\{p^{\prime }_m(w,x_k)\}, &{}\;\; \text {for }\quad 0 < x_k \le 1,\\ f_2(x_k)=0, &{}\quad \text {otherwise}. \end{array} \right. \end{aligned}$$

(4.30)

Using similar arguments we obtain

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\Vert _{L^\infty [1,2]}\sum _{0< x_k\le 1}\frac{{\varDelta } x_k}{x_k^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}} \le \mathcal {C}(\log m). \end{aligned}$$

Now, the sum on the left-hand side is $\sim \int _{x_1}^1 x^{-r-\gamma +\frac{\alpha }{2}+\frac{1}{4}}\,\text {d} x$, while the norm is bounded. So, from

$$\begin{aligned} \int _{x_1}^1 x^{-r-\gamma +\frac{\alpha }{2}+\frac{1}{4}}\,\mathrm {d} x\le \mathcal {C}\sum _{0< x_k\le 1}\frac{{\varDelta } x_k}{x_k^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}} \le \mathcal {C}(\log m) \end{aligned}$$

we deduce $r+\gamma \le \frac{\alpha }{2} + \frac{5}{4}$ and the first part of the theorem follows.

Let us now prove (3.3). Since $\left| f^{(i)}\right|$ is quasi increasing on $[0,\alpha _m]$, with $i\in \{0,1,\ldots , r-1\}$ and $\alpha _m=a_m/m^2$, we have

$$\begin{aligned} \alpha _m^{i+\gamma }\left| f^{(i)}(0)\right| \le \mathcal {C}\alpha _m^{i+\gamma }\left| f^{(i)}\left( \frac{\alpha _m}{2}\right) \right| \le \mathcal {C}\alpha _m^{i+\gamma }\left\| f^{(i)}\right\| _{L^\infty [\frac{\alpha _m}{2},\alpha _m]} . \end{aligned}$$

Recalling a formula in [3, p.15], it follows that

$$\begin{aligned} \alpha _m^{i+\gamma }\left| f^{(i)}(0)\right|\le & {} \mathcal {C}\alpha _m^{\gamma }\left\| f\right\| _{L^\infty [\frac{\alpha _m}{2},\alpha _m]} +\mathcal {C}\alpha _m^{r-1+\gamma }\left\| f^{(r-1)}\right\| _{L^\infty [\frac{\alpha _m}{2},\alpha _m]} \\\le & {} \mathcal {C}\left\| f u\right\| _{L^\infty [\frac{\alpha _m}{2},\alpha _m]} +\mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^{r-1}\left\| f^{(r-1)} \varphi ^{r-1} u\right\| _{L^\infty [\frac{\alpha _m}{2},\alpha _m]}. \end{aligned}$$

Hence (3.1) becomes

$$\begin{aligned} \left\| \mathcal {L}_{m,r}\left( w,f\right) u\right\| _\infty \le \mathcal {C}(\log m)\Vert f u\Vert _\infty + \mathcal {O}\left( \left( \frac{\sqrt{a_m}}{m}\right) ^{r-1}\right) . \end{aligned}$$

Letting $P_{m+r} \in \mathbb {P}_{m+r}$ be the polynomial of best approximation for $f\in C_u$, it follows that

$$\begin{aligned} \left\| \left[ f-\mathcal {L}_{m,r}\left( w,f\right) \right] u\right\| _\infty= & {} \left\| \left[ f-P_{m+r} - \mathcal {L}_{m,r}\left( w,f-P_{m+r}\right) \right] u\right\| _\infty \\\le & {} \mathcal {C}(\log m) E_{m+r}(f)_{u,\infty } + \mathcal {O}\left( \left( \frac{\sqrt{a_m}}{m}\right) ^{r-1}\right) , \end{aligned}$$

i.e. (3.3). $\square$

In order to prove Theorem 4 we recall some known results concerning the Hilbert transform H(f, t). The Hilbert transform related to the interval (0, a) is defined as follows

$$\begin{aligned} H(f,t)=\int _0^{a} \dfrac{f(x)}{x-t}\,\mathrm {d}x, \quad t\in (0,a),\; a>0, \end{aligned}$$

where the integral is understood in the Cauchy principal value sense. Letting $v^\rho (x)=x^\rho$ and $1<p<\infty$, the bound

$$\begin{aligned} \Vert (Hf)v^\rho \Vert _{L^p(0,1)} \le \mathcal {C}\Vert fv^\rho \Vert _{L^p(0,1)}, \quad \mathcal {C}\ne \mathcal {C}(f), \end{aligned}$$

(4.31)

holds for any f such that $fv^\rho \in L^p(0,a)$ if and only if $-\frac{1}{p} < \rho < 1 - \frac{1}{p}$ (see [23]).

The following lemma, that can be found in [21] (see also [8]), will be also useful in the proof of Theorem 4.

Lemma 10

Let $0<\theta <1,\, j$ be given by (2.3), $u \in L^p$, $1\le p < \infty$ and ${\varDelta } x_k= x_{k+1}- x_k$. Then, for an arbitrary polynomial $P\in \mathbb {P}_{l m}$ (where l is a fixed integer), there exists $\bar{\theta }\in (\theta ,1$) such that

$$\begin{aligned} \sum _{k=1}^j {\varDelta } x_k |Pu|^p(x_k) \le \mathcal {C}\int _{x_1}^{\bar{\theta } a_m}|Pu|^p(x)\, \mathrm {d}x, \end{aligned}$$

(4.32)

with $\mathcal {C}\ne \mathcal {C}(m,f)$.

We want to emphasize that the first Marcinkiewicz inequality (4.32) does not hold if the sum on the left-hand side is extended to all the zeros of $p_m(w)$, i.e. $k=1,\ldots , m$ (see [13, 14]).

Proof of Theorem 4

Taking into account Proposition 9 it remains to estimate the $L_u^p$-norm of the first term in (2.4). Letting $v^r(x)=x^r$ and

$$\begin{aligned} L_{m}^*(w,f,x)=\sum _{k=1}^{j}\ell _k(w,x)f(x_k),\quad \ell _k(w,x)=\frac{p_m(w,x)}{(x-x_k)p^{\prime }_m(w,x_k)}\frac{(a_m-x)}{(a_m-x_k)}, \end{aligned}$$

this term can be rewritten as $v^r L_{m}^*\left( w,\frac{f}{v^r}\right)$. So, using (4.1), we have

$$\begin{aligned} \left\| v^r L_{m}^*\left( w,\frac{f}{v^r}\right) u \right\| _p\le & {} \mathcal {C}\left\| \,v^r L_{m}^*\left( w,\dfrac{f}{v^r}\right) u\right\| _{L^p(I_m)} \nonumber \\= & {} \mathcal {C}\sup _{\Vert g\Vert _q=1} \left| \int _{I_m} x^r L_{m}^* \left( w,\dfrac{f}{x^r},x\right) u(x)g(x) \,\mathrm {d}x \right| \nonumber \\=: & {} \sup _{\Vert g\Vert _q=1} |{\varGamma }(g)|, \end{aligned}$$

(4.33)

where $I_m=\left[ c\,a_m m^{-2},a_m\right]$ and

$$\begin{aligned} {\varGamma }(g)= & {} \int _{I_m} x^r \sum _{k=1}^{j}\dfrac{\ell _k(w,x)}{x_k^r} f(x_k) g(x) u(x)\,\mathrm {d}x\\= & {} \sum _{k=1}^{j} \dfrac{f(x_k)u(x_k)}{p_m^{\prime }(w,x_k)(a_m-x_k)u(x_k)x_k^r}\int _{I_m}\dfrac{(a_m-x) p_m(w,x) x^r g(x) u(x)}{x-x_k} \,\mathrm {d}x. \end{aligned}$$

By using (4.11), we have

$$\begin{aligned} |{\varGamma }(g)|\le & {} \mathcal {C}\sum _{k=1}^{j} \dfrac{{\varDelta } x_k |f u|(x_k)}{(a_m-x_k)^{\frac{3}{4}}x_k^{r+\gamma -\frac{\alpha }{2} -\frac{1}{4}}} \left| \int _{I_m}\dfrac{(a_m-x) p_m(w,x) x^r g(x) u(x)}{x-x_k} \,\mathrm {d}x\right| \\\le & {} \dfrac{\mathcal {C}}{a_m^{3/4}} \sum _{k=1}^{j} \dfrac{{\varDelta } x_k |fu|(x_k)}{x_k^{r+\gamma -\frac{\alpha }{2} -\frac{1}{4}}} \left| \int _{I_m}\dfrac{(a_m-x) p_m(w,x) x^r g(x) u(x)}{x-x_k} \,\mathrm {d}x\right| . \end{aligned}$$

Denoting by $G(x_k)$ the integral on the right-hand side and using the Hölder inequality, we get

$$\begin{aligned} |{\varGamma }(g)| \le \dfrac{\mathcal {C}}{a_m^{3/4}}\left( \sum _{k=1}^{j} {\varDelta } x_k |fu|^p(x_k) \right) ^{1/p}\left( \sum _{k=1}^{j}{\varDelta } x_k \left| \dfrac{G(x_k)}{x_k^{r+\gamma -\alpha /2 -1/4}}\right| ^{\,q} \right) ^{1/q}.\qquad \end{aligned}$$

(4.34)

For any polynomial $0<Q\in \mathbb {P}_{lm}$, with l a fixed integer, we can write

$$\begin{aligned}&G(x_k)\\&\quad =\int _{I_m}\frac{(a_m-x)p_m(w,x)Q(x)-(a_m-x_k)p_m(w,x_k)Q(x_k)}{x-x_k} \frac{x^r g(x) u(x)}{Q(x)}\,\mathrm {d}x \end{aligned}$$

and G(t) is a polynomial of degree $m+lm$. Thus, using Lemma 10, we have

$$\begin{aligned} B= \left( \sum _{k=1}^{j} {\varDelta } x_k\left| \dfrac{G(x_k)}{x_k^{r+\gamma -\alpha /2 -1/4}}\right| ^{\,q} \right) ^{1/q} \le \left( \int _{I_{\bar{\theta }m}} \left| \dfrac{G(t)}{t^{r+\gamma -\alpha /2-1/4}}\right| ^{\,q}\,\mathrm {d}t\right) ^{1/q}, \end{aligned}$$

where $0<\theta <\bar{\theta }<1$ and $I_{\bar{\theta }m}=[c\,a_m m^{-2},\bar{\theta }a_m]$. Then we obtain

$$\begin{aligned} B\le & {} \left( \int _{I_{\bar{\theta }m}} \left| t^{-r-\gamma +\alpha /2+1/4} \int _{I_m}\frac{(a_m-x)p_m(w,x)x^r (gu)(x)}{x-t}\,\mathrm {d}x\right| ^{q}\mathrm {d}t\right) ^{1/q} \nonumber \\&+ \left( \int _{I_{\bar{\theta }m}} \left| t^{-r-\gamma +\alpha /2+1/4}(a_m-t)p_m(w,t)Q(t)\int _{I_m}\frac{x^r (gu)(x)}{(x-t)Q(x)}\,\mathrm {d}x\right| ^{q}\mathrm {d}t\right) ^{1/q} \nonumber \\=: & {} B_1 + B_2. \end{aligned}$$

(4.35)

Let us estimate the term $B_1$. By (4.7) we have

$$\begin{aligned} \left| \widetilde{A}(x)\right| :=|p_m(w,x)|\sqrt{w(x)\sqrt{x(a_m-x)}}\le \mathcal {C},\quad x\in I_m. \end{aligned}$$

Moreover, under the assumptions (3.5), $t^{-r-\gamma +\frac{\alpha }{2}+\frac{1}{4}}\in L^q$ and $t^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}\in L^p$, we can apply (4.31), obtaining

$$\begin{aligned} B_1\le & {} \left( \int _{I_{m}}\left| \,t^{-r-\gamma +\frac{\alpha }{2}+\frac{1}{4}}\int _{I_{m}}\frac{(a_m-x)^{\frac{3}{4}}\widetilde{A}(x)x^{r+\gamma -\frac{\alpha }{2}-\frac{1}{4}}g(x)}{x-t}\,\mathrm {d}x\right| ^{q}\mathrm {d}t\right) ^{1/q} \nonumber \\\le & {} \mathcal {C}\left( \int _{I_m} \left| (a_m-x)^{\frac{3}{4}}\widetilde{A}(t)g(t) \right| ^{q}\,\mathrm {d}t\right) ^{1/q} \nonumber \\\le & {} \mathcal {C}a_m^{\frac{3}{4}} \Vert g\Vert _{L^q(I_m)}. \end{aligned}$$

(4.36)

In order to estimate $B_2$ we can construct a polynomial $Q\in \mathbb {P}_{lm}$ such that $Q(x) \sim x^r u(x)$ for $x \in I_m$. So, using (4.7), we have

$$\begin{aligned} \left| t^{-r-\gamma +\alpha /2+1/4}(a_m-t)p_m(w,t)Q(t)\right| \le \mathcal {C}a_m^{3/4}. \end{aligned}$$

Then, by (4.31), we get

$$\begin{aligned} B_2 \le \mathcal {C}a_m^{3/4} \left( \int _{I_{m}} \left| \int _{I_m}\frac{x^r (gu)(x)}{(x-t)Q(x)}\,\mathrm {d}x\right| ^{q}\mathrm {d}t\right) ^{1/q} \le \mathcal {C}a_m^{3/4} \Vert g\Vert _{L^q(I_m)}. \end{aligned}$$

(4.37)

So, taking into account (4.36), (4.37) and (4.35), it follows that

$$\begin{aligned} \left( \int _{I_{\bar{\theta }m}} \left| \dfrac{G(t)}{t^{r+\gamma -\alpha /2-1/4}}\right| ^{\,q}\,\mathrm {d}t\right) ^{1/q} \le \mathcal {C}a_m^{3/4} \Vert g\Vert _{L^q(I_m)}, \end{aligned}$$

and combining this last inequality with (4.34) and (4.33), the first term in (2.4) is bounded by

$$\begin{aligned} \mathcal {C}\left( \sum _{k=1}^{j} {\varDelta } x_k |fu|^p(x_k) \right) ^{1/p}. \end{aligned}$$

Therefore, recalling (4.22), we have that (3.5) implies (3.4).

In order to prove that (3.4) implies (3.5) we use arguments similar to those in the proof of Theorem 4. Therefore, we are going to show only the main steps. Considering the function $f_1$ as in (4.27) we can write

$$\begin{aligned} \Vert \mathcal {L}_{m,r}^*(w,f_1)u\Vert _{L^p[c\,a_m m^{-2},1]} \le \Vert \mathcal {L}_{m,r}^*(w,f_1)u\Vert _p \le \mathcal {C}\left( \sum _{1\le x_k \le 2} {\varDelta } x_k |f_1 u|^p(x_k) \right) ^{1/p}, \end{aligned}$$

where $[c\,a_m m^{-2},1]=[0,1]\cap I_m$. Recalling the expression of $\mathcal {L}_{m,r}^*(w,f_1,x)$ for $x\in [c\,a_m m^{-2},1]$, with

$$\begin{aligned} \left| \widetilde{A}(x)\right| =|p_m(w,x)|\sqrt{w(x)\sqrt{x(a_m-x)}}, \end{aligned}$$

we obtain

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma - \frac{\alpha }{2} - \frac{1}{4}}\Vert _{L^p[a_m m^{-2},1]} \sum _{1\le x_k \le 2} \frac{{\varDelta } x_k |f_1 u|(x_k)}{x_k^{r+ \gamma - \frac{\alpha }{2} - \frac{1}{4}}}\le \mathcal {C}\left( \sum _{1\le x_k \le 2} {\varDelta } x_k |f_1 u|^p(x_k) \right) ^{1/p}. \end{aligned}$$

(4.38)

Now, to simplify the notation, we set

$$\begin{aligned} a_k=\frac{({\varDelta } x_k)^{1/q}}{x_k^{r+ \gamma - \frac{\alpha }{2} - \frac{1}{4}}}, \,\, b_k=({\varDelta } x_k)^{1/p}|f_1 u|(x_k), \,\, \Vert \underline{b}\,\Vert _p^* =\left( \sum _{1\le x_k\le 2}b_k^p\right) ^{1/p}. \end{aligned}$$

Hence (4.38) becomes

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma - \frac{\alpha }{2} - \frac{1}{4}}\Vert _{L^p[c\,a_m m^{-2},1]}\sum _{1\le x_k \le 2}a_k \frac{b_k}{\Vert \underline{b}\,\Vert _p^*} \le \mathcal {C}, \quad \mathcal {C}\ne \mathcal {C}(m,f). \end{aligned}$$

Taking the supremum on $\underline{b}$, we get

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma - \frac{\alpha }{2} - \frac{1}{4}}\Vert _{L^p[c\,a_m m^{-2},1]}\left( \sum _{1\le x_k \le 2}|a_k|^q\right) ^{1/q} \le \mathcal {C}, \end{aligned}$$

which implies $r+\gamma - \frac{\alpha }{2} - \frac{1}{4} > -\frac{1}{p}$, taking into account that, by (4.7), $\left| \widetilde{A}(x)\right| \le \mathcal {C}, x\in [c\,a_m m^{-2},1]$.

Analogously, considering the function $f_2$ defined in (4.30), we get

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma - \frac{\alpha }{2} - \frac{1}{4}}\Vert _{L^p[1,2]}\left( \sum _{0 < x_k \le 1}|a_k|^q\right) ^{1/q} \le \mathcal {C}, \end{aligned}$$

i.e.

$$\begin{aligned} \Vert \widetilde{A} v^{r+\gamma - \frac{\alpha }{2} - \frac{1}{4}}\Vert _{L^p[1,2]}\left( \sum _{0 < x_k \le 1} \frac{{\varDelta } x_k}{x_k^{(r+\gamma -\alpha /2-1/4)q}}\right) ^{1/q} \le \mathcal {C}, \end{aligned}$$

whence $-r-\gamma + \frac{\alpha }{2} +\frac{1}{4} > -\frac{1}{q}$ that is $r+\gamma - \frac{\alpha }{2} - \frac{1}{4} < 1 -\frac{1}{p}$, which completes the proof. $\square$

Proof of Theorem 5

Since $\left| f^{(i)}\right|$ is quasi increasing on $[0,\alpha _m]$, with $i\in \{0,1,\ldots , r-1\}$ and $\alpha _m=a_m/m^2$, using the Hölder inequality, we have

$$\begin{aligned} \alpha _m^{i+\gamma +1/p}\left| f^{(i)}(0)\right|\le & {} \mathcal {C}\alpha _m^{i+\gamma +1/p}\left| f^{(i)}\left( \frac{\alpha _m}{2}\right) \right| \,\le \, \mathcal {C}\alpha _m^{i+\gamma +1/p-1}\int _{\frac{\alpha _m}{2}}^{\alpha _m}\left| f^{(i)}(x)\right| \mathrm {d}x \\\le & {} \mathcal {C}\alpha _m^{i+\gamma }\left\| f^{(i)}\right\| _{L^p[\frac{\alpha _m}{2},\alpha _m]} . \end{aligned}$$

Recalling a formula in [3, p.15], it follows that

$$\begin{aligned} \alpha _m^{i+\gamma +1/p}\left| f^{(i)}(0)\right|\le & {} \mathcal {C}\alpha _m^{\gamma }\left\| f\right\| _{L^p[\frac{\alpha _m}{2},\alpha _m]} +\mathcal {C}\alpha _m^{r-1+\gamma }\left\| f^{(r-1)}\right\| _{L^p[\frac{\alpha _m}{2},\alpha _m]} \\\le & {} \mathcal {C}\left\| f u\right\| _{L^p[\frac{\alpha _m}{2},\alpha _m]} +\mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^{r-1}\left\| f^{(r-1)} \varphi ^{r-1} u\right\| _{L^p[\frac{\alpha _m}{2},\alpha _m]}. \end{aligned}$$

So inequality (3.7) becomes

$$\begin{aligned} \left\| \mathcal {L}_{m,r}^*\left( w,f\right) u\right\| _p\le & {} \mathcal {C}\left[ \Vert fu\Vert _p + \left( \frac{\sqrt{a_m}}{m} \right) ^{1/p} \int _0^{\frac{\sqrt{a_m}}{m}} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t^{1+1/p}}\,\mathrm {d}t \right. \nonumber \\&\left. +\left( \frac{\sqrt{a_m}}{m}\right) ^{r-1}\left\| f^{(r-1)} \varphi ^{r-1} u\right\| _{L^p[0,\alpha _m]}\right] . \end{aligned}$$

(4.39)

Let $P_M \in \mathbb {P}_M$, with $M=\left\lfloor \frac{\theta m}{\theta +1}\right\rfloor$, be a polynomial of quasi best approximation for $f\in L^p_u$, and set $Q=\mathcal {L}_{m,r}^*(w,P_M)$. Hence, by (4.39), we get

$$\begin{aligned} \Vert [f\!-\!\mathcal {L}^*_{m,r}(w,f)]u\Vert _p\le & {} \mathcal {C}\left[ \left\| \left( f\!-\!Q\right) u\right\| _p {+} \left( \frac{\sqrt{a_m}}{m} \right) ^{1/p} \!\!\int _0^{\frac{\sqrt{a_m}}{m}} \dfrac{{\varOmega }_{\varphi }^s(f-Q,t)_{u,p}}{t^{1+1/p}}\,\mathrm {d}t \right. \nonumber \\&\left. + \left( \frac{\sqrt{a_m}}{m}\right) ^{r-1}\left\| \left( f-Q\right) ^{(r-1)} \varphi ^{r-1} u\right\| _{L^p[0,\alpha _m]}\right] \end{aligned}$$

Now, to estimate the first summand at the right-hand side we can use (2.12) and (2.8). For the second summand we note that

$$\begin{aligned} {\varOmega }_{\varphi }^s(f-Q,t)_{u,p}\le {\varOmega }_{\varphi }^s(f,t)_{u,p}+t^s \left\| Q^{(s)} \varphi ^{s} u\right\| _{p}. \end{aligned}$$

So, by (2.13), taking into account that

$$\begin{aligned} \int _0^{\frac{\sqrt{a_m}}{m}} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t}\,\mathrm {d}t \le \left( \frac{\sqrt{a_m}}{m} \right) ^{1/p} \int _0^{\frac{\sqrt{a_m}}{m}} \dfrac{{\varOmega }_{\varphi }^s(f,t)_{u,p}}{t^{1+1/p}}\,\mathrm {d}t, \end{aligned}$$

inequality (3.8) follows. $\square$

Proof of Corollary 6

Proceeding as in the proof of Theorem 5, since $\left| f^{(i)}\right|$ is quasi increasing on $[0,\alpha _m]$, with $i\in \{0,1,\ldots , r-1\}$ and $\alpha _m=a_m/m^2$, and $f\in W_r^p(u)$, we get

$$\begin{aligned} \alpha _m^{i+\gamma +1/p}\left| f^{(i)}(0)\right|\le & {} \mathcal {C}\alpha _m^{\gamma }\left\| f\right\| _{L^p[\frac{\alpha _m}{2},\alpha _m]} +\mathcal {C}\alpha _m^{r+\gamma }\left\| f^{(r)}\right\| _{L^p[\frac{\alpha _m}{2},\alpha _m]} \nonumber \\ \!\le & {} \! \mathcal {C}\left\| f u\right\| _{L^p[\frac{\alpha _m}{2},\alpha _m]} \!+\!\mathcal {C}\left( \frac{\sqrt{a_m}}{m}\right) ^{r}\left\| f^{(r)} \varphi ^r u\right\| _{L^p[\frac{\alpha _m}{2},\alpha _m]}.\quad \end{aligned}$$

(4.40)

Hence, by (3.7), (2.6) and (4.40), we obtain (3.9).

Now, in order to prove (3.10), let $Q=\mathcal {L}_{m,r}^*(w,P_M)$, where $P_M \in \mathbb {P}_M$, with $M=\left\lfloor \frac{\theta m}{\theta +1}\right\rfloor$, is a polynomial of quasi best approximation for $f\in L^p_u$. By (3.9) we have

$$\begin{aligned}&\left\| \left[ f-\mathcal {L}_{m,r}^*\left( w,f\right) \right] u\right\| _p \le \mathcal {C}\left\| \left( f-Q\right) u\right\| _p+ \left\| \mathcal {L}_{m,r}^*\left( w,f-Q\right) u\right\| _p \\&\quad \le \mathcal {C}\left\| \left( f-Q\right) u\right\| _p + \left( \frac{\sqrt{a_m}}{m}\right) ^{r}\left\| f^{(r)} \varphi ^r u\right\| _p +\left( \frac{\sqrt{a_m}}{m}\right) ^{r}\left\| Q^{(r)} \varphi ^r u\right\| _p. \end{aligned}$$

Using Lemma 1, (2.10) and (2.6), we obtain (3.10). $\square$