# De la Vallée Poussin filtered polynomial approximation on the half line

D.Occorsio, W.Themistoclakis

###### Abstract

On the half line we introduce a new sequence of near–best uniform
approximation polynomials, easily computable by the values of the approximated
function at a truncated number of Laguerre zeros. Such approximation
polynomials come from a discretization of filtered Fourier–Laguerre partial
sums, which are filtered by using a de la Vallée Poussin (VP) filter. They
have the peculiarity of depending on two parameters: a truncation parameter
that determines how many of the $n$ Laguerre zeros are considered, and a
localization parameter, which determines the range of action of the VP filter
that we are going to apply. As $n\to\infty$, under simple assumptions on such
parameters and on the Laguerre exponents of the involved weights, we prove
that the new VP filtered approximation polynomials have uniformly bounded
Lebesgue constants and uniformly convergence at a near–best approximation
rate, for any locally continuous function on the semiaxis.
The theoretical results have been validated by the numerical experiments. In
particular, they show a better performance of the proposed VP filtered
approximation versus the truncated Lagrange interpolation at the same nodes,
especially for functions a.e. very smooth with isolated singularities. In such
cases we see a more localized approximation as well as a good reduction of the
Gibbs phenomenon.

##  1 Introduction

The paper deals with the approximation of locally continuous functions $f$
defined on the positive half-line. The function $f$ we consider can
exponentially grow at infinity or/and have an algebraic singularity at the
origin. Measuring the approximation error with a suitable weighted uniform
norm, the Weierstrass approximation theorem may be applied and $f$ can be
approximated by polynomials at the desired precision.

The topic has been extensively studied by many authors, which were looking for
a polynomial approximating sequence having optimal Lebesgue constants in
spaces of locally continuous functions equipped with uniform weighted norms
[9, 15, 3, 6, 8, 10].Based on Laguerre zeros, we mention the classical
Lagrange interpolating polynomial for which in [9] it has been proved that the
associated weighted Lebesgue constants diverge as $n^{\frac{1}{6}}$. In the
same paper a ”remedium” was proposed to this bad behavior, by introducing a
suitable additional node. In this case, sufficient and necessary conditions
have been stated to get optimal Lebesgue constants growing with minimal order
$\log n$. A truncated version of the latter Lagrange polynomial has been
introduced and studied in [3, 6] by involving only a very reduced number of
function evaluations. Also in this case we get optimal Lebesgue constants,
under suitable sufficient conditions.

In the present paper we introduce new approximation polynomials based on the
same data of the truncated Lagrange-Laguerre interpolation but able to provide
uniformly bounded Lebesgue constants and a near-best convergence order for any
locally continuous function on the semiaxis. Such polynomials are defined by
discretizing some VP means, namely delayed arithmetic means of the Fourier-
Laguerre partial sums.

We recall that in literature such continuous VP means have been introduced as
an useful tool to prove several theoretical results [13, 14].

In the Jacobi case it is known that by discretizing VP means we get near-est
approximation operators that preserve the polynomials up to certain degree,
have uniformly bounded Lebesgue constants [18] and, in some cases, are
interpolating too. Here we show that, in the Laguerre case, the interpolation
property as well as the polynomial preservation are not ensured. But, all the
same, we prove the uniform convergence at a near-best approximation rate with
uniformly bounded Lebesgue constants, under suitable assumptions.

The outline of the paper is the following. In Section 2 the necessary
notations and preliminary results are introduced. Section 3 concerns the
definition and the properties of the new VP discrete approximation
polynomials. Section 4 deals with the numerical experiments and, finally,
Section 5 regards the conclusions.

##  2 Notations and preliminary results

Along all the paper the notation ${\mathcal{C}}$ will be used several times to
denote a positive constant having different values in different formulas. We
will write ${\mathcal{C}}\neq\mathcal{C}(a,b,\ldots)$ in order to say that
$\mathcal{C}$ is independent of the parameters $a,b,\ldots$, and
$\mathcal{C}=\mathcal{C}(a,b,\ldots)$ to say that $\mathcal{C}$ depends on
$a,b,\ldots$. Moreover, if $A,B>0$ are quantities depending on some
parameters, we will write $A\sim B,$ if there exists an absolute constant
${\mathcal{C}}>0$, independent on such parameters, such that
${\mathcal{C}}^{-1}B\leq A\leq{\mathcal{C}}B$.

For all $\gamma\geq 0,$ we will consider the weight $u(x)=x^{\gamma}e^{-x/2},$
and the space $C_{u}$ of all locally continuous functions on the half line
(i.e., continuous in $]0,\infty[$) such that

$\lim_{x\to\infty}(fu)(x)=0,\qquad\mbox{and, in the case $\gamma>0$,
also}\qquad\lim_{x\to 0}(fu)(x)=0.$

We will take the space $C_{u}$ equipped with the norm

$\|f\|_{C_{u}}=\|fu\|:=\sup_{x\geq 0}\left|(fu)(x)\right|,$

and for any compact set $D\subset[0,+\infty)$, we will use
$\|f\|_{D}=\sup_{x\in D}|f(x)|$.

Note that functions in $C_{u}$ may grow exponentially as $x\to+\infty$, and
may have an algebraic singularity at the origin.

Denoting by ${\mathbb{P}}_{n}$ the space of the algebraic polynomials of
degree at most $n\geq 0$, we will consider the error of best approximation of
$f\in C_{u}$ in ${\mathbb{P}}_{n}$

$E_{n}(f)_{u}:=\inf_{P_{n}\in{\mathbb{P}}_{n}}\|(f-P_{n}){u}\|,$ (1)

It is known that [4, 5, 11, 12]

$\lim_{n\to\infty}E_{n}(f)_{{u}}=0,\qquad\forall f\in C_{u}.$

For smoother functions, we consider the Sobolev-type spaces of order
$r\in{\mathbb{N}}$

$W_{r}(u)=\left\\{f\in C_{u}\ :\ f^{(r-1)}\in AC((0,+\infty))\mbox{ and }\
\|f^{(r)}\varphi^{r}u\|<+\infty\right\\},$

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

$\|f\|_{W_{r}(u)}:=\|fu\|+\|f^{(r)}\varphi^{r}u\|.$

For any function $f\in W_{r}(u)$ the following estimate holds [1, 4]

$E_{n}(f)_{{u}}\leq{\mathcal{C}}\frac{\|f\|_{W_{r}(u)}}{\sqrt{n^{r}}},\quad{\mathcal{C}}\neq{\mathcal{C}}(n,f).$
(2)

###  2.1 Truncated Lagrange interpolation at the zeros of Laguerre polynomials

Let $w_{\alpha}(x)=e^{-x}x^{\alpha}$ be the Laguerre weight of parameter
$\alpha>-1$ and let $\\{p_{n}(w_{\alpha})\\}_{n}$ be the corresponding
sequence of orthonormal Laguerre polynomials with positive leading
coefficients. Denoting by $x_{k}:=x_{n,k},\ k=1,\dots,n,$ the zeros of
$p_{n}(w_{\alpha})$ in increasing order, we recall that (see [17])

$\frac{{\mathcal{C}}}{n}<x_{1}<x_{2}<\ldots<x_{n}<4n+2\alpha-{\mathcal{C}}n^{\frac{1}{3}},\qquad{\mathcal{C}}\neq{\mathcal{C}}(n).$
(3)

In the sequel, we will be interested in taking only a part of these zeros.
More precisely, throughout the paper, the integer $j:=j(n)$ will denote the
index of the last zero of $p_{n}(w_{\alpha})$ we consider. It is defined by

$j=\min_{k=1,2,..,n}\left\\{k:x_{k}\geq 4n\rho\right\\},$ (4)

with $0<\rho<1$ arbitrarily fixed,

We recall that, inside the segment $[0,x_{j}]$, the distance between two
consecutive zeros of $p_{n}(w_{\alpha})$ can be estimated as follows

$\Delta x_{k}\sim\Delta x_{k-1}\sim\sqrt{\frac{x_{k}}{n}},\qquad\quad\Delta
x_{k}=x_{k+1}-x_{k},\qquad k=1,2,\dots,j.$

With $j$ defined in (4), let $L_{n+1}^{*}(w_{\alpha},f)$ be the truncated
Lagrange polynomial [6] defined by

$L_{n+1}^{*}(w_{\alpha},f,x):=\sum_{k=1}^{j}f(x_{k})\ell_{n,k}(x),$ (5)

where

$\ell_{n,k}(x)=\frac{p_{n}(w_{\alpha},x)}{p^{\prime}_{n}(x_{k})(x-x_{k})}\frac{4n-x}{4n-x_{k}},\qquad
k=1,2,\dots,j,$ (6)

are the fundamental Lagrange polynomials corresponding to the zeros of
$p_{n}(w_{\alpha},x)(4n-x)$.

It is known that the norm of the operator $L_{n+1}^{*}(w_{\alpha}):f\in
C_{u}\to L_{n+1}^{*}(w_{\alpha},f)\in C_{u}$, i.e. the weighted Lebesgue
constant

$\|L_{n+1}^{*}(w_{\alpha})\|_{C_{u}}=\sup_{\|fu\|=1}\|L_{n+1}^{*}(w_{\alpha},f)u\|=\sup_{x\geq
0}\sum_{k=1}^{j}|\ell_{n,k}(x)|\frac{u(x)}{u(x_{k})},$ (7)

diverges at least as $\log n$ when $n\to\infty$. To be more precise, the
following result holds [6]

###### Theorem 2.1

[6] Let $w_{\alpha}(x)=e^{-x}x^{\alpha},\ \alpha>-1$ and
$u(x)=e^{-\frac{x}{2}}x^{\gamma},\ \gamma\geq 0.$ Under the assumption

$\max\left(0,\frac{\alpha}{2}+\frac{1}{4}\right)\leq\gamma\leq\frac{\alpha}{2}+\frac{5}{4},$
(8)

we have

$\|L_{n+1}^{*}(w_{\alpha},f)u\|\leq{\mathcal{C}}\|fu\|\log n,\quad\forall f\in
C_{u},\qquad{\mathcal{C}}\neq{\mathcal{C}}(n,f).$ (9)

Moreover, as regards the approximation of $f\in C_{u}$ provided by the
polynomial $L_{n+1}^{*}(w_{\alpha},f)$ we recall the following

###### Theorem 2.2

[6, Th.2.2] For any $f\in C_{u},$ under the assumption (8), we have

$\|(f-L_{n+1}^{*}(w_{\alpha},f))u\|\leq{\mathcal{C}}(\log
n)\left[E_{N}(f)_{u}+e^{-An}\|fu\|\right],$

where $N=\left\lfloor\left(\frac{\rho}{1+\rho}\right)n\right\rfloor\sim n$,
and $A$ and ${\mathcal{C}}$ are positive constants independent of $n$ and $f$.

###  2.2 Ordinary and truncated Gauss-Laguerre rules

According to the previous notation, the Gauss-Laguerre rule on the nodes in
(3) is given by

$\int_{0}^{+\infty}f(x)w_{\alpha}(x)dx=\sum_{k=1}^{n}f(x_{k})\lambda_{n,k}(w_{\alpha})+R_{n}(f),\qquad
n\in{\mathbb{N}},$ (10)

where $\lambda_{n,k}(w_{\alpha})=\left[\sum_{j=0}^{n}p_{j}(x_{k})\right]^{-1}$
are the Christoffel numbers w.r.t. $w_{\alpha},$ and the remainder term
satisfies

$R_{n}(f)=0,\qquad\forall f\in{\mathbb{P}}_{2n-1}.$ (11)

For any $f\in C_{u}$, under the assumption $\alpha-\gamma>-1$, the following
estimate holds

$|R_{n}(f)|\leq\
{\mathcal{C}}E_{2n-1}(f)_{u},\qquad{\mathcal{C}}\neq{\mathcal{C}}(n,f).$

In [7] the “truncated” Gauss-Laguerre rule was introduced by using only the
first $j$ zeros of $p_{n}(w_{\alpha})$, with $j$ defined in (4), i.e.

$\int_{0}^{+\infty}f(x)w_{\alpha}(x)dx=\sum_{k=1}^{j}f(x_{k})\lambda_{n,k}(w_{\alpha})+R_{n,j}(f),$
(12)

where

$R_{n,j}(f)=0,\qquad\forall
f\in\mathcal{P}_{n-1}^{*}:=\left\\{p\in{\mathbb{P}}_{n-1}:p(x_{k})=0,\quad
k>j\right\\}.$ (13)

In [6] the following estimate was proved under the assumption
$\alpha-\gamma>-1$

$|R_{n,j}(f)|\leq{\mathcal{C}}\
\left(E_{N}(f)_{u}+e^{-\mathcal{B}n}\|fu\|\right),\qquad\forall f\in C_{u},$

where $N=\left\lfloor\left(\frac{\rho}{1+\rho}\right)n\right\rfloor\sim n$,
and $\mathcal{B},{\mathcal{C}}>0$ are constants independent of $n,f$.

###  2.3 Cesaro and de la Vallée Poussin means

For any given $n\in{\mathbb{N}}_{0}=\\{0,1,2,...\\}$, let
$S_{n}(w_{\alpha},f)\in{\mathbb{P}}_{n}$ the $n-$th Fourier sum of $f$ in the
orthonormal system $\\{p_{i}(w_{\alpha})\\}_{i=0,1,..}$, namely

$S_{n}(w_{\alpha},f,x)=\sum_{i=0}^{n}c_{i}(f)p_{i}(w_{\alpha},x)$

with Fourier coefficients defined as

$c_{i}(f)=\int_{0}^{+\infty}f(t)p_{i}(w_{\alpha},t)w_{\alpha}(t)dt.$ (14)

For all $n\in{\mathbb{N}}=\\{1,2,....\\}$, we recall the Cesaro means are
defined as arithmetic means of the first $n-1$ Fourier sums, i.e.

$\sigma_{n}(w_{\alpha},f,x)=\frac{1}{n}\sum_{k=0}^{n-1}S_{k}(w_{\alpha},f,x)=\sum_{i=0}^{n-1}\mu_{n,i}^{m}c_{i}(f)p_{i}(w_{\alpha},x),$
(15)

In [16] simple necessary and sufficient conditions have been stated in order
that the map $\sigma_{n}(w_{\alpha}):C_{u}\to C_{u}$ is uniformly bounded
w.r.t. $n\in{\mathbb{N}}$.

###### Theorem 2.3

[16, Theorems 1 and 5]
For all $\alpha>-1$, let $w_{\alpha}(x)=e^{-x}x^{\alpha}$ and
$u(x)=x^{\beta}\sqrt{w_{\alpha}(x)}$. The map $\sigma_{n}(w_{\alpha}):C_{u}\to
C_{u}$ is uniformly bounded w.r.t. $n\in{\mathbb{N}}$, i.e. we have

$\|\sigma_{n}(w_{\alpha},f)u\|\leq{\mathcal{C}}\|fu\|,\quad\forall f\in
C_{u},\quad\forall
n\in{\mathbb{N}},\qquad{\mathcal{C}}\neq{\mathcal{C}}(n,f),$ (16)

if and only if the following bounds are satisfied

$-\min\left\\{\frac{\alpha}{2},\frac{1}{4}\right\\}<\beta<1+\min\left\\{\frac{\alpha}{2},\frac{1}{4}\right\\},\qquad\mbox{and}\qquad-\frac{1}{2}\leq\beta\leq\frac{7}{6}$
(17)

Now, besides $n\in{\mathbb{N}}$, let us take an additional parameter
$m\in{\mathbb{N}}$ such that $0<m<n$, and, for a given function $f$, let us
consider the VP means, namely the following delayed arithmetic means of
Fourier sums

$\mathcal{V}_{n}^{m}(w_{\alpha},f,x)=\frac{1}{2m}\sum_{k=n-m}^{n+m-1}S_{k}(w_{\alpha},f,x),$
(18)

From a computational point of view, this is equivalent to

$\mathcal{V}_{n}^{m}(w_{\alpha},f,x)=\sum_{i=0}^{n+m-1}\mu_{n,i}^{m}c_{i}(f)p_{i}(w_{\alpha},x),$
(19)

that is a filtered version of $S_{n+m-1}(w_{\alpha},f)$ with filter
coefficients $\mu_{n,i}^{m}$ defined as

$\mu_{n,i}^{m}:=\left\\{\begin{array}[]{cl}1&\mbox{if}\quad
i=0,\ldots,n-m,\\\\[7.22743pt] \displaystyle\frac{n+m-i}{2m}&\mbox{if}\quad
n-m+1\leq i\leq n+m-1.\\\ \end{array}\right.$ (20)

The filtered VP polynomial
$\mathcal{V}_{n}^{m}(w_{\alpha},f)\in{\mathbb{P}}_{n+m-1}$ can be also
rewritten in integral form as

$\mathcal{V}_{n}^{m}(w_{\alpha},f,x)=\int_{0}^{+\infty}f(t)v_{n}^{m}(x,t)w_{\alpha}(t)dt,$
(21)

where

$v_{n}^{m}(x,t)=\frac{1}{2m}\sum_{r=n-m}^{n+m-1}K_{r}(x,t),\qquad\mbox{and}\qquad
K_{r}(x,t)=\sum_{i=0}^{r}p_{i}(w_{\alpha},x)p_{i}(w_{\alpha},t)$

are the VP and Darboux kernels, respectively.

Note that $v_{n}^{m}(x,t)=v_{n}^{m}(t,x)$ is a polynomial of degree $n+m-1$ in
both the variables that can be represented as

$v_{n}^{m}(x,t)=\sum_{i=0}^{n+m-1}\mu_{n,i}^{m}p_{i}(w_{\alpha},x)p_{i}(w_{\alpha},t).$

An advantage of taking VP means instead of Cesaro means is given by the
following invariance property

$\mathcal{V}_{n}^{m}(w_{\alpha},P)=P,\quad\forall P\in{\mathbb{P}}_{n-m}.$
(22)

Such preservation property allows us to investigate the approximation provided
by $\mathcal{V}_{n}^{m}(w_{\alpha},f)$ in $C_{u}$, by estimating the norm of
the operator map $\mathcal{V}_{n}^{m}(w_{\alpha}):C_{u}\to C_{u}$, that is

$\|\mathcal{V}_{n}^{m}(w_{\alpha})\|_{C_{u}}=\sup_{\|fu\|=1}\|\mathcal{V}_{n}^{m}(w_{\alpha},f)u\|=\sup_{x\geq
0}\left[u(x)\int_{0}^{+\infty}|v_{n}^{m}(x,t)|\frac{w_{\alpha}(t)}{u(t)}dt\right].$
(23)

To this aim, we introduce the following notation concerning the degree
parameters $m,n\in{\mathbb{N}}$ defining the VP mean:

$\displaystyle m\sim n$ iff $\displaystyle
m<n\leq{\mathcal{C}}m,\qquad\qquad\mbox{holds with ${\mathcal{C}}>1$
independent of $n,m$},$ (24) $\displaystyle m\approx n$ iff $\displaystyle
m<c_{1}m\leq n\leq c_{2}m,\quad\mbox{holds with $c_{2}\geq c_{1}>1$
independent of $m,n$.}$ (25)

Note that $m\sim n$ agrees with the notation introduced in Section 2 while
$m\approx n$ is a stronger condition that ensures $(n-m)\sim n$ holds too.

If we take the additional parameter $m$ such that $m\sim n$ then from Theorem
2.3 we easily deduce the uniform boundedness of VP operator
$\mathcal{V}_{n}^{m}(w_{\alpha})$ in $C_{u}$ for suitable choices of the
weights $w_{\alpha}$ and $u$. More precisely, we have the following

###### Theorem 2.4

Let the weight function $w_{\alpha}(x)=e^{-x}x^{\alpha}$ and
$u(x)=e^{-\frac{x}{2}}x^{\gamma},$ satisfy the following bounds

$\max\left\\{\frac{\alpha}{2}-\frac{1}{4},\
0\right\\}<\gamma<\min\left\\{\frac{\alpha}{2}+\frac{7}{6},\
\alpha+1\right\\}$ (26)

then, for any couple of positive integers $m\sim n$, the map
$\mathcal{V}_{n}^{m}(w_{\alpha}):C_{u}\to C_{u}$ is uniformly bounded w.r.t.
$n$ and $m$, i.e.

$\|\mathcal{V}_{n}^{m}(w_{\alpha},f)u\|\leq{\mathcal{C}}\|fu\|,\quad\forall
f\in C_{u},\quad\forall m\sim n,\qquad{\mathcal{C}}\neq{\mathcal{C}}(n,m,f).$
(27)

Proof of Theorem 2.4 Taking into account that

$\mathcal{V}_{n}^{m}(w_{\alpha},f)=\frac{n+m}{2m}\sigma_{n+m}(w_{\alpha},f)-\frac{n-m}{2m}\sigma_{n-m}(w_{\alpha},f),$

the statement follows by using the assumption (24) that implies

$\frac{n+m}{2m}\leq\frac{{\mathcal{C}}+1}{2},\qquad\mbox{and}\qquad\frac{n-m}{2m}\leq\frac{{\mathcal{C}}-1}{2},\qquad{\mathcal{C}}\neq{\mathcal{C}}(n,m),$

and applying Theorem 2.3 with $\beta=\gamma-\frac{\alpha}{2}$, that satisfies
(17) by virtue of (26). $\diamondsuit$

Hence, concerning the error estimate the following result can be deduced

###### Theorem 2.5

Let the weight function $w_{\alpha}(x)=e^{-x}x^{\alpha}$ and
$u(x)=e^{-\frac{x}{2}}x^{\gamma},$ satisfy (26). Then, for any $f\in C_{u}$
and for all $m\sim n$, we have

$E_{n+m-1}(f)_{u}\leq\|[f-\mathcal{V}_{n}^{m}(w_{\alpha},f)]u\|\leq{\mathcal{C}}E_{n-m}(f)_{u},\qquad{\mathcal{C}}\neq{\mathcal{C}}(n,m,f),$
(28)

Moreover, if $m\approx n$ then we get

$\|f-\mathcal{V}_{n}^{m}(w_{\alpha},f)\|_{C_{u}}\sim
E_{n}(f)_{u},\qquad\forall f\in C_{u}$ (29)

with the constants in $\sim$ independent of $n,m,f$

Proof of Theorem 2.5

The first inequality in (28) trivially follows from
$\mathcal{V}_{n}^{m}(w_{\alpha},f)\in{\mathbb{P}}_{n+m-1}$.

The second inequality in (28) can be deduced by means of (22) and (27) that
yield

$\|[f-\mathcal{V}_{n}^{m}(w_{\alpha},f)]u\|\leq\|(f-P)u\|+\|\mathcal{V}_{n}^{m}(w_{\alpha},\
f-P)u\|\leq{\mathcal{C}}\|(f-P)u\|,\qquad\forall P\in{\mathbb{P}}_{n-m}$

and the statement follows by taking, at both the sides, the infimum w.r.t.
$P\in{\mathbb{P}}_{n-m}$. $\diamondsuit$

##  3 VP filtered discrete approximation

In the previous section, we have seen that for any $f\in C_{u}$ and suitable
choices of $w_{\alpha}$, the polynomials
$\\{\mathcal{V}_{n}^{m}(w_{\alpha},f)\\}_{m\approx n}$ provide a near–best
approximation of $f$. However, their construction requires the knowledge of
the Fourier coefficients of the function $f$. To overcome this problem, in
this section we introduce the polynomial ${V}_{n}^{m}(w_{\alpha},f)$, that we
define as a discrete version of $\mathcal{V}_{n}^{m}(w_{\alpha},f)$. To be
more precise, with $j=j(n)$ defined in (4), we apply the Gauss-Laguerre rule
in (12) to the coefficients $c_{i}(f)$ in (14), getting

$c_{n,i}^{(j)}(f)=\sum_{k=1}^{j}f(x_{k})p_{i}(w_{\alpha},x_{k})\lambda_{n,k}(w_{\alpha})$
(30)

Hence, similarly to (19) we define the following discrete VP filtered
approximation polynomials

${V}_{n}^{m}(w_{\alpha},f,x)=\sum_{i=0}^{n+m-1}\mu_{n,i}^{m}c_{n,i}^{(j)}(f)p_{i}(w_{\alpha},x),\qquad
0<m<n$ (31)

that can be equivalently written as

${V}_{n}^{m}(w_{\alpha},f,x)=\sum_{k=1}^{j}f(x_{k})\Phi_{n,k}^{m}(x),\qquad
x\geq 0$ (32)

with $j=j(n)$ defined in (4), and

$\Phi_{n,k}^{m}(x)=\lambda_{n,k}(w_{\alpha})v_{n}^{m}(x,x_{k})=\lambda_{n,k}(w_{\alpha})\sum_{i=0}^{n+m-1}\mu_{n,i}^{m}p_{i}(w_{\alpha},x_{k})p_{i}(w_{\alpha},x).$
(33)

In Fig. 1 we show, as an example, the graphics of the fundamental polynomial
$\Phi_{n,k}^{m}(x)$, for $n$ and $k$ fixed, and $m$ varying. As we can see for
increasing $m$ the oscillations of $\Phi_{n,k}^{m}(x)$ progressively dampen.
The major oscillating behavior is presented by the fundamental Lagrange
polynomial $\ell_{n,k}(x)$ that are plotted in the same figure.

Figure 1:  Plots of the fundamental polynomials $\ell_{n,k}(x)$ and
$\Phi_{n,k}^{m}(x)$ for $\alpha=-0.5$, $n=15$, $k=7$, and $m=\lfloor\theta
n\rfloor$

We remark that, similarly to $\mathcal{V}_{n}^{m}(w_{\alpha},f)$, for any
function $f$ we have ${V}_{n}^{m}(w_{\alpha},f)\in{\mathbb{P}}_{n+m-1}$, but
in the case $f=P\in{\mathbb{P}}_{n-m}$ the invariance property (22) does not
apply to $V_{n}^{m}(w_{\alpha},f)$.

It can be easily proved that the norm of the operator map
${V}_{n}^{m}(w_{\alpha}):C_{u}\to C_{u}$ takes the following form

$\|{V}_{n}^{m}(w_{\alpha})\|_{C_{u}}=\sup_{\|fu\|=1}\|{V}_{n}^{m}(w_{\alpha},f)\|=\sup_{x\geq
0}\sum_{k=1}^{j}\left|\Phi_{n,k}^{m}(x)\right|\frac{u(x)}{u(x_{k})}.$ (34)

As the number of nodes increases, the behavior of such norm is an important
measure of the conditioning, due to the linearity of $V_{n}^{m}(w_{\alpha})$.
Under the same assumptions of Theorem 2.4, the next theorem states that also
the discrete filtered VP operator ${V}_{n}^{m}(w_{\alpha}):C_{u}\to C_{u}$ is
uniformly bounded w.r.t any $n\sim m$.

###### Theorem 3.1

If the weights $w_{\alpha}(x)=e^{-x}x^{\alpha}$ and
$u(x)=e^{-\frac{x}{2}}x^{\gamma}$ satisfy (26) then, for any $m\sim n$, the
map ${V}_{n}^{m}(w_{\alpha}):C_{u}\to C_{u}$ is uniformly bounded w.r.t. $n$
and $m$, i.e.

$\|{V}_{n}^{m}(w_{\alpha},f)u\|\leq{\mathcal{C}}\|fu\|,\quad\forall f\in
C_{u},\qquad{\mathcal{C}}\neq{\mathcal{C}}(n,m,f).$ (35)

Proof of Theorem 3.1

By Remez-type polynomial inequality [2], since
${V}_{n}^{m}(w_{\alpha},f)\in{\mathbb{P}}_{n+m-1}$, we have

$\displaystyle\|{V}_{n}^{m}(w_{\alpha},f)u\|$ $\displaystyle\leq$
$\displaystyle{\mathcal{C}}\max_{x\in\mathcal{A}_{n}}\left|{V}_{n}^{m}(w_{\alpha},f,x)u(x)\right|,$
$\displaystyle\quad\mathcal{A}_{n}$ $\displaystyle=$
$\displaystyle[a_{n,m},b_{n,m}],\quad a_{n,m}\sim\frac{c}{n+m-1},\quad
b_{n,m}\sim 4(n+m)$

by which

$\|{V}_{n}^{m}(w_{\alpha},f)u\|\leq{\mathcal{C}}\|fu\|\left(\max_{x\in\mathcal{A}_{n}}\sum_{k=1}^{j}\left|\Phi_{n,k}^{m}(x)\right|\frac{u(x)}{u(x_{k})}\right)$
(36)

Hence, our aim is to prove that

$\Sigma_{n,m}(x):=\sum_{k=1}^{j}\left|\Phi_{n,k}^{m}(x)\right|\frac{u(x)}{u(x_{k})}\leq{\mathcal{C}},\qquad\forall
x\in\mathcal{A}_{n},\qquad{\mathcal{C}}\neq{\mathcal{C}}(n,m,x)$ (37)

Start from (33) and use

$\lambda_{n,k}(w_{\alpha})\sim\Delta x_{k}w_{\alpha}(x_{k}),\quad\Delta
x_{k}=x_{k+1}-x_{k},$ (38)

to obtain

$\Sigma_{n,m}(x)\leq u(x)\sum_{k=1}^{j}\Delta
x_{k}\frac{w_{\alpha}(x_{k})}{u(x_{k})}\left|v_{n}^{m}(x,x_{k})\right|,\qquad\forall
x\in\mathcal{A}_{n},$ (39)

with $j$ defined in (4).

Now, let us recall that the following Marcinkiewicz-type inequality was proved
in [15, Lemma 2.5] for an arbitrary polynomial $P\in{\mathbb{P}}_{nl}$ with
$l\in{\mathbb{N}}$

$\sum_{k=1}^{j}\Delta
x_{k}|(Pu)(x_{k})|\leq{\mathcal{C}}\int_{x_{1}}^{4m\rho_{1}}|P(y)|u(y)dy,\quad{\mathcal{C}}\neq{\mathcal{C}}(m,P).$
(40)

where $0<\rho<\rho_{1}<1$, and ${\mathcal{C}}\neq{\mathcal{C}}(n)$.

By applying (40) to the the polynomial
$P(y)=v_{n}^{m}(x,y)\in{\mathbb{P}}_{2n}$ and the weights in (39), we get

$\Sigma_{n,m}(x)\leq{\mathcal{C}}\int_{x_{1}}^{4n\rho_{1}}\frac{w_{\alpha}(t)}{u(t)}\left|v_{n}^{m}(x,t)\right|dt\leq{\mathcal{C}}\int_{0}^{+\infty}\frac{w_{\alpha}(t)}{u(t)}\left|v_{n}^{m}(x,t)\right|dt.$
(41)

Consequently, by (23) and Theorem 2.4, we deduce

$\Sigma_{n,m}(x)\leq{\mathcal{C}}u(x)\int_{0}^{+\infty}\frac{w_{\alpha}(t)}{u(t)}\left|v_{n}^{m}(x,t)\right|dt\leq{\mathcal{C}}\|\mathcal{V}_{n}^{m}(w_{\alpha})\|_{C_{u}}\leq{\mathcal{C}},$

i.e. (37) holds and the theorem follows. $\diamondsuit$

Finally, by the next theorem we state that for suitable weight functions, the
VP polynomials $V_{n}^{m}(w_{\alpha},f)$, similarly to their continuous
version $\mathcal{V}_{n}^{m}(w_{\alpha},f)$ uniformly converge to $f$ in the
space $C_{u}$, with a near–best convergence order. Hence, in conclusion, the
VP filtered polynomials $V_{n}^{m}(w_{\alpha},f)$ are simpler to compute than
the polynomials $\mathcal{V}_{n}^{m}(w_{\alpha},f)$ and share with the latter
a near–best convergence rate in the spaces $C_{u}$.

###### Theorem 3.2

Let the weight function $w_{\alpha}(x)=e^{-x}x^{\alpha}$ and
$u(x)=e^{-\frac{x}{2}}x^{\gamma},$ satisfy (26). Then, for any $f\in C_{u}$
and for all $m\sim n$, we have

$\|(f-{V}_{n}^{m}(w_{\alpha},f))u\|\leq{\mathcal{C}}\left(E_{q}(f)_{u}+e^{-An}\|fu\|\right),\quad
q=\min\left\\{n-m,\ \left\lfloor
n\frac{\rho}{(1+\rho)}\right\rfloor\right\\},$ (42)

with $0<\rho<1$ given by (4) and ${\mathcal{C}},A>0$ independent of $n,m,f.$
Moreover, if $m\approx n$ then we get

$\|f-V_{n}^{m}(w_{\alpha},f)\|_{C_{u}}\sim E_{n}(f)_{u},\qquad\forall f\in
C_{u}$ (43)

with the constants in $\sim$ independent of $n,m,f$.

Proof of Theorem 3.2

Let $P^{*}\in{\mathbb{P}}_{n-m}$ be the polynomial of best approximation of
$f\in C_{u}$, i.e., let

$E_{n-m}(f)_{u}=\inf_{P\in{\mathbb{P}}_{n-m}}\|f-P\|_{C_{u}}=\|f-P^{*}\|_{C_{u}}.$
(44)

Hence, consider the following decomposition

$\|(f-{V}_{n}^{m}(w_{\alpha},f))u\|\leq\|[f-P^{*}]u\|+\|{V}_{n}^{m}(w_{\alpha},f-P^{*})u\|+\|[P^{*}-{V}_{n}^{m}(w_{\alpha},P^{*})]u\|.$

By applying Theorem 3.1 and (44) we deduce

$\|(f-{V}_{n}^{m}(w_{\alpha},f))u\|\leq{\mathcal{C}}E_{n-m}(f)_{u}+\|[P^{*}-{V}_{n}^{m}(w_{\alpha},P^{*})]u\|$
(45)

In order to estimate the last addendum, we note that using (22) and (10)–(11)
we can write

$\displaystyle P^{*}(x)$ $\displaystyle=$
$\displaystyle\mathcal{V}_{n}^{m}(w_{\alpha},P^{*},x)=\int_{0}^{+\infty}P^{*}(t)v_{n}^{m}(x,t)w_{\alpha}(t)dt$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}P^{*}(x_{k})v_{n}^{m}(x,t)\lambda_{n,k}(w_{\alpha})=\sum_{k=1}^{n}P^{*}(x_{k})\Phi_{n,k}^{m}(x)$

On the other hand, by (32), we have

$V_{n}^{m}(w_{\alpha},P^{*},x)=\sum_{k=1}^{j}P^{*}(x_{k})\Phi_{n,k}^{m}(x),\qquad
j=\min\\{k\ :\ x_{k}\geq 4n\rho\\}$

with $0<\rho<1$ arbitrarily fixed. Thus, for any $x\geq 0$, it is

$\displaystyle|P^{*}(x)-V_{n}^{m}(w_{\alpha},P^{*},x)|u(x)=\left|\sum_{k=j+1}^{n}P^{*}(x_{k})\Phi_{n,k}^{m}(x)\right|u(x)$
(46) $\displaystyle\leq$
$\displaystyle\sum_{k=j+1}^{n}|P^{*}(x_{k})||\Phi_{n,k}^{m}(x)|u(x)$
$\displaystyle\leq$
$\displaystyle\max_{k>j}|(P^{*}u)(x_{k})|\left(\sum_{k=j+1}^{n}\left|\Phi_{n,k}^{m}(x)\right|\frac{u(x)}{u(x_{k})}\right)$
$\displaystyle\leq$
$\displaystyle\|P^{*}u\|_{(4n\rho,+\infty)}\left(\max_{x\geq
0}\sum_{k=1}^{n}\left|\Phi_{n,k}^{m}(x)\right|\frac{u(x)}{u(x_{k})}\right)$
$\displaystyle=$ $\displaystyle\|P^{*}u\|_{(4n\rho,+\infty)}\
\|{V}_{n}^{m}(w_{\alpha})\|_{C_{u}}$ $\displaystyle\leq$
$\displaystyle{\mathcal{C}}\|P^{*}u\|_{(4n\rho,+\infty)}$

where in the last two lines we have applied (34) and Theorem 3.1,
respectively.

Finally, we recall that for sufficiently large $n$, we have for any fixed
positive $0<\rho<1$ [6, Proposition 2.1]

$\|fu\|_{[4n\rho,+\infty)}\leq
E_{N}(f)_{u}+{\mathcal{C}}e^{-An}\|fu\|,\qquad\forall f\in C_{u}$ (47)

where $N=\left\lfloor\frac{\rho}{1+\rho}n\right\rfloor,$ and ${\mathcal{C}},A$
positive constants independent of $m,f.$

Consequently, by (44) and (47), we get

$\displaystyle\|P^{*}u\|_{(4n\rho,+\infty)}$ $\displaystyle\leq$
$\displaystyle\|(P^{*}-f)u\|_{(4n\rho,+\infty)}+\|fu\|_{(4n\rho,+\infty)}$
$\displaystyle\leq$
$\displaystyle{\mathcal{C}}E_{n-m}(f)_{u}+E_{M}(f)_{u}+{\mathcal{C}}e^{-An}\|fu\|$

and the statement follows combining last bound with (45) and (46).
$\diamondsuit$

##  4 Numerical experiments

In this section, we propose some tests to assess the performance of the
truncated VP approximation by making comparisons with the truncated Lagrange
approximation, under different points of view. To be more precise we will
propose tests about

  1. 1.

the approximation of functions with different smoothness degree,

  2. 2.

the behavior of the weighted Lebesgue constants for different choices of the
parameters $\alpha$ and $\gamma$.

In what follows we will denote by $Y$ a sufficiently large uniform mesh of
points in a finite interval $(0,a)$ with $a>0$ sufficiently large. For any
$x\in Y$, we consider the following weighted and unweighted error functions

$e_{n,m}^{VP}(f,x)=|V_{n}^{m}(w_{\alpha},f,x)-f(x)|u(x),\quad\widetilde{e}_{n,m}^{VP}(f,x)=|V_{n}^{m}(w_{\alpha},f,x)-f(x)|$
$e_{N}^{Lag}(f,x)=|L_{N+1}^{*}(w_{\alpha},f,x)-f(x)|u(x),\quad\widetilde{e}_{N}^{Lag}(f,x)=|L_{N+1}^{*}(w_{\alpha},f,x)-f(x)|$

where $L_{N+1}^{*}(w_{\alpha},f,x)$ is the truncated interpolating polynomial
interpolating $f$ at the nodes $\\{x_{N,k}\\}_{k=1}^{N}\cup\\{4N\\}$, being
$\\{x_{N,k}\\}_{k=1}^{N}$ the zeros of $p_{N}(w_{\alpha})$. We will consider
two cases: $N=n$ and $N=n+m-1$. In the first case, $L_{n+1}^{*}(w_{\alpha},f)$
and $V_{n}^{m}(w_{\alpha},f)$ require the same number of function evaluations,
but they have different degrees. In the second case,
$L_{n+m}^{*}(w_{\alpha},f)$ and $V_{n}^{m}(w_{\alpha},f)$ have the same
degree, but $L_{n+m}^{*}(w_{\alpha},f)$ uses a greater number of function
evaluations.

Moreover, we will compute the maximum weighted errors given by

$\mathcal{E}_{n,m}^{VP}:=\max_{y\in
Y}\biggm{(}e_{n,m}^{VP}(f,y)\biggm{)},\qquad\mbox{and}\mathcal{E}_{N+1}^{Lag}:=\max_{x\in
Y}\biggm{(}e_{N}^{Lag}(f,y)\biggm{)},$ (48)

where $N\in\\{n,n+m-1\\}$

###  4.1 Approximation of functions

In the following we will consider six examples concerning the approximation of
six test functions having different smoothness.
For increasing values of $n$, we report the values of the maximum weighted
errors in tables where the first two columns display the values of $n$ and $m$
defining the VP polynomial $V_{n}^{m}(w_{\alpha},f)$, the 3rd column shows the
number of function evaluations for computing both the polynomials
$V_{n}^{m}(w_{\alpha},f)$ and $L_{n+1}^{*}(w_{\alpha},f)$, in the 4th and 5th
columns we find the errors $\mathcal{E}_{n,m}^{VP}$ and
$\mathcal{E}_{n+1}^{Lag}$, and, finally, in the 6th and 7th columns the number
of function evaluations required for $L_{n+m}^{*}(w_{\alpha},f)$ and the
corresponding error $\mathcal{E}_{n+m}^{Lag}$ are reported.
Moreover, in order to show the quality of the pointwise approximation, we plot
the unweighted error functions $\tilde{e}^{VP}_{n,m}(f,x)$ and
$\tilde{e}^{Lag}_{n+1}(f,x)$ for fixed $n$ and $m$, letting x variable in a
meaningful compact interval.

###### Example 1

Let

$\alpha=-0.4,\quad\gamma=0.05\quad f_{1}(x)=\exp{(x/4)}$

The function $f_{1}$ is analytic and, according to the theory, the maxima
weighted errors quickly decrease, as shown in Table 1. The unweighted error
functions $\tilde{e}^{VP}_{n,m}(f_{1},x)$ and $\tilde{e}^{Lag}_{n+1}(f_{1},x)$
are plotted in Figure 2.

$n$ | $m$ | # f.eval | $\mathcal{E}_{n,m}^{VP}$ | $\mathcal{E}_{n}^{Lag}$ | # f.eval | $\mathcal{E}_{n+m}^{Lag}$
---|---|---|---|---|---|---
20 | 6 | 19 | 8.00e-08 | 1.80e-05 | 25 | 2.21e-07
70 | 7 | 58 | 2.70e-14 | 2.76e-14 | 61 | 8.90e-14
120 | 12 | 79 | 5.83e-14 | 5.77e-14 | 83 | 9.14e-14
170 | 51 | 96 | 4.49e-14 | 5.86e-14 | 110 | 3.02e-13
220 | 22 | 110 | 6.25e-14 | 6.55e-14 | 115 | 7.07e-13
Table 1:  Maxima weighted errors induced by Lagrange and VP polynomials for
Example 1 Figure 2:  Example 1: Plots of $\tilde{e}^{VP}_{n,m}(f_{1},x)$ and
$\tilde{e}^{Lag}_{n+1}(f_{1},x)$ for $n=620$, $m=186$.

###### Example 2

Consider

$\displaystyle\alpha=0.5,\quad\gamma=0.5\quad
f_{2}(x)=\frac{|x-1|^{\frac{11}{2}}}{100+x^{2}}$

In this case, $f_{2}$ belongs to an intermediate space between $W_{1}(u)$ and
$W_{2}(u)$. According to [5, (5.2.7)], the theoretical convergence order is
$\mathcal{O}\left(\frac{1}{(n-m)^{\frac{11}{4}}}\right)$ for
$\mathcal{E}_{n,m}^{VP}$, and $\mathcal{O}\left(\frac{\log
N}{N^{\frac{11}{4}}}\right)$ for $\mathcal{E}_{N}^{Lag}$. As shown in Table 2,
all the numerical errors agree with the theoretical estimates and are
comparable as a consequence of the very slow divergence of the logarithmic
factor. In particular, VP approximation provides a slightly better
approximation than the interpolating polynomial in the case $N=n$. This trend
is little bit reversed when $N=n+m$.

However, in Figure 3, the graphics of the pointwise error function taken for
$n=420,\ m=126$, shows that $V_{n}^{m}(w_{\alpha},f_{2})$ provides a pointwise
approximation better than the one offered by $L_{n+1}^{*}(w_{\alpha},f_{2})$.

$n$ | $m$ | # f.eval | $\mathcal{E}_{n,m}^{VP}$ | $\mathcal{E}_{n}^{Lag}$ | # f.eval | $\mathcal{E}_{n+m}^{Lag}$
---|---|---|---|---|---|---
20 | 6 | 19 | 1.10e-04 | 4.17e-02 | 25 | 2.92e-02
220 | 22 | 96 | 9.78e-08 | 9.23e-08 | 101 | 5.17e-08
420 | 42 | 134 | 1.27e-08 | 1.38e-08 | 140 | 8.54e-09
620 | 310 | 163 | 1.54e-09 | 2.36e-09 | 200 | 1.42e-09
820 | 82 | 188 | 2.47e-09 | 2.53e-09 | 197 | 1.63e-09
1020 | 102 | 210 | 1.31e-09 | 1.31e-09 | 220 | 1.01e-09
1220 | 122 | 230 | 7.28e-10 | 7.39e-10 | 241 | 5.75e-10
Table 2:  Maxima weighted errors induced by interpolating and VP polynomials
for Example 2. Figure 3:  Example 2: Plots of $\tilde{e}^{VP}_{n,m}(f_{2},x)$
and $\tilde{e}^{Lag}_{n+1}(f_{2},x)$ for $n=420$, $m=126$.

###### Example 3

Consider the case

$\displaystyle
f_{3}(x)=\frac{1}{1+100(x-3)^{2}},\quad\alpha=-0.4,\quad\gamma=0.05\quad$

$n$ | $m$ | # f.eval | $\mathcal{E}_{n,m}^{VP}$ | $\mathcal{E}_{n}^{Lag}$ | # f.eval | $\mathcal{E}_{n+m}^{Lag}$
---|---|---|---|---|---|---
20 | 6 | 17 | 2.24e-01 | 2.25e-01 | 20 | 1.13e-01
220 | 22 | 64 | 9.04e-02 | 8.95e-02 | 67 | 1.57e-01
420 | 378 | 89 | 6.98e-02 | 8.34e-02 | 123 | 8.31e-02
620 | 558 | 108 | 6.21e-02 | 7.05e-02 | 149 | 4.12e-02
820 | 738 | 124 | 3.72e-02 | 4.93e-02 | 172 | 2.56e-02
1020 | 102 | 139 | 5.86e-02 | 5.82e-02 | 146 | 5.66e-02
1220 | 1098 | 152 | 2.84e-02 | 3.64e-02 | 209 | 2.87e-02
Table 3:  Maxima weighted errors induced by interpolating and VP polynomials
for Example 3.

Note that $f_{3}\in W_{r}(u)$, $\forall r>0$, and the slow convergence we see
in Table 3 depends on the growth of the norms in the error estimates. For
instance, with $r=5$, $\|f_{3}\|_{W_{5}(u)}\sim 0.5e+12$. Similarly to the
previous example, the maxima errors attained by VP and Lagrange approximation
are almost comparable among them. However, also in this case in Figure 4 are
displayed the absolute pointwise errors taken for $n=420,\ m=378$, and except
a small range around the peak point $x=3$, the errors by
$V_{n}^{m}(w_{\alpha},f)$ are smaller than those by
$L_{n+1}^{*}(w_{\alpha},f)$.

Figure 4:  Example 3: Plots of $\tilde{e}^{VP}_{n,m}(f_{3},x)$ and
$\tilde{e}^{Lag}_{n+1}(f_{3},x)$ for $n=420$ and $m=378$

###### Example 4

Let be

$\displaystyle\alpha=0.6,\quad\gamma=0.6\quad f_{4}(x)=|\cos(\pi-x)|$

$n$ | $m$ | # f.eval | $\mathcal{E}_{n,m}^{VP}$ | $\mathcal{E}_{n}^{Lag}$ | # f.eval | $\mathcal{E}_{n+m}^{Lag}$
---|---|---|---|---|---|---
20 | 2 | 19 | 1.37e-01 | 1.36e-01 | 21 | 1.16e-01
220 | 22 | 79 | 3.24e-02 | 3.25e-02 | 83 | 3.95e-02
420 | 84 | 108 | 2.57e-02 | 2.63e-02 | 121 | 1.65e-02
620 | 62 | 137 | 1.75e-02 | 1.73e-02 | 141 | 1.23e-02
820 | 82 | 155 | 1.69e-02 | 1.67e-02 | 152 | 1.63e-02
1020 | 102 | 173 | 1.76e-02 | 1.74e-02 | 182 | 1.25e-02
1220 | 488 | 190 | 8.31e-03 | 8.01e-03 | 225 | 9.35e-03
Table 4:  Maxima weighted errors induced by interpolating and VP polynomials
for Example 4.

In this case we have $f_{4}\in W_{1}(u)$, and the degree of approximation is
very poor, since, according to (2), VP and Lagrange approximation rates are
$\mathcal{O}\left(\frac{1}{\sqrt{n-m}}\right),\mathcal{O}\left(\frac{\log
n}{\sqrt{n}}\right)$, respectively. The numerical results displayed in Table 4
and Figure 5 agree with the theory and, as the previous cases, we see that the
maxima errors are almost comparable but, concerning the local approximation,
VP approximation error behaves better away from the singularity $x=\pi/2$.

Figure 5:  Example 4: Plots of $\tilde{e}^{VP}_{n,m}(f_{3},x)$ and
$\tilde{e}^{Lag}_{n+1}(f_{3},x)$ for $n=500$ and $m=150$.

###### Example 5

Now consider the following function plotted in Figure 6 (left–hand side)

$f(x)=\frac{1}{1+100(x-0.5)^{2}}+\frac{1}{1+1000\sqrt{x^{2}+0.5}},\quad\gamma=0.5$

To evidence the improvement in the pointwise approximation achieved by VP
polynomials, on the right–hand side of Fig. 6 we report the plots of both
Lagrange and VP polynomials when $n=400$, $\alpha=0$ and $m=200.$

Figure 6:  Example 5: graphic of the function $f_{5}$ (left) and plots of the
VP and Lagrange polynomials (right) for $\alpha=0$, $n=400$ and $m=200.$

###### Example 6

This test highlights the feature offered by VP polynomials when we want to
approximate functions of bounded variation having a jump discontinuity.
Consider the function

$f_{6}(x)=\begin{cases}x,&x<1\\\ x+2,&x\geq 1\end{cases}$

Figure 7 displays the graphics of $f_{6}u$ with
$u(x)=e^{-\frac{x}{2}}\sqrt{x}$ (black line) and the approximations by the
weighted Lagrange polynomial $uL_{n+1}^{*}(w_{\alpha},f_{6})$ (orange line)
and the weighted $uV_{n}^{m}(w_{\alpha},f_{6})$ (blue line), for $n=620$ and
$m=310$.

As expected, the Gibbs phenomenon appears. In fact, as the graphics in Figure
7 show, the Lagrange polynomial presents damped oscillations that persist even
when moving away from the jump while the VP polynomial induces a reduction of
the Gibbs phenomenon as $m$ increases. This is confirmed by Figure 8
displaying the unweighted absolute error induced by Lagrange polynomial and VP
polynomial

Figure 7:  $n=620$, $\alpha=0.5$, $\gamma=0.5$: the function $f_{6}u$ and
weighted VP polynomials for $m=298$, $m=496$ (left), and the weighted Lagrange
polynomial of degree $n$ (right).  Figure 8:  $n=620$, $m=310$, $\alpha=0.5$.

###  4.2 Behavior of the weighted Lebesgue constants

In what follows we show the behavior of the Lebesgue constants related to
Lagrange and VP approximation for different choices of $\alpha$ and $\gamma$,
as $n$ increases. From the theory sufficient conditions relating $\alpha$ and
$\gamma$ are known to ensure the optimal logarithmic grow in the Lagrange case
(cf. (8)), and the uniform boundedness in the VP case (cf. (26)). In the
latter case, it is also required that $m\approx n$ to get a near best
approximation order. To this aim, we take $m=\lfloor n\theta\rfloor$, being
$\theta\in(0,1)$ chosen as $\theta\in\\{0.1,0.2,\dots,0.9.\\}$. Hence for
these values of $\theta$ and $n$ increasing, the behavior of the Lebesgue
constants is displayed in Figures 9-10 for several couples of
$(\alpha,\gamma)$ both in the ranges to obtain optimal Lebesgue constants for
VP and Lagrange operators.

Figure 9: Lebesgue constants for $\alpha=\gamma=0.5$ (left), for for
$\alpha=\gamma=1$ (right).

Figure 10: Lebesgue constants for $\alpha=-0.5,\gamma=0$ (left),
$\alpha=0.7,\gamma=0.6$ (right).

In conclusion, in Figure 11 we consider the case when $\alpha$ and $\gamma$ do
not satisfy the bounds in (26) concerning VP approximation, but they satisfy
those in (8) concerning Lagrange. In this case, even if not predicted by
Theorem 3.1 we see the Lebesgue constants of VP operator are still bounded for
any $\theta$.

Empirically we have detected that the same behavior occurs for other possible
choices of couples of $(\alpha,\gamma)$ s.t.
$\frac{\alpha}{2}+\frac{7}{6}<\gamma\leq\frac{\alpha}{2}+\frac{5}{4}.$ This
observation leads us to conjecture that the hypotheses of Theorem 3.1 can be
extended by replacing the upper bound
$\min\\{\frac{\alpha}{2}+\frac{7}{6},\alpha+1\\}$ by
$\frac{\alpha}{2}+\frac{5}{4}$.

Figure 11: Lebesgue constants for $\alpha=0.5,\gamma=1.43$ (left),
$\alpha=0,\gamma=5/4$ (right).

##  5 Conclusions

We have introduced and studied a new truncated polynomial operator
$V_{n}^{m}(w_{\alpha}):C_{u}\to C_{u}$, involving the values of $f$ at only
the first $j<<n$ zeros of the Laguerre polynomial $p_{n}(w_{\alpha})$.
Differently from $L_{n+1}^{*}(w_{\alpha})$, we have proved that the map
$V_{n}^{m}(w_{\alpha}):C_{u}\to C_{u}$ is uniformly bounded under suitable
conditions on the weights $w_{\alpha}$ and $u$ and in view of this theoretical
improvement, the maxima errors attained by VP approximation are better on
average than those achieved by Lagrange one. However, the differences between
the numerical errors in norm are not so large, since the factor $\log n$ does
not have a strong impact in the error estimate. The situation is instead quite
different about the pointwise errors, especially in approximating smooth
functions presenting some peaks or isolated pathologies (see Examples 4-5).
Indeed, in such cases the additional parameter $m$ can be suitably modulated
to reduce the Gibbs phenomenon. We observe a drastic reduction of the
oscillations in VP polynomials as $m$ increases, and even more w.r.t the
Lagrange polynomial, the latter swinging more than anyone. Such reduction is
valuable also in Example 6 where $f_{6}$ is a bounded variation function. In
these cases it is well–known that Lagrange polynomials present overshoots and
oscillations not only close to the singularities, but also in the smooth part
of the function. The graphics in Fig. 7 show how the Gibbs phenomenon can be
reduced by VP polynomials, for suitable values of the parameter
$\theta\in(0,1).$ We conclude observing that the range where $\alpha,\gamma$
vary for the boundedness of VP polynomials, is partially overlapping with that
the norm of the Lagrange operator logarithmically behaves (compare (8) and
(26)), and in any case the VP range is larger than the Lagrange one. However,
numerical results (see Fig. 11) allow us to conjecture that the range
determined by Thm. 3.1 about the parameters $\alpha,\gamma$ could actually be
stretched. More precisely, we conjecture that the hypothesis
$\max\\{0,\frac{\alpha}{2}-\frac{1}{4}\\}<\gamma\leq\frac{\alpha}{2}+\frac{5}{4}$
could replace (26).

Acknowledgments. This work was partially supported by INdAM-GNCS. The research
has been accomplished within RITA (Research ITalian network on Approximation)

## References

  * [1] M. C. De Bonis, G. Mastroianni, and M. Viggiano.  K-functionals, moduli of smoothness and weighted best approximation on the semiaxis.  1999\.
  * [2] Mhaskar H. N. and E. B. Saff.  Extremal problems for polynomials with Laguerre weights.  Approx. Theory IV, Academic Press, 1983.
  * [3] C. Laurita and G. Mastroianni.  Lp-convergence of lagrange interpolation on the semiaxis.  Acta Math. Hungar, 120(3):249–273, 2008.
  * [4] G. Mastroianni.  Polynomial inequalities, functional spaces and best approximation on the real semiaxis with laguerre weights.  Electronic Transactions on Numerical Analysis, 14:142–151, 2002\.
  * [5] G. Mastroianni and G. V. Milovanovic.  Interpolation Processes Basic Theory and Applications.  Springer Monographs in Mathematics. Springer Verlag, Berlin, 2009.
  * [6] G. Mastroianni and G. V. Milovanovi.̧  Some numerical methods for second-kind Fredholm integral equations on the real semiaxis.  IMA Journal of Numerical Analysis, 29(4):1046–1066, 2009.
  * [7] G. Mastroianni and G. Monegato.  Truncated quadrature rules over $(0,+\infty)$ and Nyström-type methods.  SIAM Journal on Numerical Analysis, 41:1870 –1892, 2003.
  * [8] G. Mastroianni and I. Notarangelo.  Some fourier-type operators for functions on unbounded intervals.  Acta Math. Hungar., 127((4)):347 –375, 2010.
  * [9] G. Mastroianni and D. Occorsio.  Lagrange interpolation at Laguerre zeros in some weighted uniform spaces.  Acta Math. Hungar., 91((1-2)):27–52, 2001.
  * [10] G. Mastroianni and D. Occorsio.  Some quadrature formulae with nonstandard weights.  Jour. of Comput. and Appl. Math., 235:602–614, 2010.
  * [11] G. Mastroianni and J. Szabados.  Polynomial approximation on infinite intervals with weights having inner zeros.  Acta Math. Hungar, 96(3):221–258, 2002.
  * [12] G. Mastroianni and J. Szabados.  Direct and converse polynomial approximation theorems on the real line with weights having zeros,.  volume 283 of Pure Appl. Math. (Boca Raton), pages 287–306. 2007\.
  * [13] G. Mastroianni and W. Themistoclakis.  Uniform approximation on $[-1,1]$ via discrete de la vallée poussin means.  Acta Sci. Math., 74:147–170, 2008.
  * [14] G. Mastroianni and W. Themistoclakis.  Pointwise estimates for polynomial approximation on the semiaxis.  Jour. of Approx. Theory, 162(11):2078–2105, 2010.
  * [15] G. Mastroianni and P. Vértesi.  Fourier sums and lagrange interpolation on $(0,+\infty)$ and $(-\infty,+\infty)$.  volume 283 of Pure Appl. Math. (Boca Raton), pages 307–344. 2007\.
  * [16] E. Poiani.  Mean Cesàro summability of Laguerre and hermite series.  Transactions of the American Mathematical Society, 1972.
  * [17] G. Szegö.  Orthogonal Polynomials, volume 4th Ed.  American Mathematical Society, Providence, RI, 1975.
  * [18] W. Themistoclakis.  Uniform approximation on $[-1,1]$ via discrete de la vallée poussin means.  Numer. Alg., 60:593–612, 2012.