New estimates related with
the best polynomial approximation

Jorge Bustamante$^\ast $

March 21, 2023; accepted: May 2, 2023; published online: July 5, 2023.

In some old results, we find estimates the best approximation $E_{n,p}(f)$ of a periodic function satisfying $f^{(r)}\in \mathbb {L}^p_{2\pi }$ in terms of the norm of $f^{(r)}$ (Favard inequality). In this work, we look for a similar result under the weaker assumption $f^{(r)}\in \mathbb {L}^q_{2\pi }$, with $1{\lt}q{\lt}p{\lt}\infty $. We will present inequalities of the form $E_{n,p}(f)\leq C(n)\Vert D^{(r)}f\Vert _q$, where $D^{(r)}$ is a differential operator. We also study the same problem in spaces of non-periodic functions with a Jacobi weight.

MSC. 42A10, 41A10.

Keywords. Favard inequality; best approximation.

$^\ast $Benemerita Universidad Autonoma de Puebla, Faculty of Physics and Mathematics, Puebla, Mexico, e-mail: jbusta@fcfm.buap.mx.

1 Introduction

Let $C_{2\pi }$ denote the Banach space of all real $2\pi $-periodic continuous functions $f$ defined on the real line $\mathbb {R}$ with the sup norm

\begin{equation*} \Vert f\Vert _\infty =\max \limits _{x\in [-\pi ,\pi ]}\mid f(x)\mid . \end{equation*}

For $1\leq p {\lt}\infty $, the Banach space $\mathbb {L}^p_{2\pi }$ consists of all $2\pi $-periodic, $p$-th power Lebesgue integrable (class of) functions $f$ on $\mathbb {R}$ with the norm

\begin{equation*} \Vert f\Vert _p =\Big(\tfrac {1}{2\pi }\int _{-\pi }^\pi \mid f(x) \mid ^pdx\Big)^{1/p}. \end{equation*}

In order to simplify we write $X^p_{2\pi }=\mathbb {L}^p_{2\pi }$ for $1\leq p {\lt}\infty $ and $X^\infty _{2\pi }=C_{2\pi }$.

Let $\mathbb {T}_n$ denote the family of all trigonometric polynomials of degree non greater that $n$. For $n\in \mathbb {N}_0$ and $f\in X^p_{2\pi }$, the best approximation is defined by

\begin{equation*} E_{n,p}(f)=\inf \limits _{T\in \mathbb {T}_n} \Vert f- T\Vert _p. \end{equation*}

By $W^r_{p,2\pi }$ we mean the family of all functions $f\in X^p_{2\pi }$ such that $f,\ldots , f^{(r-1)}$ are absolutely continuous and $f^{(r)}\in X^p_{2\pi }$.

The following result is known (for instance, see [ 4 , p. 166 ] where other notations were used).

Theorem 1

If $1\leq p \leq \infty $, $r,n\in \mathbb {N}$ and $g\in W^r_{p,2\pi }$, then

\begin{equation} \label{stechkinderivada} E_{n,p}(g)\leq \tfrac {F_r}{(n+1)^r} E_{n,p}(g^{(r)}), \end{equation}

where $F_r$ is the Favard constant.

Here we want to consider the following problem. Is there an estimate similar to 1, if $1\leq q{\lt}p\leq \infty $, $r,n\in \mathbb {N}$, $g\in X^p_{2\pi }$ and $g\in W^r_{q,2\pi }$ (with the necessary adjustment)?

In [ 1 ] , Ganzburg considered a similar problem, when the derivative of a continuous functions is not continuous.

Theorem 2 Ganzburg [ 1 ]

If $f\in C_{2\pi }$ is absolutely continuous and $f'\in \mathbb {L}^1_{2\pi }$, then

\begin{equation*} E_{n,\infty }(f) \le \tfrac {1}{2} E_{n,1}(f’). \end{equation*}

In section 2 we extend the result of Ganzburh to the case when $1{\lt} q{\lt}p{\lt}\infty $, $f\in X^p_{2\pi }$ and $f\in W^r_{q,2\pi }$.

For $1\leq p{\lt}\infty $, we denote by $\mathbb {L}^p_{\alpha ,\beta }$ the space of all measurable functions $f$ satisfying

\begin{equation*} \Vert f\Vert _{p,\alpha ,\beta }=\left\{ \int _{-1}^1 \mid f(x)\mid ^p \varrho _{\alpha ,\beta }(x)dx\right\} ^{1/p} {\lt} \infty , \end{equation*}

where $\varrho _{\alpha ,\beta }(x)=(1-x)^\alpha (1+x)^\beta $ and $\alpha ,\beta {\gt}-1$.

For $n\in \mathbb {N}_0$ and $f\in L^p_{(\alpha ,\beta )}$, define the best approximation of $f$ of order $n$ by

\begin{equation*} E_{n}(f)_{p,\alpha ,\beta }=\inf \Big\{ \Vert f-P_n\Vert _{p,\alpha ,\beta }\, ; \, P_n\in \mathbb {P}_n \Big\} , \end{equation*}

here $\mathbb {P}_n$ is the family of all algebraic polynomials of degree not bigger than $n$.

In section 3, for $1{\lt}q{\lt}p$ and functions $f\in L^p_{(\alpha ,\beta )}$ such that $\mathcal{D}^r_{\alpha ,\beta }(f) \in L^q_{(\alpha ,\beta )}$ (see 8), we estimate $ E_{n}(f)_{p,\alpha ,\beta }$ in terms of $E_{n}(\mathcal{D}^r_{\alpha ,\beta }(f))_{q,\alpha , \beta }$. We will assume that $\alpha \geq \beta \geq -1/2$.

2 The periodical case

Let

\begin{equation} \label{Berfunt} \mathfrak {B}_r(t)=\sum \limits _{k=1}^\infty \tfrac {\cos (kt -r\pi /2)}{k^r} \end{equation}

be the Bernoulli kernel.

Theorem 3

Assume $1{\lt} q{\lt}p{\lt}\infty $ and let $s$ satisfy

\begin{equation} \label{tomalas} \tfrac {1}{q}-\tfrac {1}{p}=1-\tfrac {1}{s}. \end{equation}

If $r\in \mathbb {N}$, $f\in X^p_{2\pi }$ and $f\in W^r_{q,2\pi }$, then

\begin{equation*} E_{n,p}(f)\leq 2\, E_{n,s}(\mathfrak {B}_r) \, \, E_{n,q}(f^{(r)}). \end{equation*}

for each $n\in \mathbb {N}$.

Proof. Fix polynomials $T_n,B_n\in \mathbb {T}_n$ such that

\begin{equation*} E_{n,q}(f^{(r)})=\Vert f^{(r)}-T_n\Vert _q \quad \quad {\rm and} \quad \quad \Vert \mathfrak {B}_r-B_n\Vert _s=E_{n,s}(\mathfrak {B}_r). \end{equation*}

Define

\begin{align*} M_n(f,x)& =\tfrac {1}{\pi }\int _{-\pi }^\pi f^{(r)}(t)B_{n}(x-t)dt,\\ N_n(f,x)& =\tfrac {1}{\pi }\int _{-\pi }^\pi T_n(t)\Big(\mathfrak {B}_r(x-t)-B_{n}(x-t)\Big)dt, \end{align*}

and

\begin{equation*} L_n(f,x)= \tfrac {1}{2\pi }\int _0^{2\pi }f(t)dt + M_n(f,x)+N_n(f,x). \end{equation*}

Since $M_n(f,x)$ and $N_n(f,x)$ are convolution with trigonometric polynomials of degree not bigger than $n$, one has $M_n(f), N_n(f), L_n(f)\in \mathbb {T}_n$.

Recall that (see [ 4 , p. 30 ] )

\begin{equation*} f(x)=\tfrac {1}{2\pi }\int _{-\pi }^{\pi }f(t)dt + \tfrac {1}{\pi }\int _{-\pi }^\pi f^{(r)}(t)\mathfrak {B}_r(x-t)dt. \end{equation*}

Therefore

\begin{align*} & L_{n}(f,x)-f(x)=\\ & = \tfrac {1}{\pi }\int _{-\pi }^\pi f^{(r)}(t)B_{n}(x\! -\! t)\! +\! T_n(t)\Big(\mathfrak {B}_r(x\! -\! t)\! -\! B_{n}(x\! -\! t)\Big)- \Big(f^{(r)}(t)\mathfrak {B}_r(x\! -\! t)\Big)dt\\ & = \tfrac {1}{\pi }\int _{-\pi }^\pi \Big(T_n(t)-f^{(r)}(t)\Big)\Big(\mathfrak {B}_r(x-t)-B_{n}(x-t)\Big)dt. \end{align*}

That is,

\begin{equation} \label{principalnote} \tfrac {1}{2}(L_{n}(f,x)-f(x)) =\Big(T_n-f^{(r)}\Big) * \Big(\mathfrak {B}_r-B_{n}\Big) (x). \end{equation}

Now an application of the Young inequality yields

\begin{equation*} \hspace{10mm}\tfrac {1}{2}\Vert L_{n}(f)-f\Vert _p\leq \Vert T_n-f^{(r)}\Vert _q \, \, \Vert \mathfrak {B}_r-B_{n}\Vert _s = E_{n,s}(\mathfrak {B}_r) \, \, E_{n,q}(f^{(r)}). \hspace{10mm}\square \end{equation*}

In theorem 4 we extend the result of Ganzburg presented in the introduction. We consider $r{\gt}1$, because $\mathfrak {B}_1$ is not a continuous function. On the other hand the series 2 converges uniformly for $r\geq 2$.

Theorem 4

If $r\in \mathbb {N}$, $r{\gt}1$, $1{\lt}p\leq \infty $, $f\in X^p_{2\pi }$ and $f\in W^r_1$, then

\begin{equation*} E_{n,\infty }(f)\leq \, 2 E_{n,p}(\mathfrak {B}_r) \, \, E_{n,1}(f^{(r)}). \end{equation*}

for each $n\in \mathbb {N}$.

Proof. It is known (see [ 4 , p. 36 ] ) that if $f\in X^p_{2\pi }$ and $h\in \mathbb {L}^1$, then $f*h\in X^p_{2\pi }$ and

\begin{equation*} \Vert f*h\Vert _p \leq \Vert h\Vert _1\, \Vert f\Vert _p. \end{equation*}

If $f\in X^p_{2\pi }$ and $f\in W^r_1$, we fix polynomials $T_n,B_n\in \mathbb {T}_n$ such that

\begin{equation*} E_{n,1}(f^{(r)})=\Vert f^{(r)}-T_n\Vert _1 \quad \quad {\rm and} \quad \quad \Vert \mathfrak {B}_r-B_n\Vert _p=E_{n,p}(\mathfrak {B}_r). \end{equation*}

It follows from 4 that

\begin{equation*} \hspace{10mm}E_{n,p}(f)\leq 2 \Vert T_n-f^{(r)}\Vert _1 \, \, \Vert \mathfrak {B}_r-B_{n}\Vert _p =2 E_{n,p}(\mathfrak {B}_r) \, \, E_{n,1}(f^{(r)}). \hspace{10mm}\square \end{equation*}

Remark 5

We have not found a study of the quantities $E_{n,s}(\mathfrak {B}_r)$ for $s{\gt}1$. But, if $r,s{\gt}1$ and $S_n(x)$ is the partial sum of the series 2, then

\begin{align*} E_{n,s}(\mathfrak {B}_r) & \leq \Vert \mathfrak {B}_r-S_n\Vert _s =\Big(\tfrac {1}{2\pi }\int _{-\pi }^\pi \Big| \sum \limits _{k=n+1}^\infty \tfrac {\cos (kx)}{k^r}\Big|^sdx\Big)^{1/s} \\ \nonumber & \leq \Big(\tfrac {1}{2\pi }\int _{-\pi }^\pi \Big(\sum \limits _{k=n+1}^\infty \tfrac {1}{k^r}\Big)^sdx\Big)^{1/s} = \sum \limits _{k=n+1}^\infty \tfrac {1}{k^r}\leq \tfrac {1}{(r-1)(n+1)^{r-1}}. \end{align*}

â–¡

3 The case of non-periodic functions

Let $\{ J_n^{(\alpha ,\beta )}\} $ be the orthogonal system of Jacobi polynomials on $[-1, 1]$ with respect to the weight $\varrho _{\alpha ,\beta }(x)=(1-x)^\alpha (1+x)^\beta $, normalized by the conditions $J_n^{(\alpha ,\beta )}(1)=1$.

If $\alpha \geq \beta \geq -1/2$ and $1\leq p\leq \infty $, it is known that the exists a generalized translation $\tau _t : \mathbb {L}^p_{\alpha ,\beta }\to \mathbb {L}^p_{\alpha ,\beta }$ with the following properties

\begin{equation*} \Vert \tau _t(f)\Vert _{p,\alpha ,\beta }\leq \Vert f\Vert _{p,\alpha ,\beta },\quad \quad t\in [-1,1],\, \, f\in \mathbb {L}^p_{\alpha ,\beta }, \end{equation*}

and

\begin{equation} \label{coefconvu} c_k(\tau _t(f))= c_k(f)\, \, J_k^{(\alpha ,\beta )}(t),\quad \quad k\in \mathbb {N}_0, \end{equation}

where

\begin{equation*} c_k(g)=c_k^{(\alpha ,\beta )}(g) = \int _{-1}^1 g(x)J^{(\alpha ,\beta )}_k (x)\varrho _{ \alpha ,\beta }(x)dx, \quad \quad k \in \mathbb {N}_0. \end{equation*}

for all $g\in \mathbb {L}^1_{\alpha ,\beta }$.

The generalized translation allows us to define a convolution in $\mathbb {L}^p_{\alpha ,\beta }$ by setting

\begin{equation*} (f*g)(x)=\int _{-1}^1\tau _s(f,x)g(s)\varrho _{\alpha , \beta }(s)\, ds. \end{equation*}

If $f\in L^p_{(\alpha ,\beta )}$ and $g\in L^1_{(\alpha ,\beta )}$, for almost every $x\in [-1,1]$ the integral

\begin{equation} \label{AConvolution} (f*g)(x)=\int _{-1}^1 \tau _y(f,x)\, g(y)\, \varrho ^{(\alpha ,\beta )}(y)dy \end{equation}

exists, $f*g\in L^1_{(\alpha ,\beta )}$, $f*g=g*f$ and

\begin{equation} \label{convoinequality} \Vert f*g\Vert _p\, \leq \, \Vert g\Vert _1 \Vert f\Vert _p. \end{equation}

Moreover

\begin{equation*} c_k(f*g)= c_k(f) \, \, c_k(g), \quad \quad k\in \mathbb {N}_0. \end{equation*}

For the properties of the translation quoted above see [ 2 ] and [ 3 ] .

We also need the following property (see [ 5 ] ). Assume $1\leq r,q{\lt} \infty $, $r^{-1}+q^{-1}{\gt} 1$, and $p^{-1}= r^{-1}+q^{-1}-1$. If $\varphi \in L^r_{(\alpha ,\beta )}$ and $g\in L^q _{(\alpha ,\beta )}$, then $\varphi * g \in L^p_{(\alpha ,\beta )}$ and

\begin{equation} \label{Rafalson5} \Vert \varphi * g\Vert _{p,\alpha ,\beta } \leq \Vert \varphi \Vert _{r,\alpha ,\beta }\, \, \Vert g\Vert _{q,\alpha ,\beta }. \end{equation}

As in Rafalson [ 5 ] , let $\Omega ^r_{\alpha ,\beta }$ be the family of all functions $f\in C^{2r-1}(-1,1)$ such that, for $0\leq k\leq r-1$, the function

\begin{equation*} \psi ^k_r(f,x):=\Big((1-x)^{\alpha +r} (1+x)^{\beta +r} f^{(r)}(x)\Big)^{(k)} \end{equation*}

is absolutely continuous in $(-1,1)$ and $\psi ^k_r(f,\pm 1)=0$.

For $f\in \Omega ^r_{\alpha ,\beta }$, define

\begin{equation} \label{Diferealhabeta} \mathcal{D}^r_{\alpha ,\beta }(f,x) =\tfrac {1}{(1-x)^\alpha } \tfrac {1}{(1+x)^\beta } \Big((1-x)^{\alpha +r} (1+x)^{\beta +r} f^{(r)}(x)\Big)^{(k)}. \end{equation}

For $r\in \mathbb {N}$ and $t\in (-1,1)$, consider the function

\begin{equation*} \Phi ^r_{\alpha ,\beta }(t)=C_r\int _{-1}^t\tfrac {(t-z)^{r-1}} {(1-z)^{1+\alpha }(1+z)^{r+\beta }}\int _{-1}^z (1-u)^{\alpha } (1+z)^{r+\beta -1}dudz, \end{equation*}

where

\[ C_r=C(r,\alpha ,\beta ):=\tfrac {(-1)^r}{2^{\alpha +\beta +r}} \tfrac {\Gamma (r+\alpha +\beta +1)}{\Gamma ^2(r) \Gamma (\alpha +1)\Gamma (r+\beta )}. \]

It is known [ 5 , Lemma 8 ] that, if $f\in \Omega ^r_{\alpha ,\beta }$, then

\begin{equation} \label{Rafa8} f(x)-S_{r-1}^{(\alpha ,\beta )}(f,x)= \mathcal{D}^r_{\alpha ,\beta }(f)*\Phi ^r_{\alpha ,\beta }(x) \end{equation}

almost everywhere with respect to the Lebesgue measure. Here $S_{r}^{(\alpha ,\beta )}(f)$ is the partial sum of order $r$ of the Jacobi expansion of $f$.

Let us recall a known result.

Theorem 6

(Rafalson, [ 5 , Th. 1 ] ) Assume $\alpha \geq \beta \geq -1$, $r,n\in \mathbb {N}$, $n\geq r-1$, and one of the following conditions hold:

\[ \hspace{-20mm}{\rm (i)} \hspace{20mm}r{\gt}\alpha +1, \quad and \quad 1\leq p\leq \infty , \]

\[ \hspace{-20mm}{\rm (ii)} \hspace{20mm} r=\alpha +1, \quad and \quad 1\leq p {\lt} \infty , \]

\[ \hspace{-10mm}{\rm (iii)} \hspace{20mm}r{\gt} \alpha +1, \quad and \quad 1\leq r {\lt} \tfrac {1+\alpha }{1+\alpha -r}. \]

If $f\in \Omega ^r_{\alpha ,\beta }$, then

\[ E_{n}(f)_{p,\alpha ,\beta }\leq E_{n}(\Phi ^r_{\alpha ,\beta })_{p,\alpha ,\beta }\, \, \, E_{n}(\mathcal{D}^r_{\alpha ,\beta }(f))_{p,\alpha , \beta }. \]

We present an analogous of theorem 3 in the frame of Jacobi spaces.

Theorem 7

Assume $\alpha \geq \beta \geq -1/2$, $1{\lt} q{\lt}p{\lt}\infty $ and chose $s$ from the condition $\eqref{tomalas}$. If $r\in \mathbb {N}$, $f\in \Omega ^r_{\alpha ,\beta }$ and $\mathcal{D}^r_{\alpha ,\beta }(f)\in L^q_{(\alpha ,\beta )}$, then

\begin{equation*} E_{n}(f)_{p,\alpha ,\beta }\leq E_{n}(\Phi ^r_{\alpha ,\beta })_{s,\alpha ,\beta }\, \, \, E_{n}(\mathcal{D}^r_{\alpha ,\beta }(f))_{q,\alpha , \beta }. \end{equation*}

for each $n\in \mathbb {N}$, $n\geq r-1$.

Proof. Let $P_n\in \mathbb {P}_n$ ($B_n\in \mathbb {P}_n$) be the polynomial of the best approximation of order $n$ of $\mathcal{D}^r_{\alpha ,\beta }(f)$ ($\Phi ^r_{\alpha ,\beta }$) in the norm of $L^q_{(\alpha ,\beta )}$ ($L^s_{(\alpha ,\beta )}$). That is,

\begin{equation*} E_{n}(\mathcal{D}^r_{\alpha ,\beta }(f))_{q,\alpha , \beta }=\Vert \mathcal{D}^r_{\alpha ,\beta }(f)-P_n\Vert _{q,\alpha ,\beta } \end{equation*}

and

\begin{equation*} E_{n}(\Phi ^r_{\alpha ,\beta })_{s,\alpha ,\beta }=\Vert \Phi ^r_{\alpha ,\beta }-B_n\Vert _{s,\alpha ,\beta } \end{equation*}

Define

\begin{align*} M_n(g,x)& =\int _{-1}^1 (\mathcal{D}^r_{\alpha ,\beta }(f) (t)\tau _t (B_{n},x)\varrho ^{(\alpha ,\beta )}(t)dt, \\ N_n(g,x)& =\int _{-1}^1 P_n(t)\tau _t\Big(\Phi ^r_{\alpha ,\beta }-B_n\Big)(x) \varrho ^{(\alpha ,\beta )}(t)dt, \end{align*}

and

\begin{equation*} L_n(g,x)= S_{r-1}^{(\alpha ,\beta )}(f,x) + M_n(g,x)+N_n(g,x). \end{equation*}

Since $M_n(g)$ and $N_n(g)$ are convolutions with algebraic polynomials of degree non greater than $n$ and $n\geq r-1$, it follows from 4 that $M_n(g), N_n(g)$, $L_n(g)\in \mathbb {P}_n$.

Taking into account 9, we obtain

\begin{align*} & L_{n}(f,x)-f(x) = L_n(f,x)-S_{r-1}^{(\alpha ,\beta )}(x)- (f-S_{r-1}^{(\alpha ,\beta )}(f))(x) \\ & = L_n(f,x)-S_{r-1}^{(\alpha ,\beta )}(x)- (\mathcal{D}^r_{\alpha ,\beta }(f)*\Phi ^r_{\alpha ,\beta } (x)) \\ & = \! \int _{-1}^1 \! \! \Big(\mathcal{D}^r_{\alpha ,\beta }(f,t)\tau _t(B_n) \! +\! P_n(t)\tau _t(\Phi ^r_{\alpha ,\beta } \! -\! B_n) \! -\! \mathcal{D}^r_{\alpha ,\beta }(f,t)\tau _t( \Phi ^r_{\alpha ,\beta })\Big)(x)\varrho ^{(\alpha ,\beta )}(t)dt \\ & =\int _{-1}^1 \Big(P_n-\mathcal{D}^r_{\alpha ,\beta } (f)\Big)(t) \tau _t\Big(\Phi ^r_{\alpha ,\beta }-B_n\Big)(x)dt \\ & =\Big((P_n-\mathcal{D}^r_{\alpha ,\beta } (f))* \tau _t(\Phi ^r_{\alpha ,\beta }-B_n)\Big)(x). \end{align*}

Now, it follows from 7 that

\begin{align*} \Vert L_{n}(f)-f\Vert _{p,\alpha ,\beta }& \leq \Vert \Phi ^r_{\alpha ,\beta }-B_n\Vert _{s,\alpha ,\beta }\, \, \, \Vert P_n-\mathcal{D}^r_{\alpha ,\beta } (f))\Vert _{q,\alpha ,\beta } \\ \hspace{40mm} & =E_{n} (\Phi ^r_{\alpha ,\beta })_{s,\alpha ,\beta }\, \, \, E_{n}(\mathcal{D}^r_{\alpha ,\beta }(f))_{q,\alpha , \beta }. \hspace{23mm}\square \end{align*}

Bibliography

1: M. I. Ganzburg, On best uniform approximation of periodic functions by trigonometric polynomials, J. Concr. Appl. Math., 8 (2010) no. 4, pp. 631–637.
2: G. Gasper, Positivity and the convolution structure for Jacobi series, Ann. Math., 93 (1971), pp. 112–118. https://doi.org/10.2307/1970755 $\includegraphics[scale=0.1]{ext-link.png}$
3: G. Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. Math., 95 (1972) no. 2, pp. 261–280. https://doi.org/10.2307/1970800 $\includegraphics[scale=0.1]{ext-link.png}$
4: N.P. Korneichuk, Exact Constants in Approximation Theory, Cambridge Univ. Press, Cambridge, 1991.
5: S. Rafalson, An extremal relation of the theory of approximation of functions by algebraic polynomials, J. Approx. Theory, 110 (2001) no. 2, pp. 146–170. https://doi.org/10.1006/jath.2001.3558 $\includegraphics[scale=0.1]{ext-link.png}$

This work is licensed under a Creative Commons Attribution 4.0 International License.

New estimates related with the best polynomial approximation

1 Introduction

2 The periodical case

3 The case of non-periodic functions

Bibliography

New estimates related with
the best polynomial approximation