Elements of Approximation Theory

Constructive Approximation and Examples

Abstract

In this survey we introduce the general Theory of Approximation to functions in (quasisemi)normed spaces. The exposition starts with an explanation of the main problem: we consider a family of subspaces (our approximants), and we obtain a description of the subspace(s) that are approximated by this family with a given approximation order. We also introduce some of the most common tools used to solve these problems.

Approximation Theory gets heavily improved when some efforts are put into the effective construction of the approximants on each given example, rather than simply stating its existence—this is what we call *Constructive Approximation*.

The fact that we can handle actual functions allows us to obtain yet more properties of the approximants. It is implicit throughout the exposition how Approximation Theory benefits from other branches of Mathematics, but also how Constructive Approximation can be used to prove results from those other fields.

Let \((X,\lVert\cdot\rVert_{X})\) be a (quasisemi)normed linear space. Consider a countable family of spaces in \(X\), \(\{X_{n}\}_{n}\) with associated error functionals \(E(\cdot,X_{n})_{X}=\inf_{g\in X_{n}}\lVert\cdot-g\rVert_{X}\), satisfying the following properties:

- (i)
Homogeneity: \(\lambda X_{n}=X_{n}\) for all \(n\) and \(\lambda\in\mathbb{R}\)

- (ii)
\(C\)-linearity: There exists \(C\geq 1\) (independent of \(n\)) such that \(X_{n}+X_{n}=X_{Cn}\) for all \(n\).

- (iii)
Local (near)best approximation: Given \(f\in X\), there exists an element of (near)best approximation to \(f\) from \(X_{n}\) for all \(n\).

We say \(g\in Y\) is an element of best approximation to \(f\in X\), if \(\lVert f-g\rVert_{X}=E(f,Y)_{X}\). A near best approximation element to \(f\) from \(Y\) is by definition any function \(g\) such that \(\lVert f-g\rVert_{X}\leq\tau E(f,Y)_{X}\) for some value \(\tau>0\). In that case, we will often refer to such a function \(g\) as a \(\tau\)-near best approximation element.

- (iv)
Global best approximation: \(\lim_{n}E(f,X_{n})_{X}=0\) for all \(f\in X\).

We will call such a set \(\{X_{n}\}\) a *family of approximants*. For any such family, and given parameter values \(\alpha,q>0\), consider the subspaces \({\cal A}_{q}^{\alpha}(X,X_{n})=\Big{\{}f\in X:\lvert f\rvert_{{\cal A}_{q}^{\alpha}(X,X_{n})}<\infty\Big{\}}\) for the associated following (quasi)seminorms:

\begin{align}
\label{Haar}
\lvert\cdot\rvert_{{\cal A}_{q}^{\alpha}(X,X_{n})} & \label{Haar}=\Big{[}\sum_{n=1}^{\infty}\tfrac{1}{n}\big{(}n^{\alpha}E(\cdot,X_{n})_{X}\big{)}^{q}\Big{]}^{1/q}\quad\text{for }0<q<\infty \\
\lvert\cdot\rvert_{{\cal A}_{\infty}^{\alpha}(X,X_{n})} & \label{infinityHaar}=\sup_{n\geq 1}\big{\{}n^{\alpha}E(\cdot,X_{n})_{X}\big{\}}\\
\end{align}

We call these *Approximation Spaces* associated to the family of approximants \(\{X_{n}\}\). They consist on those functions \(f\in X\) that are approximated in \(X\) by elements of \(X_{n}\) with error of order \({\cal O}(\alpha)\) (i.e. , there exists \(C>0\) such that \(E(f,X_{n})_{X}\leq Cn^{-\alpha}\) for all \(n\)) and *smoothness* \(q\).

If the sequence \(\big{(}E(f,X_{n})_{X}\big{)}_{n}\) is monotone decreasing, then the (quasi)seminorms \(\lvert\cdot\rvert_{{\cal A}_{q}^{\alpha}(X,X_{n})}\) are equivalent to the following:

\begin{align}
\label{discreteHaar}
\lvert f\rvert_{{\cal A}_{q}^{\alpha}(X,X_{n})} & \label{discreteHaar}\asymp\Big{\{}\sum_{k=0}^{\infty}\big{(}2^{k\alpha}E(f,X_{2^{k}})_{X}\big{)}^{q}\Big{\}}^{1/q}\text{ for }0<q<\infty \\
\lvert f\rvert_{{\cal A}_{\infty}^{\alpha}(X,X_{n})} & \label{infinitydiscreteHaar}\asymp\sup_{k\geq 0}\,2^{k\alpha}E(f,X_{2^{k}})_{X}\\
\end{align}

For any \(k>0\) and any \(2^{k-1}\leq n<2^{k}\), we have the estimates

\begin{equation} 2^{k-1}E(f,X_{2^{k-1}})_{X}\leq n^{\alpha}E(f,X_{n})_{X}\leq 2^{k}E(f,X_{2^{k}})_{X};\nonumber \\ \end{equation}therefore,

\begin{align} \sum_{n=2^{k-1}}^{2^{k}-1}\tfrac{1}{n}\big{(}n^{\alpha}E(f,X_{n})_{X}\big{)}^{q} & \leq\sum_{n=2^{k-1}}^{2^{k}-1}\tfrac{1}{n}\big{(}2^{k\alpha}E(f,X_{2^{k}})_{X}\big{)}^{q}\notag \\ & \leq\sum_{n=2^{k-1}}^{2^{k}-1}\frac{1}{2^{k+1}}\big{(}2^{k\alpha}E(f,X_{2^{k}})_{X}\big{)}^{q}\notag \\ & =\big{(}2^{k\alpha}E(f,X_{2^{k}})_{X}\big{)}^{q},\notag \\ \end{align}and similarly,

\begin{equation} \sum_{n=2^{k-1}}^{2^{k}-1}\tfrac{1}{n}\big{(}n^{\alpha}E(f,X_{n})_{X}\big{)}^{q}\geq\big{(}2^{(k-1)\alpha}E(f,X_{2^{k-1}})_{X}\big{)}^{q}.\nonumber \\ \end{equation}Adding all the terms of the series that defines the seminorm associated to the spaces \({\cal A}_{q}^{\alpha}(X,X_{n})\), and applying the estimates above, we get the desired result. ∎

Under the same hypothesis as before, and any value \(\alpha>0\), the following inclusion is verified for all \(0<q<p\leq\infty\):

\begin{equation} {\cal A}_{q}^{\alpha}(X,X_{n})\subset{\cal A}_{p}^{\alpha}(X,X_{n}).\nonumber \\ \end{equation}This is just an application of the well known inclusions \(\ell_{q}\subset\ell_{p}\) for \(0<q<p\leq\infty\): Consider the measure space \((\mathbb{N},\bar{\mu})\), where \(\bar{\mu}(n)=1\) for all \(n\), and consider for each function \(f\in X\) the (measurable in \((\mathbb{N},\bar{\mu})\)) functions \(\varphi_{\alpha}:\mathbb{N}\ni k\mapsto 2^{k\alpha}E(f;X_{2^{k}})_{X}\in\mathbb{R}^{+}\); then,

\begin{equation} \lvert f\rvert_{{\cal A}_{q}^{\alpha}(X,X_{n})}\asymp\Big{\{}\sum_{k=0}^{\infty}\big{(}2^{k\alpha}E(f;X_{2^{k}})_{X}\big{)}^{q}\Big{\}}^{1/q}=\Big{(}\int_{\mathbb{N}}\varphi_{\alpha}^{q}\,d\bar{\mu}\Big{)}^{1/q}\nonumber \\ \end{equation}If \(f\in{\cal A}_{q}^{\alpha}\), then certainly \(\varphi_{\alpha}\in\ell_{q}\cap\ell_{\infty}\), and therefore;

\begin{equation} \lvert f\rvert_{{\cal A}_{p}^{\alpha}(X,X_{n})}\asymp\Big{(}\int_{\mathbb{N}}\varphi_{\alpha}^{p}\,d\bar{\mu}\Big{)}^{1/p}\leq\lVert\varphi_{\alpha}\rVert_{\ell_{\infty}}\lVert\varphi_{\alpha}\rVert_{\ell_{q}}\asymp\lvert f\rvert_{{\cal A}_{\infty}^{\alpha}(X,X_{n})}\lvert f\rvert_{{\cal A}_{q}^{\alpha}(X,X_{n})}\nonumber \\ \end{equation}∎

In the following sections we will learn to find descriptions of these spaces in terms of classical spaces. The main tools used in this sense are given in the following order:

- •
The (quasi)seminorms (\ref{Haar}), (\ref{infinityHaar}), (\ref{discreteHaar}) and (\ref{infinitydiscreteHaar}) are discretizations of \(p\)-norms over \(\mathbb{R}^{+}\) with the Haar measure \(dx/x\). We have included some useful results related to these measures in §\ref{hardy}.

- •
§\ref{BestApprox} deals with the existence and properties of general (abstract) best and near-best approximations.

- •
Both §\ref{interpspaces} and §\ref{JacksonBernstein} introduce us to Interpolation Spaces and how these are related to our problem of approximation via Jackson and Bernstein’s inequalities.

Given \(\alpha>0\) and \(1\leq q<\infty\), the following inequalities hold for each nonnegative measurable function \(\phi\):

\begin{align}
\label{Hardy1}
\int_{0}^{\infty}\Big{(}t^{-\alpha}\int_{0}^{t}\phi(s)\tfrac{ds}{s}\Big{)}^{q}\tfrac{dt}{t} & \label{Hardy1}\leq\frac{1}{\alpha^{q}}\int_{0}^{\infty}\big{[}t^{-\alpha}\phi(t)\big{]}^{q}\tfrac{dt}{t} \\
\int_{0}^{\infty}\Big{(}t^{\alpha}\int_{t}^{\infty}\phi(s)\tfrac{ds}{s}\Big{)}^{q}\tfrac{dt}{t} & \leq\frac{1}{\alpha^{q}}\int_{0}^{\infty}\big{[}t^{\alpha}\phi(t)\big{]}^{q}\tfrac{dt}{t}\\
\end{align}

For \(q=\infty\), the integral is replaced by the \(L_{\infty}\) norms:

\begin{aligned} \sup_{t>0}\Big{\{}t^{-\alpha}\int_{0}^{t}\phi(s)\tfrac{ds}{s}\Big{\}} & \leq\tfrac{1}{\alpha}\lVert t^{-\alpha}\phi(t)\rVert_{L_{\infty}(0,\infty)} & \\ \sup_{t>0}\Big{\{}t^{\alpha}\int_{t}^{\infty}\phi(s)\tfrac{ds}{s}\Big{\}} & \leq\tfrac{1}{\alpha}\lVert t^{\alpha}\phi(t)\rVert_{L_{\infty}(0,\infty)} & \\ \end{aligned}Let us prove the estimate (\ref{Hardy1}). For any value \(\lambda>0\) we estimate first the interior integral using Hölder’s Inequality; let