Notes on Averaged Taylor Approximation

Averaged Taylor Polynomial

Definition

The Taylor polynomial of order $m$ , centered at $y$ and evaluated at $x$ , is given by

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where $\alpha = (\alpha_1,\dots,\alpha_n)$ is an n-tuple of non-negative integers, $x \in \mathbb{R}^n$ and

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However, if $u$ is in a Sobolev space, $D^\alpha u$ may not exist in the usual pointwise sense. We accomplish this by taking an “average” over $y$ on a ball.

Definition

Suppose $u \in W_p^{m-1}(\Omega)$ , with $p \ge 1$ , and $B \subset\subset \Omega$ . The corresponding Taylor polynomial of order $m$ of $u$ averaged over $B$ is defined as

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where $T_y^m u(x)$ is defined as $\eqref{eq:taylor}$ , $B$ is a ball centered at $x_0$ with radius $\rho$ , and $\phi \in C_0^\infty(B)$ is nonnegative and normalized by $\int_B \phi(y)\,dy=1$ . For estimates uniform in the ball, choose $\phi(y)=\rho^{-n}\widehat\phi((y-x_0)/\rho)$ from a fixed normalized smooth function $\widehat\phi$ supported in the unit ball.

Remark

Such a polynomial is not unique, due to the choice of cut-off function $\phi$ .

Proposition

$Q^m u$ is a polynomial of degree less than $m$ in $x$ .

Proof

The definition of $Q^m u$ makes sense because $u \in W_p^{m-1}(\Omega)$ . If we write

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where $\gamma = (\gamma_1,\dots,\gamma_n)$ , $\beta = (\beta_1,\dots,\beta_n)$ and $a_{[\gamma,\beta]}$ are constants, $Q^m u(x)$ can be written as

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Note

The degree of $Q^m u$ is at most $m-1$ .

Though we use the assumption that $u$ is in a Sobolev space $W_p^{m-1}(\Omega)$ , we can extend the definition of $Q^m u$ to $L^1(B)$ by integrating by parts:

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It is equivalent to $\eqref{eq:avg_taylor}$ if $u \in W_p^{m-1}(\Omega)$ .

Proposition

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where $\psi_\lambda \in C_0^\infty(\mathbb{R}^n)$ and $\mathrm{supp}(\psi_\lambda) \subset \overline{B}$ .

Proof

This follows from $\eqref{eq:avg_taylor2}$ if we define

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Corollary

If $\Omega$ is a bounded domain in $\mathbb{R}^n$ , then for any nonnegative integer $k$ , $Q^m$ is a bounded map of $L^1(B)$ into $W^{k}_\infty(\Omega)$ . With the cut-off function fixed, there exists a constant $C = C(m,n,k,\phi,\Omega)$ such that for all $u \in L^1(B)$ ,

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Proof

This follows from $\eqref{eq:avg_taylor3}$ and the fact that both $\sup_{y \in B} |\psi_\lambda(y)|$ and $\sup_{x \in \Omega} |D^\alpha x^\lambda |$ are bounded.

Proposition

For any $\alpha$ such that $|\alpha| \leqslant m-1$ ,

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Proof

It’s easy to verify this proposition for all $u \in C^\infty(\Omega)$ . The proof is completed via a density argument.

Error Representation

Definition

$\Omega$ is star-shaped with respect to the ball $B$ if, for all $x \in \Omega$ , the closed convex hull of $\{x\} \cup B$ is a subset of $\Omega$ .

From now on, we assume that $\Omega$ is star-shaped with respect to the ball $B$ . Let $C_x$ denote the convex hull of $\{x\} \cup B$ .

The integral form of the Taylor remainder for $f \in C^m([0,1])$ is given by

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Let $u$ be a $C^m$ function on $\Omega$ . For $x \in \Omega$ and $y \in B$ , define $f(s) = u(y + s(x-y))$ . Then, we obtain

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Hence,

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Take $s=1$ , we have $f(1) = u(x)$ . By using the variable substitution $t \to 1-t$ in the integral, we obtain

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Definition

The $m$ -th order remainder term of $Q^m u$ is given by

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Using the definition of $Q^m$ and cut-off function, we obtain

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Proposition

The remainder $R^m u$ satisfies

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where $z = x + t(y-x)$ , $k_{\alpha}(x,z) = \frac{1}{\alpha !}(x-z)^{\alpha} k(x,z)$ and

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The proof is omitted.

Definition

Suppose $\Omega$ has diameter $d$ and is star-shaped with respect to a ball $B$ . Then the chunkiness parameter of $\Omega$ is defined as $\gamma = \frac{d}{\rho_{\max}}$ , where $\rho_{\max}$ is the supremum of the radii of all such balls $B$ ,

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Corollary

The ball $B$ can be chosen so that the function $k(x,z)$ in the remainder representation above satisfies the following estimate:

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where $\gamma$ is the chunkiness parameter of $\Omega$ .

Proof

Choose a ball $B$ such that $\Omega$ is star-shaped with respect to $B$ and the radius of $B$ satisfies $\rho \geqslant \frac12 \rho_{\max}$ . Then

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Bounds for Riesz Potentials

Lemma

If $f \in L^p(\Omega)$ for $1<p<\infty$ and $m > n/p$ , then

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This inequality also holds for $p=1$ if $m \geqslant n$ .

Proposition

If $1<p<\infty$ and $m > n/p$ , or $p=1$ and $m \geqslant n$ , there exists a constant $C = C(m,n,\gamma,p)$ such that

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for all $u \in W^m_p(\Omega)$ .

Proof

First, we assume that $u \in C^m(\Omega) \cap W^m_p(\Omega)$ . We can use the pointwise representation of $R^mu(x)$ .

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The proof can be completed via a density argument.

Lemma

Let $f \in L^p(\Omega)$ for $p \geqslant 1$ and $m \geqslant 1$ and let

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Then

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Lemma (Bramble-Hilbert)

Let $B$ be a ball in $\Omega$ such that $\Omega$ is star-shaped with respect to $B$ and such that its radius $\rho > \frac 12 \rho_{\max}$ . Let $Q^m u$ be the Taylor polynomial of order $m$ of $u$ averaged over $B$ where $u \in W_p^m(\Omega)$ and $p \geqslant 1$ . Then

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where $d = \mathrm{diam}(\Omega)$ .

Note

Bramble-Hilbert lemma is an important result for the analysis of the approximation properties of finite elements.

Corollary

Under the assumptions of the Bramble–Hilbert lemma, the following full-norm estimate holds, where $P^{m-1}$ denotes the polynomials of total degree at most $m-1$ :

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Proof

Choose $v=Q^m u$ and sum the seminorm estimates of the Bramble–Hilbert lemma over derivative orders $j=0,\dots,k$ .