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I am looking for a reference for the $L_p$ error of the difference of a Sobolev function and its convolution with a band-limited mollifier.

The type of estimate that is quoted in a paper without a source is as follows:

Consider $f\in W_p^k(\mathbb{R})$, and $\phi$ is a band-limited function ($\widehat{\phi}$ is supported on the interval $[-\pi/(2h), \pi/(2h)]$ for some small $h>0$). Their estimate was $$\|f-(f\ast\phi)\|_{L_p(\mathbb{R})}\leq Ch^k|f|_{W_p^k} = Ch^k\|f^{(k)}\|_{L_p}$$

The author simply says by the "usual error estimate from mollification by a band-limited mollifier."

Any book or paper that proves a similar estimated would be appreciated.

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1 Answer 1

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Some references for the rate of $L^p$ approximation in terms of Sobolev norms are given here. Some insightful remarks on the rate of convergence of mollified functions are found here.

Below I give a proof of the estimate (not in your interpretation, which misses important points).

Let $\varphi$ be as in the paper you are reading. For $h>0$ let $\varphi_h(x)=h^{-1}\varphi(x/h)$. That the support of $\widehat{\varphi_h}$ is contained in $\{\xi: |\xi|\le (\pi+\varepsilon)/h\}$ is not of much help. What really matters is that $\widehat {\varphi_h}=1$ on $\{\xi: |\xi|\le c/h\}$ where $c=\pi-\varepsilon$. Since the Fourier transform of $f*\varphi_h$ agrees with $\widehat f$ when $|\xi|\le c/h$, the transform of $f-f*\varphi_h$ vanishes for such $\xi$.

As usual, the case $p=2$ is easier to deal with. Let $M$ be the supremum of $|\widehat \varphi| $ (which can be arranged to be $1$). Then
$$ \begin{split} \|f-f*\varphi_h\|_{L^2}^2 & = \int_{|\xi|\ge c/h} |\widehat f_1(\xi)|^2 d\xi \\ & \le (h/c)^{2k} \int_{\mathbb R} |\xi|^{2k} |\widehat f_1(\xi)|^2 d\xi \\ &\le (h/c)^{2k} M \int_{\mathbb R} |\xi|^{2k} |\widehat f(\xi)|^2 d\xi \\ &= (h/c)^{2k} M |f|_{W^{k,2}}^2 \end{split} \tag1$$

The idea for the general case $1<p<\infty$ is about the same: $f-f*\varphi_h$ has only high frequencies of $f$ (those above $c/h$), which are magnified in $f^{(k)}$ by factors of at least $(c/h)^k$. But to relate the Fourier transform to the $L^p$ norm, we need the Littlewood-Paley decomposition of $f$: $$f=\sum_{j\in \mathbb Z} f_j$$ where $ f_j =f*\varphi_{2^{-j}}-f*\varphi_{2^{-j+1}}$. The important point here is that $\widehat{f_j}$ is supported in a roughly dyadic annulus of size $2^{j}$. These frequencies get magnified by about $2^{jk}$ in the $k$th derivative. The Littlewood-Paley theorem says that $\|f\|_{L^p}$ is comparable to the $L^p$ norm of the square function of $f$: $$\|f\|_{L^p}\approx \left\|\left(\sum_{j\in\mathbb Z} |f_j|^2\right)^{1/2}\right\|_{L^p} \tag{LP}$$ Use (LP) for $f^{(k)}$ and for $f-f*\varphi_h$: $$ \|f^{(k)}\|_{L^p} \approx \left\|\left(\sum_{j\in\mathbb Z} 2^{2jk}|f_j|^2\right)^{1/2} \right\|_{L^p} \gtrsim (c/h)^k \left\|\left(\sum_{2^j\ge c/h} |f_j|^2\right)^{1/2} \right\|_{L^p} \tag2$$ and $$ \|f-f*\varphi_h\|_{L^p} \lesssim \left\|\left(\sum_{2^j\ge c/h} |f_j|^2\right)^{1/2} \right\|_{L^p} \tag3$$ because the low frequencies ($2^j<c/h$) cancel out in $ f-f*\varphi_h$. Comparing (2) and (3) we conclude with $\|f-f*\varphi_h\|_{L^p}\lesssim h^k \|f^{(k)}\|_{L^p}$.

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