91Ó°ÊÓ

Use mollifiers to construct an approximation of the function \(f\). Consider a mollifier \(\phi_{\delta}\) defined as \[\phi_{\delta}(x) = \frac{1}{\delta}\phi\left(\frac{x}{\delta}\right),\] where \(\phi\) is a smooth function with compact support and \(\int_{\mathbf{R}}\phi(x)dx = 1\). Note that this is a smooth function with compact support and integral equal to one when scaling by \(\delta\). Now, consider the convolution of \(f\) and \(\phi_{\delta}\) as \[g_{\delta}(x) = (f * \phi_{\delta})(x) = \int_{\mathbf{R}} f(y) \phi_{\delta}(x-y) dy.\] Using the properties of convolution, we know that \(g_{\delta}\) is a continuous function.

We need to show that the set of non-zero elements of \(g_{\delta}(x)\) is bounded. Since \(\phi\) is a smooth function with compact support, there exists an interval \([-M,M]\) such that \(\phi(x) = 0\) for all \(x\not\in[-M,M]\). Let \(A = \{x \in \mathbf{R}: f(x) \phi_{\delta}(x)\neq 0\}\). For our function \(g_{\delta}(x)\) to be nonzero, it must be true that \(\phi_{\delta}(x-y)\neq 0\) for some values inside the integral. Thus, \(x - y \in A\), which implies that \(y \in [x - M, x + M]\). Therefore, the set where \(g_{\delta}(x)\) is non-zero must also be bounded.

To show that the \(p\)-norm of the difference between \(f\) and \(g_{\delta}\) is less than some \(\varepsilon > 0\), we need to investigate the difference \(\|f-g_{\delta}\|_{p}\). Since \(f \in \mathcal{L}^{p}(\mathbf{R})\), we know that the integral \(\int_{\mathbf{R}} |f(y)|^p dy\) is finite. Using Hölder's inequality, we find that \[\int_{\mathbf{R}} |f(y)|^p \left|\phi_{\delta}(x-y)-1\right| dy \leq \left(\int_{\mathbf{R}} |f(y)|^p dy\right)^{\frac{1}{p}} \left(\int_{\mathbf{R}} |\phi_{\delta}(x-y)-1|^q dy\right)^{\frac{1}{q}},\] where \(q = \frac{p}{p-1}\). Since \(\phi_{\delta}\) is smooth, we can make this inequality arbitrarily small by choosing \(\delta\) small enough. This implies that for every \(\varepsilon>0\), there exists \(\delta\) such that \(\|f-g_{\delta}\|_{p}<\varepsilon\). So, we can find a continuous function \(g\) (given by the convolution \(g_{\delta}\)) that approximates our function \(f\) in the desired sense: the set where \(g\) is non-zero is bounded and the \(p\)-norm of the difference between \(f\) and \(g\) is less than \(\varepsilon\).