Convolution averages a function against translated kernels. Mollifiers use this operation to turn rough functions into smooth approximations.
Convolution definition
Convolution combines translation and integration.
Definition. For suitable functions \(f,g\) on \(\mathbb R^d\), define \[ (f*g)(x)=\int_{\mathbb R^d} f(x-y)g(y)\,\mathrm{d}y. \]
Proposition. If \(f\in L^p\) and \(g\in L^1\), then \(f*g\) is defined almost everywhere and \(\|f*g\|_p\le\|f\|_p\|g\|_1\).
Proof. Minkowski's integral inequality gives \(\|f*g\|_p\le\int |g(y)|\|f(\cdot-y)\|_p\,\mathrm{d}y=\|f\|_p\|g\|_1\). The \(p=\infty\) case is direct from the essential supremum.
Proposition (algebraic identities). Whenever absolute integrability justifies the operations, \[ f*g=g*f,\qquad (f*g)*h=f*(g*h),\qquad \tau_a(f*g)=(\tau_af)*g=f*(\tau_ag). \]
Proof. Commutativity follows from \(y\mapsto x-y\). Associativity follows from Fubini: \[ \begin{aligned} ((f*g)*h)(x) &=\iint f(x-z-y)g(y)h(z)\,\mathrm{d}y\,\mathrm{d}z\\ &=\int f(x-u)\left(\int g(u-z)h(z)\,\mathrm{d}z\right)\mathrm{d}u. \end{aligned} \] The translation identities are direct substitutions.
Smoothness by convolution
Smooth kernels transfer smoothness to the convolution.
Proposition. If \(g\in C_c^\infty(\mathbb R^d)\) and \(f\in L^1_{\mathrm{loc}}\), then \(f*g\in C^\infty\), and \(\partial^\alpha(f*g)=f*(\partial^\alpha g)\).
Proof. Write \[ (f*g)(x)=\int f(y)g(x-y)\,\mathrm{d}y. \] On a compact set of \(x\)-values, the integrand uses \(y\) in one fixed compact enlargement of \(\operatorname{supp}g\), where \(f\) is integrable. For a coordinate direction, \[ \frac{g(x+he_i-y)-g(x-y)}h \longrightarrow\partial_i g(x-y), \] and the mean value theorem bounds the quotient by \(\|\partial_i g\|_\infty\) on that compact support. Dominated convergence yields \(\partial_i(f*g)=f*(\partial_i g)\). Iteration gives every multi-index derivative, continuously.
Approximate identities
A mollifier concentrates mass near the origin while preserving total mass one.
Definition. A standard mollifier is \(\rho_\varepsilon(x)=\varepsilon^{-d}\rho(x/\varepsilon)\), where \(\rho\in C_c^\infty\), \(\rho\ge0\), and \(\int\rho=1\).
Theorem. If \(f\in L^p(\mathbb R^d)\), \(1\le p\lt \infty\), then \(\rho_\varepsilon*f\to f\) in \(L^p\). If \(f\) is bounded and uniformly continuous, convergence is uniform.
Proof. Write \(\rho_\varepsilon*f-f=\int\rho_\varepsilon(y)(f(\cdot-y)-f)\,\mathrm{d}y\). For \(L^p\), use Minkowski and continuity of translations in \(L^p\). For uniform convergence, use the modulus of continuity and the fact that \(\rho_\varepsilon\) is supported near zero.
Theorem (pointwise recovery at Lebesgue points). If \(f\in L^1_{\mathrm{loc}}\) and \(\rho\) is bounded, compactly supported, and nonnegative with integral one, then \[ (\rho_\varepsilon*f)(x)\longrightarrow f(x) \] at every Lebesgue point \(x\) of \(f\), hence almost everywhere.
Proof. If \(\operatorname{supp}\rho\subset B(0,R)\), then \[ \begin{aligned} |(\rho_\varepsilon*f)(x)-f(x)| &\le\int\rho_\varepsilon(y)|f(x-y)-f(x)|\,\mathrm{d}y\\ &\le\|\rho\|_\infty\varepsilon^{-d} \int_{B(0,R\varepsilon)}|f(x-y)-f(x)|\,\mathrm{d}y. \end{aligned} \] The last expression is a constant multiple of the mean oscillation of \(f\) over \(B(x,R\varepsilon)\), which tends to zero at a Lebesgue point.
Support and boundary effects
Convolution expands support by the support of the kernel.
Proposition. \(\operatorname{supp}(f*g)\subset \operatorname{supp}f+\operatorname{supp}g\) whenever the convolution is defined and the supports are closed in the usual sense.
Proof. Interpret support as essential support for \(L^p\) functions. If \(x\) lies outside the closed Minkowski sum, a neighborhood \(U\) of \(x\) is disjoint from it. For \(z\in U\) and almost every \(y\), \(g(y)\ne0\) implies \(y\) lies in the essential support of \(g\), while \(f(z-y)\ne0\) implies \(z-y\) lies in that of \(f\). Both cannot occur, so the integrand vanishes almost everywhere and \(f*g=0\) on \(U\).
Weak derivatives commute with mollification
Mollification regularizes without destroying distributional derivative identities.
Proposition. If \(f\in L^1_{\mathrm{loc}}\) has weak derivative \(\partial^\alpha f\in L^1_{\mathrm{loc}}\), then on every region where the convolution is defined, \[ \partial^\alpha(\rho_\varepsilon*f) =\rho_\varepsilon*(\partial^\alpha f) =(\partial^\alpha\rho_\varepsilon)*f. \]
Proof. For a test function \(\varphi\), Fubini and the definition of weak derivative give \[ \int(\rho_\varepsilon*f)\,\partial^\alpha\varphi =(-1)^{|\alpha|} \int(\rho_\varepsilon*\partial^\alpha f)\,\varphi. \] Thus the weak derivative of the smooth convolution is \(\rho_\varepsilon*(\partial^\alpha f)\). The earlier differentiation-under-the-integral formula gives the kernel-derivative representation.