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Diffstat (limited to 'convolution.tex')
| -rw-r--r-- | convolution.tex | 28 |
1 files changed, 16 insertions, 12 deletions
diff --git a/convolution.tex b/convolution.tex index eee6e16..66120f2 100644 --- a/convolution.tex +++ b/convolution.tex @@ -42,14 +42,18 @@ as this will facilitate our proof of the convolution formula. We consider a $\sigma$-finite measure space $(X,\SigmaAlgebra{A},\mu)$, a separable Fréchet space $Y$ (over $\CC$) and the task is to define the integral of functions $f \vcentcolon X \to Y$. -Recall that a measure space is said to be \emph{$\sigma$-finite} +Recall that a measure space is said to be +\emph{$\sigma$-finite} +%\index{sigma-finite@$sigma$-finite} TODO: fix @ +%\nomenclature[A]{$\mathcal{A}'$}{commutant of $\mathcal{A}$} if it can be exhausted by a countable number of measurable subsets of finite measure. -By \emph{Fréchet space} we mean a complete Hausdorff locally convex (topological vector) space +By \emph{Fréchet space}\index{Fréchet space} +we mean a complete Hausdorff locally convex (topological vector) space which possesses countable neighborhood bases. We will make use of a countable family $P@@$ of seminorms that generates the topology of $@@Y$. A topological space is called \emph{separable} if it contains a countable dense subset. -A function $f \vcentcolon X \to Y$ will be called \emph{simple} +A function $f \vcentcolon X \to Y$ will be called \emph{simple}\index{simple function} if it is of the form $\sum_{i=1}^n \chi_{A_i} y_i$ where $n \in \NN$, $A_i \in \SigmaAlgebra{A}$ with $\mu(A_i) < \infty$, and $y_i \in Y$. Naturally, the \emph{integral} of $f$ is defined to be the vector $\int f = \sum_{i=1}^n \mu(A_i) y_i \in Y$. @@ -60,14 +64,14 @@ if it is the $\mu$-almost everywhere pointwise limit of simple functions. Suppose $(X,\SigmaAlgebra{A},\mu)$ is a $\sigma$-finite measure space, and $Y@@$ is a separable Fréchet space whose topology is generated by a family $P@@$ of seminorms. -A strongly measurable function $f \vcentcolon X \to Y$ is called \emph{(generalized Bochner) integrable} + A strongly measurable function $f \vcentcolon X \to Y$ is called \emph{(generalized Bochner) integrable} if there exists a sequence $(f_n)$ of simple functions such that \begin{equation} \label{equation:bochner-integrable} \lim_{n \to \infty} \int_X p \circ (f_n - f) \, d\mu = 0 \qquad \forall p \in P. \end{equation} -In this case, the \emph{(generalized Bochner) integral} of $f$ is defined by + In this case, the \emph{(generalized Bochner) integral}\index{Bochner integral!generalized} of $f$ is defined by \begin{equation} \label{equation:bochner-integral} \int_X f \ d\mu \defequal @@ -226,7 +230,7 @@ Denote by $\TestFunctions{\RR^n}$ the vector space of all functions $f \vcentco such that the derivatives $\partial^{\alpha} f$ exist and are continuous for all multi-indices $\alpha \in \NN^n$. Recall that the space $\SchwartzFunctions{\RR^n}$ of \emph{Schwartz functions} is defined to be the vector space \begin{equation*} - \SchwartzFunctions{\RR^n,X} \defequal \braces{f \in \TestFunctions{\RR^n} \vcentcolon \norm{f}_{\alpha,\beta} < \infty \ \forall \alpha,\beta \in \NN^n} + \SchwartzFunctions{\RR^n} \defequal \braces{f \in \TestFunctions{\RR^n} \vcentcolon \norm{f}_{\alpha,\beta} < \infty \ \forall \alpha,\beta \in \NN^n} \end{equation*} equipped with the locally convex topology induced by the family of seminorms \begin{equation*} @@ -247,7 +251,7 @@ is defined in the same way as in \cref{definition:convolution-distribution-test- \end{equation*} \begin{proposition}{Vector-Valued Convolution Formula}{vector-valued-convolution-formula} - Let $v \in \TemperedDistributions{\RR^n\!,X}$ be a tempered distribution with values in a separable Fréchet space $X$, and + Let $v \in \TemperedDistributions{\RR^n\!,Y}$ be a tempered distribution with values in a separable Fréchet space $Y$, and let $f \in \SchwartzFunctions{\RR^n}$ be a Schwartz test function. Then one has \begin{equation*} (v * f)(g) = \int v(\tau_x \tilde{f}@@) g(x) \, dx \qquad g \in \SchwartzFunctions{\RR^n}. @@ -256,7 +260,7 @@ is defined in the same way as in \cref{definition:convolution-distribution-test- \begin{proof} We fix a Schwartz function $g$, and consider the finite measure $\mu = \abs{g} \lambda$ on $\RR^n$, - where $\lambda(x) = dx$ is the Lebesgue measure. + where $\lambda = dx$ is the Lebesgue measure. We show that the mapping $x \mapsto \tau_x \tilde{f}$ is a generalized Bochner $\mu$-integrable function $\RR^n \to \SchwartzFunctions{\RR^n}$ using \cref{theorem:generalized-bochner}. For all $\alpha,\beta \in \NN^n$ we see by substituting $x+y$ for $y$ that @@ -275,9 +279,9 @@ is defined in the same way as in \cref{definition:convolution-distribution-test- because $g$ is Schwartz class. Hence, $x \mapsto \tau_x \tilde{f}$ defines an integrable function. - The mapping $v \vcentcolon \SchwartzFunctions{\RR^n} \to X$ is linear and continuous by definition. + The mapping $v \vcentcolon \SchwartzFunctions{\RR^n} \to Y$ is linear and continuous by definition. By \cref{theorem:integral-commutes-with-operator}, - the composite mapping $x \mapsto v(\tau_x \tilde{f})$ is a $\mu$-integrable function $\RR^n \to X$, and + the composite mapping $x \mapsto v(\tau_x \tilde{f})$ is a $\mu$-integrable function $\RR^n \to Y$, and \begin{equation} \label{equation:general-bochner-appears} \int v(\tau_x \tilde{f}) \, d\mu(x) = v \parens[\bigg]{\int \tau_x \tilde{f} \, d\mu(x)} @@ -293,11 +297,11 @@ is defined in the same way as in \cref{definition:convolution-distribution-test- and the proof is complete. \end{proof} -Let us point out that even in the special case that $X$ is a Banach space +Let us point out that even in the special case that $Y$ is a Banach space the integral on the right hand side of~\eqref{equation:general-bochner-appears} only has meaning as a generalized Bochner integral, since the integrand takes values in $\SchwartzFunctions{\RR^n}$, which is not a Banach space. We could not have performed this step with the ordinary Bochner integral. -%\nomenclature[B]{$\BoundedLinearOperators{X,Y}$}{bounded linear operators from $X$ to $Y$\nomnorefpage} +% vim: syntax=mytex |