ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Simple map
Simple function
Measurable simple complex function
Simple integral
Unsigned basic integral
P-integrable basic function
Set of P-integrable complex Borel functions
Complex Lebesgue convolution
Definition D5632
Complex Lebesgue convolution approximate identity
Formulation 0
Let $M = (\mathbb{R}^D, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that
(i) $\eta_0, \eta_1, \eta_2, \ldots : \mathbb{R}^D \to \mathbb{C}$ are each an D1921: Absolutely integrable function on $M$
(ii) $\mathfrak{L}^p = \mathfrak{L}^p(M \to \mathbb{C})$ is the D316: Set of P-integrable complex Borel functions on $M$ for $p \in [1, \infty)$
Then $\eta = \{ \eta_n \}_{n \in \mathbb{N}}$ is an approximate identity for complex Lebesgue convolution on $M$ with respect to $p$ if and only if \begin{equation} \forall \, f \in \mathfrak{L}^p : \lim_{n \to \infty} \Vert \eta_n * f - f \Vert_{L^p} = 0 \end{equation}