合成積

$ x*y:t\mapsto\int_\R x(\tau)y(t-\tau)\mathrm d\tau

別名

畳み込み

性質

$ x*y(t)=\int_\R x(\tau)y(t-\tau)\mathrm d\tau

$ = -\int_{\tau\in\R} x(t-(t-\tau))y(t-\tau)\mathrm d(t-\tau)

$ = \int_\R y(\tau)x(t-\tau)\mathrm d\tau

$ = y*x(t)

$ \forall a\in\R\setminus\{0\}:(f(a\bullet)*g(a\bullet))(t)=\frac1{|a|}(f*g)(at)

$ \int_a^bf(\tau)g(t-\tau)\mathrm d\tau=\int_\R f(\tau)g(t-\tau)\llbracket a\le\tau\le b\rrbracket\mathrm d\tau

$ = f(\bullet)\llbracket a\le\bullet\le b\rrbracket*g