Integrální transformace

Jako integrální transformace se v matematice označují některé speciální případy lineárních integrálních operátorů, což jsou lineární zobrazení $T\colon A\to B$ mezi dvěma prostory funkcí $A,\,B$ , jež se dají zapsat v podobě integrálu

(Tf)(x)=\int _{\Omega }K(t,x)f(t)\mathrm {d} t

kde $\Omega \subset \mathbb {R} ^{n}$ a $D\subset \mathbb {R} ^{n}$ jsou otevřené podmnožiny, $K\colon \Omega \times D\to \mathbb {C}$ je měřitelná funkce označovaná v tomto kontextu jako jádro transformace, $f(t)$ je libovolná funkce z prostoru $A$ a $(Tf)(x)$ je její obraz, tedy funkce z prostoru $B$ .

Příklady integrálních transformací jsou Fourierova, Laplaceova nebo vlnková transformace.

K integrální transformaci může (ale obecně nemusí) existovat inverzní transformace, převádějící obraz z prostoru $B$ zpět na vzor z prostoru $A$ . Pokud existuje, dá se vyjádřit rovněž jako integrální operátor, ale s odlišným (tzv. inverzním) jádrem $K^{-1}$ a odlišným oborem integrace $\Omega '$ .

Přehled některých často používaných transformací:

Transformace	Symbol	$K$	$\Omega$	$K^{-1}$	$\Omega '$
Spojitá Fourierova transformace	${\mathcal {F}}$	${\frac {e^{-\mathrm {i} u\cdot t}}{(2\pi )^{n/2}}}$	$\mathbb {R} ^{n}\,$	${\frac {e^{+\mathrm {i} u\cdot t}}{(2\pi )^{n/2}}}$	$\mathbb {R} ^{n}\,$
Hartleyova transformace	${\mathcal {H}}$	${\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}$	$\mathbb {R} \,$	${\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}$	$\mathbb {R} \,$
Mellinova transformace	${\mathcal {M}}$	$t^{u-1}\,$	$(0,+\infty )$ $\$	${\frac {t^{-u}}{2\pi \mathrm {i} }}\,$	$c+\mathrm {i} \mathbb {R}$ $\$
Dvojstranná Laplaceova transformace	${\mathcal {B}}$	$e^{-ut}\,$	$\mathbb {R} \,$	${\frac {e^{+ut}}{2\pi \mathrm {i} }}$	$c+\mathrm {i} \mathbb {R}$
Laplaceova transformace	${\mathcal {L}}$	$e^{-ut}\,$	$(0,+\infty )$ $\$	${\frac {e^{+ut}}{2\pi \mathrm {i} }}$	$c+\mathrm {i} \mathbb {R}$
Weierstrassova transformace	${\mathcal {W}}$	${\frac {e^{-(u-t)^{2}/4}}{\sqrt {4\pi }}}\,$	$\mathbb {R} \,$	${\frac {e^{+(u-t)^{2}/4}}{\mathrm {i} {\sqrt {4\pi }}}}$	$c+\mathrm {i} \mathbb {R}$
Abelova transformace		${\frac {2t}{\sqrt {t^{2}-u^{2}}}}\chi _{(u,\infty )}(t)$	$\mathbb {R} \,$	${\frac {-1}{\pi {\sqrt {u^{2}\!-\!t^{2}}}}}\chi _{(t,\infty )}(u){\frac {\rm {d}}{{\rm {d}}u}}$	$\mathbb {R} \,$
Hilbertova transformace	${\mathcal {H}}il$ , ${\mathcal {H}}$	${\frac {1}{\pi }}{\frac {1}{u-t}}$	$\mathbb {R} \,$	${\frac {1}{\pi }}{\frac {1}{u-t}}$	$\mathbb {R} \,$
Hankelova transformace s jádrem obsahujícím $\operatorname {J} _{\nu }(ut)$ , Besselovu funkci prvního druhu a řádu ν	${\mathcal {H}}_{\nu }$	$t\operatorname {J} _{\nu }(ut)$	$(0,+\infty )$ $\$	$u\operatorname {J} _{\nu }(ut)$	$(0,+\infty )$ $\$
Stieltjesova transformace	${\mathcal {S}}$	${\frac {1}{u+t}}$	$(0,+\infty )$ $\$

Externí odkazy

Obrázky, zvuky či videa k tématu integrální transformace na Wikimedia Commons

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