Hermitova transformace - Hermite transform - Wikipedia

V matematice Hermitova transformace je integrální transformace pojmenoval podle matematika Charles Hermite, který používá Hermitovy polynomy ${ displaystyle H_ {n} (x)}$ jako jádra transformace. To bylo poprvé představeno Lokenath Debnath v roce 1964.^[1][2][3][4]

Hermitova transformace funkce ${ displaystyle F (x)}$ je

${ displaystyle H {F (x) } = f_ {H} (n) = int _ {- infty} ^ { infty} e ^ {- x ^ {2}} H_ {n} ( x) F (x) dx}$

Inverzní Hermitova transformace je dána vztahem

${ displaystyle H ^ {- 1} {f_ {H} (n) } = F (x) = součet _ {n = 0} ^ { infty} { frac {1} {{ sqrt { pi}} 2 ^ {n} n!}} f_ {H} (n) H_ {n} (x)}$

Nějaké Hermitovy transformační páry

${ displaystyle F (x) ,}$	${ displaystyle f_ {H} (n) ,}$
${ displaystyle x ^ {m}, n> m ,}$	${ displaystyle 0}$
${ displaystyle x ^ {n} ,}$	${ displaystyle { sqrt { pi}} n!}$
${ displaystyle e ^ {ax} ,}$	${ displaystyle { sqrt { pi}} a ^ {n} e ^ {a ^ {2} / 4} ,}$
${ displaystyle e ^ {2xt-t ^ {2}}, | t | <{ frac {1} {2}} ,}$	${ displaystyle { sqrt { pi}} součet _ {n = 0} ^ { infty} (2 t) ^ {n}}$
${ displaystyle e ^ {x ^ {2}} { frac {d} {dx}} vlevo [e ^ {- x ^ {2}} { frac {d} {dx}} F (x) že jo],}$	${ displaystyle -2nf_ {H} (n) ,}$
${ displaystyle { frac {d ^ {m}} {dx ^ {m}}} F (x) ,}$	${ displaystyle f_ {H} (n + m) ,}$
${ displaystyle x { frac {d ^ {m}} {dx ^ {m}}} F (x) ,}$	${ displaystyle nf_ {H} (n + m-1) + { frac {1} {2}} f_ {H} (n + m + 1) ,}$
${ displaystyle F (x) * G (x) ,}$	${ displaystyle { sqrt { pi}} (- 1) ^ {n} vlevo [2 ^ {2n + 1} gama vlevo (n + { frac {3} {2}} vpravo) vpravo ] ^ {- 1} f_ {H} (n) g_ {H} (n) ,}$[5]
${ displaystyle H_ {m} (x) ,}$	${ displaystyle { sqrt { pi}} 2 ^ {n} n! delta _ {nm} ,}$
${ displaystyle H_ {n} ^ {2} (x) ,}$	${ displaystyle { sqrt { pi}} součet _ {r = 0} ^ {n} { binom {n} {r}} 2 ^ {r + n} (2r)! n! ,}$
${ displaystyle H_ {m} (x) H_ {p} (x) ,}$	${ displaystyle { begin {cases} { frac {{ sqrt { pi}} 2 ^ {k} m! n! p!} {(km)! (kn)! (kp)!}}, & m + n + p = 2k, k geq m, n, p 0, a { text {jinak}} end {případy}} ,}$[6]
${ displaystyle H_ {m} ^ {2} (x) H_ {n} (x), m> n ,}$	${ displaystyle { sqrt { pi}} 2 ^ {n} 2 ^ {m} n! sum _ {k = 0} ^ {n} { binom {m} {k}} { binom {n } {k}} { binom {2k} {k}} ,}$[7]
${ displaystyle H_ {n + p + q} (x) H_ {p} (x) H_ {q} (x) ,}$	${ displaystyle { sqrt { pi}} 2 ^ {n + p + q} (n + p + q)! ,}$
${ displaystyle e ^ {z ^ {2}} sin ({ sqrt {2}} xz), | 2z | <1 ,}$	${ displaystyle { begin {cases} { sqrt { pi}} sum _ {m = 0} ^ { infty} (- 1) ^ {m} (2z) ^ {2m + 1}, & n = 2m + 1 0, & n neq 2m + 1 end {případy}} ,}$
${ displaystyle (1-z ^ {2}) ^ {- 1/2} exp left [{ frac {2xyz- (x ^ {2} + y ^ {2}) z ^ {2}} { (1-z ^ {2})}} vpravo] ,}$	${ displaystyle { sqrt { pi}} součet _ {m = 0} ^ { infty} z ^ {m} H_ {m} (y) delta _ {nm} ,}$

Reference