Namespaces
Variants
Actions

Difference between revisions of "Hermite transform"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
h0470301.png
 +
$#A+1 = 9 n = 0
 +
$#C+1 = 9 : ~/encyclopedia/old_files/data/H047/H.0407030 Hermite transform
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
The integral
 
The integral
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470301.png" /></td> </tr></table>
+
$$
 +
f ( n)  = H \{ F ( x) \}  = \int\limits _ {- \infty } ^  \infty 
 +
e ^ {- x  ^ {2} } H _ {n} ( x) F ( x)  d x ,\ \
 +
n = 0 , 1 \dots
 +
$$
 +
 
 +
where  $  H _ {n} ( x) $
 +
are the [[Hermite polynomials|Hermite polynomials]]. The inversion formula is
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470302.png" /> are the [[Hermite polynomials|Hermite polynomials]]. The inversion formula is
+
$$
 +
F ( x)  = \sum _ { n= } 0 ^  \infty 
 +
\frac{1}{\sqrt \pi }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470303.png" /></td> </tr></table>
+
\frac{f ( n) }{2  ^ {n} n ! }
 +
H _ {n} ( x)  = \
 +
H  ^ {-} 1 \{ f ( n) \} ,\  - \infty < x < \infty ,
 +
$$
  
 
provided that the series converges. The Hermite transform reduces the operator
 
provided that the series converges. The Hermite transform reduces the operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470304.png" /></td> </tr></table>
+
$$
 +
R [ F ( x) ]  = e ^ {x  ^ {2} }
 +
\frac{d}{dx}
 +
 
 +
\left [ e ^ {x  ^ {2} }
 +
\frac{d}{dx}
 +
F ( x) \right ]
 +
$$
  
 
to an algebraic one by the formula
 
to an algebraic one by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470305.png" /></td> </tr></table>
+
$$
 +
H \{ R [ F ( x) ] \}  = - 2 n f ( n) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470306.png" /> and all its derivatives up to and including the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470307.png" />-th order are bounded, then
+
If $  F $
 +
and all its derivatives up to and including the $  p $-
 +
th order are bounded, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470308.png" /></td> </tr></table>
+
$$
 +
H \{ F ^ { ( p) } ( x) \}  = f ( n + p ) .
 +
$$
  
The Hermite transform has also been introduced for a special class of generalized functions (see [[#References|[2]]]). They are used to solve differential equations containing the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470309.png" />.
+
The Hermite transform has also been introduced for a special class of generalized functions (see [[#References|[2]]]). They are used to solve differential equations containing the operator $  R $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Debnath,  "On the Hermite transform"  ''Mat. Vesnik'' , '''1'''  (1964)  pp. 285–292</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Zemanian,  "Generalized integral transforms" , Wiley  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Debnath,  "On the Hermite transform"  ''Mat. Vesnik'' , '''1'''  (1964)  pp. 285–292</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Zemanian,  "Generalized integral transforms" , Wiley  (1968)</TD></TR></table>

Revision as of 22:10, 5 June 2020


The integral

$$ f ( n) = H \{ F ( x) \} = \int\limits _ {- \infty } ^ \infty e ^ {- x ^ {2} } H _ {n} ( x) F ( x) d x ,\ \ n = 0 , 1 \dots $$

where $ H _ {n} ( x) $ are the Hermite polynomials. The inversion formula is

$$ F ( x) = \sum _ { n= } 0 ^ \infty \frac{1}{\sqrt \pi } \frac{f ( n) }{2 ^ {n} n ! } H _ {n} ( x) = \ H ^ {-} 1 \{ f ( n) \} ,\ - \infty < x < \infty , $$

provided that the series converges. The Hermite transform reduces the operator

$$ R [ F ( x) ] = e ^ {x ^ {2} } \frac{d}{dx} \left [ e ^ {x ^ {2} } \frac{d}{dx} F ( x) \right ] $$

to an algebraic one by the formula

$$ H \{ R [ F ( x) ] \} = - 2 n f ( n) . $$

If $ F $ and all its derivatives up to and including the $ p $- th order are bounded, then

$$ H \{ F ^ { ( p) } ( x) \} = f ( n + p ) . $$

The Hermite transform has also been introduced for a special class of generalized functions (see [2]). They are used to solve differential equations containing the operator $ R $.

References

[1] L. Debnath, "On the Hermite transform" Mat. Vesnik , 1 (1964) pp. 285–292
[2] A.G. Zemanian, "Generalized integral transforms" , Wiley (1968)
How to Cite This Entry:
Hermite transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_transform&oldid=47217
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article