Namespaces
Variants
Actions

Difference between revisions of "Gegenbauer transform"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (fix tex)
 
Line 16: Line 16:
 
$$  
 
$$  
 
T \{ F ( t) \}  = \  
 
T \{ F ( t) \}  = \  
\int\limits _ { - } 1 ^ { + } 1
+
\int\limits _ { -1 } ^ { +1 } ( 1 - t  ^ {2} ) ^ {\rho - 1/2 } C _ {n}  ^  \rho  ( t) F ( t)  dt  =  f _ {n} ^ { \rho } ,
( 1 - t  ^ {2} ) ^ {\rho - 1/2 }
 
C _ {n}  ^  \rho  ( t) F ( t)  dt  =  f _ {n} ^ { \rho } ,
 
 
$$
 
$$
  
 
$$  
 
$$  
\rho  >  -  
+
\rho  >  - \frac{1}{2} ,\  n  =  0, 1 , \ldots\;.
\frac{1}{2}
 
,\  n  =  0, 1 , .  .  . .
 
 
$$
 
$$
  
Line 56: Line 52:
  
 
====Comments====
 
====Comments====
For any system of orthogonal polynomials one can formally consider a transform pair as above, cf.
+
For any system of orthogonal polynomials one can formally consider a transform pair as above, cf. (1) and
 
+
(2) in [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]]. The Gegenbauer transform (and, more generally, the [[Jacobi transform|Jacobi transform]]) has been considered for arguments  $  n $
and
 
 
 
in [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]]. The Gegenbauer transform (and, more generally, the [[Jacobi transform|Jacobi transform]]) has been considered for arguments  $  n $
 
 
which are arbitrarily complex. Then inversion formulas exist in the form of integrals and there is a relationship with sampling theory, cf. [[#References|[a1]]], [[#References|[a2]]].
 
which are arbitrarily complex. Then inversion formulas exist in the form of integrals and there is a relationship with sampling theory, cf. [[#References|[a1]]], [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Butzer,  R.L. Stens,  M. Wehrens,  "The continuous Legendre transform, its inverse transform, and applications,"  ''Internat. J. Math. Sci.'' , '''3'''  (1980)  pp. 47–67</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.H. Koornwinder,  G.G. Walter,  "The finite continuous Jacobi transform and its inverse"  ''J. Approx. Theory''  (To appear)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Butzer,  R.L. Stens,  M. Wehrens,  "The continuous Legendre transform, its inverse transform, and applications,"  ''Internat. J. Math. Sci.'' , '''3'''  (1980)  pp. 47–67</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.H. Koornwinder,  G.G. Walter,  "The finite continuous Jacobi transform and its inverse"  ''J. Approx. Theory''  (To appear)</TD></TR></table>

Latest revision as of 18:27, 1 January 2021


The integral transform $ T \{ F( t) \} $ of a function $ F( t) $,

$$ T \{ F ( t) \} = \ \int\limits _ { -1 } ^ { +1 } ( 1 - t ^ {2} ) ^ {\rho - 1/2 } C _ {n} ^ \rho ( t) F ( t) dt = f _ {n} ^ { \rho } , $$

$$ \rho > - \frac{1}{2} ,\ n = 0, 1 , \ldots\;. $$

Here $ C _ {n} ^ \rho $ are the Gegenbauer polynomials. If a function can be expanded into a generalized Fourier series by Gegenbauer polynomials, the following inversion formula is valid:

$$ F ( t) = \sum _ {n = 0 } ^ \infty \frac{n! ( n + \rho ) \Gamma ^ {2} ( \rho ) 2 ^ {2 \rho - 1 } }{\pi \Gamma ( n + 2 \rho ) } C _ {n} ^ \rho ( t) f _ {n} ^ { \rho } ,\ \ - 1 < t < 1. $$

The Gegenbauer transform reduces the differentiation operation

$$ R [ F ( t)] = \ ( 1 - t ^ {2} ) F ^ { \prime\prime } - ( 2 \rho - 1) tF ^ { \prime\prime } $$

to the algebraic operation

$$ T \{ R [ F ( t)] \} = - n ( n + 2 \rho ) f _ {n} ^ { \rho } . $$

References

[1] V.A. Ditkin, A.P. Prudnikov, "Operational calculus" Progress in Math. , 1 (1968) pp. 1–75 Itogi Nauk. Ser. Mat. Anal. 1966 (1967) pp. 7–82

Comments

For any system of orthogonal polynomials one can formally consider a transform pair as above, cf. (1) and (2) in Fourier series in orthogonal polynomials. The Gegenbauer transform (and, more generally, the Jacobi transform) has been considered for arguments $ n $ which are arbitrarily complex. Then inversion formulas exist in the form of integrals and there is a relationship with sampling theory, cf. [a1], [a2].

References

[a1] P.L. Butzer, R.L. Stens, M. Wehrens, "The continuous Legendre transform, its inverse transform, and applications," Internat. J. Math. Sci. , 3 (1980) pp. 47–67
[a2] T.H. Koornwinder, G.G. Walter, "The finite continuous Jacobi transform and its inverse" J. Approx. Theory (To appear)
How to Cite This Entry:
Gegenbauer transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gegenbauer_transform&oldid=47057
This article was adapted from an original article by A.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article