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Difference between revisions of "Gegenbauer transform"

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The [[Integral transform|integral transform]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043620/g0436201.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043620/g0436202.png" />,
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<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/g/g043/g043620/g0436203.png" /></td> </tr></table>
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<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/g/g043/g043620/g0436204.png" /></td> </tr></table>
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The [[Integral transform|integral transform]]  $  T \{ F( t) \} $
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of a function  $  F( t) $,
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043620/g0436205.png" /> are the [[Gegenbauer polynomials|Gegenbauer polynomials]]. If a function can be expanded into a generalized Fourier series by Gegenbauer polynomials, the following inversion formula is valid:
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$$
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T \{ F ( t) \}  = \
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\int\limits _ { - } 1 ^ { + }  1
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( 1 - t  ^ {2} ) ^ {\rho - 1/2 }
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C _ {n}  ^  \rho  ( t) F ( t)  dt  = f _ {n} ^ { \rho } ,
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$$
  
<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/g/g043/g043620/g0436206.png" /></td> </tr></table>
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$$
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\rho  > -  
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\frac{1}{2}
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,\  n  = 0, 1 , .  .  . .
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$$
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Here  $  C _ {n}  ^  \rho  $
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are the [[Gegenbauer polynomials|Gegenbauer polynomials]]. If a function can be expanded into a generalized Fourier series by Gegenbauer polynomials, the following inversion formula is valid:
 +
 
 +
$$
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F ( t)  =  \sum _ {n = 0 } ^  \infty 
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 +
\frac{n! ( n + \rho ) \Gamma  ^ {2} ( \rho ) 2 ^ {2 \rho - 1 } }{\pi \Gamma ( n + 2 \rho ) }
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C _ {n}  ^  \rho  ( t) f _ {n} ^ { \rho } ,\ \
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- 1 < t < 1.
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$$
  
 
The Gegenbauer transform reduces the differentiation operation
 
The Gegenbauer transform reduces the differentiation operation
  
<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/g/g043/g043620/g0436207.png" /></td> </tr></table>
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$$
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R [ F ( t)]  = \
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( 1 - t  ^ {2} ) F ^ { \prime\prime } - ( 2 \rho - 1) tF ^ { \prime\prime }
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$$
  
 
to the algebraic operation
 
to the algebraic operation
  
<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/g/g043/g043620/g0436208.png" /></td> </tr></table>
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$$
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T \{ R [ F ( t)] \}  = - n ( n + 2 \rho ) f _ {n} ^ { \rho } .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
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and
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043620/g0436209.png" /> 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]]].
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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]]].
  
 
====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>

Revision as of 19:41, 5 June 2020


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

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

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

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.

and

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