Gegenbauer transform

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)