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Gegenbauer Transformation

Gegenbauer Transformation - Integral Transformation $T\left\{F(t)\right\}$ ${\ displaystyle T \ left \ {F (t) \ right \}}$ $T \ left \ {F (t) \ right \}$ the functions $F(t)$ ${\ displaystyle F (t)}$ $F (t)$ :

T\left\{F(t)\right\}=\int \limits _{-1}^{+1}(1-t^{2})^{\rho -1/2}C_{n}^{\rho }(t)F(t)dt=f_{n}^{\rho },

{\ displaystyle T \ left \ {F (t) \ right \} = \ int \ limits _ {- 1} ^ {+ 1} (1-t ^ {2}) ^ {\ rho -1/2} C_ {n} ^ {\ rho} (t) F (t) dt = f_ {n} ^ {\ rho},}

T \ left \ {F (t) \ right \} = \ int \ limits _ {{- 1}} ^ {{+ 1}} (1-t ^ {2}) ^ {{\ rho -1/2 }} C_ {n} ^ {\ rho} (t) F (t) dt = f_ {n} ^ {\ rho},

\rho >-1/2,~n=0,~1,~2,~\ldots ,

{\ displaystyle \ rho> -1 / 2, ~ n = 0, ~ 1, ~ 2, ~ \ ldots,}

\ rho> -1 / 2, ~ n = 0, ~ 1, ~ 2, ~ \ ldots,

Where $C_{n}^{\rho }(t)$ ${\ displaystyle C_ {n} ^ {\ rho} (t)}$ $C_ {n} ^ {\ rho} (t)$ - Gegenbauer polynomials . If a function is expanded in a generalized Fourier series in terms of the Gegenbauer polynomials, then the inversion formula holds

F(t)=\sum _{n=0}^{\mathcal {1}}{{n!(n+\rho )\Gamma ^{2}(\rho )2^{2\rho -1}} \over {\pi \Gamma (n+2\rho )}}C_{n}^{\rho }(t)f_{n}^{\rho }(t),~-1<t<1

{\ displaystyle F (t) = \ sum _ {n = 0} ^ {\ mathcal {1}} {{n! (n + \ rho) \ Gamma ^ {2} (\ rho) 2 ^ {2 \ rho - 1}} \ over {\ pi \ Gamma (n + 2 \ rho)}} C_ {n} ^ {\ rho} (t) f_ {n} ^ {\ rho} (t), ~ -1 <t < one}

F (t) = \ sum _ {{n = 0}} ^ {{\ mathcal {1}}} {{n! (N + \ rho) \ Gamma ^ {2} (\ rho) 2 ^ {{2 \ rho -1}}} \ over {\ pi \ Gamma (n + 2 \ rho)}} C_ {n} ^ {\ rho} (t) f_ {n} ^ {\ rho} (t), ~ -1 <t <1

Gegenbauer transform reduces a differential operation

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

{\ displaystyle R \ left [F (t) \ right] = (1-t ^ {2}) F ^ {\ prime \ prime} - (2 \ rho +1) tF ^ {\ prime}}

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

to algebraic

T\left\{R\left[F(t)\right]\right\}=-n(n+2\rho )f_{n}^{\rho }

{\ displaystyle T \ left \ {R \ left [F (t) \ right] \ right \} = - n (n + 2 \ rho) f_ {n} ^ {\ rho}}

T \ left \ {R \ left [F (t) \ right] \ right \} = - n (n + 2 \ rho) f_ {n} ^ {\ rho}

Named in honor of the Austrian mathematician Leopold Gegenbauer (1849-1903).

Literature

Ditkin V.A., Prudnikov A.P. , in collection: The result of science. Ser. Maths. Mathematical analysis. 1966, M., 1967, p. 7-82.

Source - https://ru.wikipedia.org/w/index.php?title=Gegenbauer Transformation&oldid = 83895052

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