- #1
T.Rex
- 62
- 0
Hi,
I fail finding a proof (even in MathWorld, in my Mathematic dictionary or on the Web) for the following property of Chebyshev polynomials:
(T_i o T_j)(x) = (T_j o T_i)(x) = T_ij(x) when x is in ] -inf ; + inf [
Example :
T_2(x) = 2x^2-1
T_3(x) = 4x^3-3x
T_3(T_2(x)) = T_2(T_3(x)) = T_6(x)
When x is in ]-1 ; +1[ , it seems easy to use x=cosA and T_n(cosA) = cos(nA) .
But how to do when x is any real ?
Thanks,
Tony
I fail finding a proof (even in MathWorld, in my Mathematic dictionary or on the Web) for the following property of Chebyshev polynomials:
(T_i o T_j)(x) = (T_j o T_i)(x) = T_ij(x) when x is in ] -inf ; + inf [
Example :
T_2(x) = 2x^2-1
T_3(x) = 4x^3-3x
T_3(T_2(x)) = T_2(T_3(x)) = T_6(x)
When x is in ]-1 ; +1[ , it seems easy to use x=cosA and T_n(cosA) = cos(nA) .
But how to do when x is any real ?
Thanks,
Tony