shifted_chebyshev_polynomial_t

→Tensor

shifted_chebyshev_polynomial_t(x: Tensor, n: int)

source

Shifted Chebyshev T polynomial $T^*_n(x)$ on $[0, 1]$ .

Maps the standard Chebyshev T family from $[-1, 1]$ onto the unit interval $[0, 1]$ via the linear substitution $x \mapsto 2x - 1$ . Useful when the natural domain of an approximation problem is $[0, 1]$ .

Parameters

xTensor

Argument on the shifted domain

[0, 1]

; any floating-point dtype.

nint

Non-negative polynomial degree.

Returns

Tensor

$T^*_n(x) = T_n(2x - 1)$ , element-wise, same shape and dtype as x.

Notes

Definition: $T^*_n(x) = T_n(2x - 1)$ . Inherits orthogonality on $[0, 1]$ with weight $1/\sqrt{x - x^2}$ .

Examples

>>> import lucid
>>> from lucid.special import shifted_chebyshev_polynomial_t
>>> shifted_chebyshev_polynomial_t(lucid.tensor([0.0, 0.5, 1.0]), n=2)
Tensor([1.0000, -1.0000, 1.0000])