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torch.special.shifted_chebyshev_polynomial_w

function shifted_chebyshev_polynomial_w<S extends Shape>(x: Tensor<S, 'float32'>, n: number | Tensor, options?: SpecialPolynomialOptions<S>): Tensor<S, 'float32'>

Computes shifted Chebyshev polynomial of the fourth kind W*_n(x).

The shifted fourth kind W_n(x) = W_n(2x - 1) applies the rarely-used fourth Chebyshev kind to [0, 1] domain. Forms theoretical complement to V_n for asymmetric basis pairs. Almost never used in practice but included for:

Theoretical completeness: full Chebyshev family on [0, 1]
Research in orthogonal polynomial approximation theory
Academic study of weighted orthogonal polynomial families

\begin{aligned} W^*_n(x) = W_n(2x - 1) \quad x \in [0, 1] \\ \text{Opposite asymmetry from V*_n: } W^*_n(0) = 1, \quad W^*_n(1) = (-1)^n \end{aligned}

Opposite asymmetry to V*_n: W*_n(0) = 1; W*_n(1) = (-1)^n (reverse of V*_n)
Theoretical role: Completes the Chebyshev polynomial family on [0, 1]
Almost never used: Even less common than W_n on [-1, 1]

Rarely useful: Very few practical applications; primarily theoretical interest

Parameters

xTensor<S, 'float32'>: Input tensor with values in [0, 1]
nnumber | Tensor: Polynomial degree (non-negative integer). Can be scalar or Tensor
optionsSpecialPolynomialOptions<S>optional: Optional output tensor

Returns

Tensor<S, 'float32'>– Tensor with W*_n(x) values

Examples

// Shifted fourth kind (rare usage)
const x = torch.linspace(0, 1, 5);
const W_star = torch.special.shifted_chebyshev_polynomial_w(x, 2);
// W*_n forms theoretical complement to V*_n for completeness

See Also

PyTorch torch.special.shifted_chebyshev_polynomial_w()
torch.special.shifted_chebyshev_polynomial_v - Third kind (complement)
torch.special.chebyshev_polynomial_w - Standard fourth kind on [-1, 1]

shifted_chebyshev_polynomial_v