torch.special.shifted_chebyshev_polynomial_v
function shifted_chebyshev_polynomial_v<S extends Shape>(x: Tensor<S, 'float32'>, n: number | Tensor, options?: SpecialPolynomialOptions<S>): Tensor<S, 'float32'>Computes shifted Chebyshev polynomial of the third kind V*_n(x).
The shifted third kind V*_n(x) = V_n(2x - 1) transforms the asymmetric Chebyshev polynomial from [-1, 1] to [0, 1]. Preserves the asymmetric singularity structure of V_n but applies naturally to [0, 1]. Specialized use for:
- Half-interval problems: domain naturally [0, 1] with asymmetric boundary conditions
- Theoretical completeness: paired with W*_n for complete asymmetric basis on [0, 1]
- Asymmetric boundaries: V*_n(0) = (-1)^n+1; V*_n(1) = 1
- Rare usage: Less common than shifted first/second kind
- Theoretical role: Forms basis pair with W*_n for asymmetric [0, 1] problems
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 V*_n(x) valuesExamples
// Shifted third kind on [0, 1]
const x = torch.linspace(0, 1, 5);
const V_star_basis = torch.special.shifted_chebyshev_polynomial_v(x, 2);
// Natural for asymmetric [0, 1] problemsSee Also
- PyTorch torch.special.shifted_chebyshev_polynomial_v()
- torch.special.shifted_chebyshev_polynomial_w - Fourth kind (complement)
- torch.special.chebyshev_polynomial_v - Standard third kind on [-1, 1]