orthax.recurrence.ChebyshevU

class orthax.recurrence.ChebyshevU(scale: str = 'standard')Source

Recurrence relation for Chebyshev polynomials of the second kind \(U_n(x)\)

Chebyshev polynomials of the second kind are orthogonal on the interval (-1, 1) with the weight function \(w(x) = (1-x^2)^{1/2}\)

Parameters:

scale ({"standard", "monic", "normalized"}) – “standard” corresponds to the common scaling found in textbooks such as Abramowitz & Stegun. “monic” scales them such that the leading coefficient is 1. “normalized” scales them to have a weighted norm of 1.

__init__(scale: str = 'standard')Source

Methods

__init__([scale])
a(k)	a coefficients of the monic three term recurrence relation.
b(k)	b coefficients of the monic three term recurrence relation.
g(k)	Weighted norm of the kth monic orthogonal polynomial.
m(k)	Coefficient of x**k in the kth polynomial in the desired normalization.
weight(x)	Weight function defining inner product.

Attributes

domain

Lower and upper bounds for inner product defining orthogonality.