Chebyshev polynomials of the second kind
GPTKB entity
Statements (23)
| Predicate | Object |
|---|---|
| gptkbp:instance_of |
gptkb:Marxism
|
| gptkbp:are_defined_by_the_formula |
U_n(cos(θ)) = sin((n+1)θ)/sin(θ)
|
| gptkbp:first_few_polynomials |
U_0(x) = 1
U_1(x) = 2x U_2(x) = 4x^2 -1 U_3(x) = 8x^3 -4x U_4(x) = 16x^4 -12x^2 + 1 |
| gptkbp:generate_function |
(1 -2xt + t^2)^(-1/2)
|
| https://www.w3.org/2000/01/rdf-schema#label |
Chebyshev polynomials of the second kind
|
| gptkbp:is_defined_by |
U_n(cos(theta)) = sin((n+1)theta)/sin(theta)
|
| gptkbp:named_after |
gptkb:Pafnuty_Chebyshev
|
| gptkbp:offers_degree |
n
|
| gptkbp:orthogonal |
with respect to the weight function (1-x^2)^(1/2)
|
| gptkbp:orthogonality_condition |
∫_{-1}^{1} U_n(x) U_m(x) (1-x^2)^(1/2) dx = 0 for n ≠ m
|
| gptkbp:recurrence_relation |
U_n(x) = 2x U_{n-1}(x) -U_{n-2}(x)
|
| gptkbp:related_to |
gptkb:Chebyshev_polynomials_of_the_first_kind
|
| gptkbp:roots |
x_k = cos((k*pi)/(n+1)) for k = 0, 1, ..., n
|
| gptkbp:satisfy |
U_n(-1) = (-1)^n * 2^(n-1)
U_n(1) = n |
| gptkbp:used_in |
signal processing
numerical analysis approximation theory spectral methods |