Chebyshev polynomials of the first kind
GPTKB entity
Statements (63)
| Predicate | Object |
|---|---|
| gptkbp:instance_of |
gptkb:Marxism
|
| gptkbp:are_defined_by_the_formula |
T_n(cos(θ)) = cos(nθ)
|
| gptkbp:associated_with |
Chebyshev nodes
|
| gptkbp:asymptotic_behavior |
T_n(x) ~ (1/2) * (1 -x^2)^(1/2) for large n
|
| gptkbp:differential_equation |
(1-x^2)y'' -xy' + n^2y = 0
|
| gptkbp:even_function |
gptkb:true
|
| gptkbp:first_polynomial |
T_0(x) = 1
|
| gptkbp:has_produced |
Chebyshev series
|
| gptkbp:has_property |
bounded between -1 and 1
|
| https://www.w3.org/2000/01/rdf-schema#label |
Chebyshev polynomials of the first kind
|
| gptkbp:is_defined_by |
T_n(x) = cos(n * arccos(x))
|
| gptkbp:maximum_value |
1 for |x| <= 1
|
| gptkbp:minimum_value |
-1 for |x| <= 1
|
| gptkbp:named_after |
gptkb:Pafnuty_Chebyshev
|
| gptkbp:offers_degree |
n
|
| gptkbp:orthogonal_polynomial |
gptkb:true
|
| gptkbp:orthogonal_with_respect_to |
weight function w(x) = (1/sqrt(1-x^2))
|
| gptkbp:recurrence_relation |
T_n(x) = 2x T_{n-1}(x) -T_{n-2}(x) for n > 1
|
| gptkbp:roots |
x_k = cos((2k-1)π/(2n)), k = 1, 2, ..., n
|
| gptkbp:second_polynomial |
T_1(x) = x
|
| gptkbp:symmetric |
gptkb:true
|
| gptkbp:used_for |
finite element methods
spectral methods function interpolation |
| gptkbp:used_in |
gptkb:Control
gptkb:Artificial_Intelligence gptkb:Graphics_Processing_Unit gptkb:quantum_computing gptkb:cloud_computing gptkb:neural_networks gptkb:machine_learning gptkb:robotics image processing data analysis audio processing computer vision deep learning financial modeling high-performance computing statistical analysis video processing data mining edge computing signal processing optimization problems approximation of functions data compression data fitting numerical analysis control theory curve fitting approximation theory pattern recognition solving differential equations signal detection numerical integration image compression system identification function approximation signal reconstruction big data analysis signal filtering machine vision |