Chebyshevsarjojen
Chebyshev polynomials, named after the 19th-century Russian mathematician Pafnuty Chebyshev, are a sequence of orthogonal polynomials that play a significant role in numerical analysis, approximation theory, and various areas of applied mathematics. These polynomials are defined recursively and possess several important properties, including minimax approximation properties and orthogonality with respect to a weighted inner product.
The Chebyshev polynomials of the first kind, denoted as \( T_n(x) \), satisfy the recurrence relation:
\[ T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x) \]
for \( n \geq 1 \). They are defined on the interval \([-1, 1]\) and can also be expressed
\[ T_n(\cos \theta) = \cos(n\theta). \]
This connection to trigonometric functions highlights their oscillatory behavior, with \( T_n(x) \) having \( n \) extrema in the
The Chebyshev polynomials of the second kind, denoted as \( U_n(x) \), follow a similar recurrence:
\[ U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x) \]
for \( n \geq 1 \). Unlike \( T_n(x) \), \( U_n(x) \) has \( n+1 \) extrema in \([-1, 1]\) and is related
\[ U_n(\cos \theta) = \frac{\sin((n+1)\theta)}{\sin \theta}. \]
Chebyshev polynomials are widely used in numerical methods, particularly in polynomial interpolation and approximation. They minimize