Chebyshev polynomials: T_n(x)=cos(n·arccos x). T_0=1, T_1=x, T_2=2x²−1, T_3=4x³−3x. Recurrence: T_{n+1}=2xT_n−T_{n-1}. Best approximation property: among all degree-n polynomials with leading coefficient 2^{n-1}, T_n/2^{n-1} deviates least from 0 on [-1,1].
How to Use
Enter n and x:
- T_n(x): First kind
- U_n(x): Second kind
- Coefficients: Polynomial form
Best Approximation
Chebyshev interpolation at nodes x_k=cos((2k+1)π/(2n)) avoids Runge's phenomenon. The error is within a factor of (4/π)log(n)+1 of the best possible. This makes Chebyshev the gold standard for polynomial approximation.
Applications
- Spectral methods for PDEs
- Digital filter design
- Clenshaw-Curtis quadrature
- Function approximation libraries
Step-by-Step Instructions
- 1Enter degree n.
- 2Enter x value.
- 3Compute T_n(x).
- 4Compute U_n(x).
- 5View coefficients.