# Chebyshev Polynomial

Polynomials related to trigonometry.

https://en.wikipedia.org/wiki/Chebyshev_polynomials

# 1 First Kind

$T_n(\cos(t)) = \cos(n t)$

with

$T_0(x) = 1$ $T_1(x) = x$ $T_{n+2}(x) = 2 x T_{n+1}(x) - T_n(x)$

# 2 Second Kind

$U_n(\cos(t)) \sin(t) = \sin((n + 1) t)$

with

$U_0(x) = 1$ $U_1(x) = 2 x$ $U_{n+2}(x) = 2 x U_{n+1}(x) - U_n(x)$

# 2.1 Sawtooth

Making a bandlimited sawtooth waveform:

double t = INPUT;
double X, Y;
sincos(2 * M_PI * t, &Y, &X);
double U0 = 1;
double U1 = 2 * X;
double saw = 0;
for (int i = 1; i <= HARMONICS; ++i)
{
  saw += (2 * (i & 1) - 1) * U0 * Y / i;
  double U2 = (2 * X) * U1 - U0;
  U0 = U1; U1 = U2;
}
OUTPUT = saw * (2 / M_PI));

Exercise: make number of harmonics depend on sample rate and rate of change of phase input.