Chenge weight function in shifted orthogonality

Question

We know that in Chebyshev orthogonal polynomial the weight function is $$\frac{1}{\sqrt{1-x^2}}$$ in interval $[-1,1]$. Do in shifted chebyshev orthogonal as example for interval $[0,1]$ the weight function changed?

Answer 1

What you need is a scaling function $K(x)$ so that $$ \left< T_n, T_m \right> = \int_0^1 T_n(x) T_m(x) K(x) dx = 0 \quad \text{for } n \ne m, $$ which is the orthogonality criterion.

In the standard case, picking $$K(x) = \frac{1}{\sqrt{1-x^2}},$$ guarantees $$ \left< T_n, T_m \right> = \int_{-1}^1 T_n(x) T_m(x) K(x) dx = 0 \quad \text{for } n \ne m $$