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?
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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 $$ \left< T_n, T_m \right> = \int_{-1}^1 T_n(x) T_m(x) K(x) dx = 0 \quad \text{for } n \ne m $$
gt6989b
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I want to approximate sqrt(x) in interval [0,1] with chebysev polynomial but how i can change weight function of chebyshev orthogonal basis? – behmor67 Jan 27 '16 at 15:27
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@behmor67 you have to figure out which function to use to make the integral come out to $0$ – gt6989b Jan 28 '16 at 16:36