Orthonormal basis

**tttcomrader** · April 18th 2008, 06:58 AM

Assume that $\{ P_n(x) \} ^{ \infty }_{n=0}$ is an orthonormal basis with respect to $<f,g> = \int _{-1} ^{1} f(t)g(t)dt$

Find an orthonormal basis $\{ Q_n (x) \} ^ { \infty } _{n=0}$ with respect to $<f,g> = \int ^1 _0 f(t)g(t)dt$

**Opalg** · April 18th 2008, 08:05 AM

Originally Posted by tttcomrader

Assume that $\{ P_n(x) \} ^{ \infty }_{n=0}$ is an orthonormal basis with respect to $<f,g> = \int _{-1} ^{1} f(t)g(t)dt$

Find an orthonormal basis $\{ Q_n (x) \} ^ { \infty } _{n=0}$ with respect to $<f,g> = \int ^1 _0 f(t)g(t)dt$

The idea is to scrunch up the functions P_n so that they fit into the unit interval, then multiply them by a suitable constant so that they still have norm 1.

In fact, let $Q_n(x) = \frac1{\sqrt2}P_n(2x-1)$ . Then you just have to check that $\int_0^1Q_m(x)Q_n(x)\,dx$ is 1 if m=n and 0 otherwise.

Thread: Orthonormal basis

LinkBack

Thread Tools

Display