Question

Let \(\left\{\varphi_{n}(x)\right\}\) be an orthogonal set of functions on \([a, b]\) such that \(\psi_{0}(x)=1\) and \(\varphi_{1}(x)=x\), Show that \(\int_{a}^{b}(\alpha x+\beta) \varphi_{n}(x) d x=0\) for \(n=2,3, \ldots\) and any constants \(\alpha\) and \(\beta\),

First we note that \(\alpha x+\beta=(\square) \Phi_{1}(x)+(\square \quad) \Psi_{0}(x)\).

Using this together with the fact that \(\varphi_{0}\) and \(\varphi_{1}\) are orthogonal to \(\varphi_{n}\) for \(n>1\), we have the following.

$$ \begin{aligned} \int_{a}^{b}(\alpha x+\beta) \varphi_{n}(x) d x &=\int_{a}^{b} a x \psi_{n}(x) d x+\int_{a}^{b} \beta \varphi_{n}(x) d x \\ &=\int_{a}^{b}\left(\square_{0}\right) \varphi_{1}(x) \varphi_{n}(x) d x+\int_{a}^{b}\left(\square_{0}\right) \psi_{0}(x) \varphi_{n}(x) d x \\ &=\square \end{aligned} $$

A real-valued function \(f\) is said to be periodic with period \(T \neq 0\) If \(f(x+T)=f(x)\) for all \(x\) in the domain of \(f\). If \(T\) is the smallest positive value for which \(f(x+D)=f(x)\) holds, then \(T\) is called the fundamental period of \(f\). Determine the fundamental period \(T\) of the given function.

$$ f(x)=\sin \left(\frac{10}{L} x\right), L>0 } \\ {T=|\quad|} \end{array} $$

Answer 1