Orthogonality of Two Functions

[White Ink]

Homework Statement

Show that:

\varphi_{0}(x) = f_{0}(x)

and

\varphi_{1}(x) = f_{1}(x) - \frac{\left\langle\right\varphi_{0},f_{1}\rangle}{\left\|\varphi_{0}\right\|^{2}}\varphi_{0}(x)

are orthogonal on the interval [a,b].

Homework Equations

Orthogonal functions satisfy:

\left\langle\right\varphi_{m},\varphi_{n}\rangle = \int^{b}_{a}\varphi_{m}(x)\varphi_{n}(x)dx = g(m)\delta_{mn}

Where, \delta_{mn} is the Delta Kronecker.

Also:

\left\langle\right\varphi_{m},\varphi_{m}\rangle = \left\|\varphi_{m}\right\|^{2}

The Attempt at a Solution

Since m and n (0 and 1) are not equal, the Delta Kronecker is zero and therefore the proof is a matter of proving that:

\left\langle\right\varphi_{0},\varphi_{1}\rangle = 0

Having substituted the functions into the inner product formula in 2:

\left\langle\right\varphi_{0},\varphi_{1}\rangle = \int^{b}_{a}\varphi_{0}(x)\varphi_{1}(x)dx = <br /> \int^{b}_{a}f_{0}(x)\left[f_{1}(x) - \frac{\left\langle\right\varphi_{0},f_{1}\rangle}{\left\|\varphi_{0}\right\|^{2}}\varphi_{0}(x)\right]dx

Because the Delta Kronecker is zero, all I have to do is show that:

\int^{b}_{a}f_{0}(x)\left[f_{1}(x) - \frac{\left\langle\right\varphi_{0},f_{1}\rangle}{\left\|\varphi_{0}\right\|^{2}}\varphi_{0}(x)\right]dx = 0

I'm unsure as to whether I should use integration by parts to do the resulting integral because there is another integral embedded in the \varphi_{1}(x) function; which (because it is a definite integral) would be tricky to differentiate or integrate.

 

[D H]

You really don't need to bring the Kronecker delta stuff into the picture. It is completely superfluous. You do need to show that \langle \varphi_0, \varphi_1 \rangle = 0 as that is the definition of orthogonality.

Why integrate by parts? Use \langle f, g \rangle \equiv \int_a^b f(x)g(x)\,dx and \varphi_0(x) = f_0(x)