Let $e_n=e^{inx}$ and Fourier coefficients are $\alpha_n= \frac{1}{2\pi} \int_{-\pi}^{\pi}f(x)\overline{e^{inx}}$ where $\overline{e^{inx}}$ is complex conjugate and $f \in L^2[-\pi,\pi]$.
I’m trying to write a new orthonormal basis and Fourier coefficients according to this new orthonormal basis for $L^2[a,b]$.
I have written if $x \in [-\pi,\pi]$ and $y \in [a,b]$ then $y= \frac{(b-a)x}{2\pi} + \frac{a+b}{2}$. But I cannot obtain orthonormality in my integrals. Could someone please write new orthonormal basis for $[a,b]$
Thanks a lot
You can check that the Fourier system of functions $u_n$ for $n\in\mathbb Z$, defined by $$u_n(y)=\frac 1{\sqrt{b-a}}e^{ 2in\pi\frac{ y}{b-a}}$$ forms an orthonormal basis of $L^2([a,b])$.