Iam a physics student, I don't cover any proper course in harmonic analysis, On the way of studying wavelets I make some guesses, But I don't know they are true or not. They are marked as 1 and 2
1) $f (x)=\int db \psi (x-b) f (b)db $
If this
$\psi (x-b) $ is $e^{ik (x-b)} $ , $f (x)$ look like
2)$f (x) = \int db e^{i(x-b)k} f (b) $
Definitely this is not in the form of fourier transform
What I need to know is, can we decompose a function $f (x)$ as shown above? Are this translated $e^{ikx}$ functions form a complete basis for the decomposition of a function?