Let $f$ be a square integrable function, strictly positive almost everywhere. Consider the family of functions $f_a=f(x+a)$, where $a$ is any real number. I want to prove that if a function is orthogonal to all this family then it must be zero. The hint I was given is to use Fourier transform, that is to show that $(g,e^{ika}\hat f)=0$ implies that $g=0$. But from here I am not sure how to go on... I see that $(g,e^{ika}\hat f)$ is proportional to the inverse Fourier transform of $g^{*}\hat f$ and therefore $g^{*}\hat f$ is zero almost everywhere, but can I say that $\hat f$ is also positive, since it is the Fourier transform of a positive function?
A problem on Fourier transforms and orthogonality
3
$\begingroup$
functional-analysis
fourier-analysis
hilbert-spaces
-
0The exercise seems to be from a book, but as Julian Aguirre showed, it's not true. I guess there is a missing hypothesis. – 2012-07-20
-
0I would imagine this to be true if, in addition, $\hat{f}(\omega) \neq 0$ a.e.? – 2012-07-20
-
0I am sorry - I have just discovered there was a misprint in the problem. The hypothesis is that $\hat f>0$ a.e., not $f$. This makes the proof straightforward. – 2012-07-20