I know coefficients of some function in basis $p\_j,j=1...K$ where


$p\_{j}(x)=\sum\_{s\in Z}a\_{s,j}\exp\left(-2\pi i(j+sK)x\right)$


With respect to inner product $(f,g)=\int\_{0}^{1}f(x)\overline{g(x)}dx$ this basis is orthogonal.
What kind of basis is it? Can I somehow use Fourier analysis framework for this basis?


One more question about this basis. How to solve this integral equation?
$$a\_{j}=\int\_{-\infty}^{\infty}g(x)p\_{j}(x)dx$$
$g-$unknown function. Could you give me some hint, or some links?



some strange orthogonal basis and  an integral equation with it
If $a\_{s,j}$ is in $\ell\_2$ for any fixed $j$, $p\_j$ forms an orthogonal basis for some finite dimensional sub-space of $L^2(\mathbb{T})$. If $f(x) = \sum\_{j=0}^K b\_j p\_j(x)$, then its fourier coefficients are just given by $\hat{f}(\xi) = a\_{s,j} b\_j$ where $j = \xi \mod K$ and $s = \lfloor \xi / K \rfloor$, so surely normal Fourier analytic techniques apply. 


I don't quite understand your question about "What kind of basis is it?" Can you clarify?


