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?