Weighted Sobolev orthogonal systems with two discrete points and Fourier series with respect to them

We consider the properties of systems $\Phi_1$ orthogonal with respect to a weighted discrete-continuous Sobolev inner product of the form $\langle f,g \rangle_S = f(a)g(a)+f(b)g(b)+\displaystyle\int_a^b f'(t)g'(t)w(t)dt$. The completeness of systems $\Phi_1$ in the Sobolev space $W^1_{L^2_w}$ and the relation of $\Phi_1$ to systems orthogonal in weighted Lebesgue spaces $L^2_u$ are studied. We also analyze properties of the Fourier series with respect to systems $\Phi_1$. In particular, conditions for the uniform convergence of Fourier series to functions from $W^1_{L^2}$ are obtained.