Sobolev orthogonal systems with two discrete points and Fourier series

We consider properties of systems $\Phi_1$ orthogonal with respect to a discrete-continuous Sobolev inner product of the form $\langle f,g \rangle_S = f(a)g(a)+f(b)g(b)+\int_a^b f'(t)g'(t)dt$. In particular, we study completeness of the $\Phi_1$ systems in the Sobolev space $W^1_{L^2}$. Additionally, we analyze properties of the Fourier series with respect to $\Phi_1$ systems and prove that these series converge uniformly to functions from $W^1_{L^2}$.