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 $${{\langle f,g\rangle }_{S}}$$ = $$f(a)g(a)$$ + $$f(b)g(b)$$ + $$\int_a^b f{\kern 1pt} '(t)g{\kern 1pt} '(t)w(t)dt$$ . The completeness of systems $${{\Phi }_{1}}$$ in the Sobolev space $$W_{{L_{w}^{2}}}^{1}$$ and the relation of $${{\Phi }_{1}}$$ to systems orthogonal in weighted Lebesgue spaces $$L_{u}^{2}$$ 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 $$W_{{{{L}^{2}}}}^{1}$$ are obtained.