Abstract
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.
