Abstract:
We deal with pseudodifferential operators with smooth symbols and
Weierstras’s theorem in weighted Sobolev spaces. We describe the zeros,
critical points, zero location and nth root asymptotics of Sobolev orthogonal
polynomials, we also show the convergence in the mean and necessary
conditions for weighted mean convergence of Fourier series in orthogonal
polynomials.We consider Sobolev embeddings , concentration-compactness,
alternative, Gagliardo-Nirenberg, composition, products, Bourgain-Brezis-
Mironescu theorem concerning limiting embeddings and Hitchhiker’s guide
of fractional Sobolev spaces, we also determine the best constants for
Sobolev inequalities for higher order fractional derivatives and how to
recognize constant functions connections with Sobolev spaces. The
structures of the relative asymptotics, asymptotic properties and Fourier
series of orthogonal polynomials with a discrete and non-discrete
Gegenbauer-Sobolev inner products are investigated,we also show the
W , -convergence of Fourier–Sobolev expansions
