Let {q (α,β,m) n (x)} n≥0 be the orthonormal polynomials respect to the Sobolev-type inner product f, g α,β,m = m k=0 1 −1We obtain necessary and sufficient conditions for the uniform boundedness of the partial sum operators related to this sequence of polynomials in the Sobolev space W p,m α,β . As a consequence we deduce the convergence of such partial sums in the norm of W p,m α,β .2010 Mathematics Subject Classification. Primary: 42A20. Secondary: 33C47. Key words and phrases. Sobolev-type inner product, Sobolev polynomials, Jacobi polynomials, partial sum operator.
We extend Newman and Keldysh theorems to the behavior of sequences of functions in H p ðÞ which explain geometric properties of discs in these spaces. Through Keldysh's theorem we obtain asymptotic results for extremal polynomials in Sobolev spaces.
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