Fourier series of Jacobi-Sobolev polynomials

Let {q(n)((alpha,beta,m)) (chi)}(n >= 0) be the orthonormal polynomials with respect to the Sobolev- type inner product (f, g)(alpha,beta,m) = Sigma(m)(k=0) integral(1)(-1) f ((k))(x)(g(k))(x) dwa(alpha+k), beta+k(x), a, beta > -1, m >= 1, where dwa, b(x) = (1 -x) a(1 + x) b dx. We 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-alpha,beta(p,m) As a consequence, we deduce the convergence of such partial sums in the norm of W-alpha,beta(p,m).