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ABSTRACT. For the Legendre-Sobolev orthonormal polynomials B,t(z) = B,,(z ; M, N) depending on the parameters M > 0, N >_ 0, weighted and uniform estimates on the orthogonality interval are obtained. It is shown that, unlike the Legendre orthonormal polynomials, the polynomials B,t(z) for M > 0, N > 0 decay at the rate of n -z/2 at the points 1 and -1. The values of B'(:t:I) are calculated.KEY WORDS: Legendre-Sobolev polynomials, orthogonality with respect to the Legendre-Sobolev inner product, Legendre polynomials.

Let J.l be the Jacobi measure on the interval [-I, I] and introduce the discrete Sobolev-type inner productwhere c E (1,00) and M, N are non negative constants such that M + N > O. The main purpose of this paper is to study the behaviour of the Fourier series in terms of the polynomials associated to the Sobolev inner product. For an appropriate function f, we prove here that the Fourier-Sobolev series converges to f on the interval (-I, I) as well as to f( c) and the derivative of the series converges to f' (c). The term appropriate means here, in general, the same as we need for a function f(x) in order to have convergence for the series of f(x) associated to the standard inner product given by the measure J.l. No additional conditions are needed.

In this paper we obtain some estimates in [-1, 1] for orthogonal polynomials with respect to an inner product of Sobolev-type f, g) fg d/zo + where p(2c + 2) (1 x2) dx + M[6(x + 1) + 6(x-1)] d#0 2(2+1)p2(a + 1) d#l N[6(x + l) + 6(x-1)], M,N > O and a>-I Finally, the asymptotic behavior of such polynomials in [-1, 1] is analyzed.