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Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product. (English) Zbl 1014.42019
Summary: Let $$\mu$$ be a finite positive Borel measure supported in $$[-1,1]$$ and introduce the discrete Sobolev-type inner product $\langle f,g\rangle = \int^1_{-1} f(x)g(x)d\mu(x)+\sum^K_{k=1} \sum^{N_k}_{i=0} M_{k,i} f^{(i)}(a_k)g^{(i)}(a_k),$ where the mass points $$a_k$$ belong to $$[-1,1]$$, $$M_{k,i}\geq 0$$, $$i = 0,\dots,N_k-1$$, and $$M_{k,N_k} >0$$. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure $$\mu$$ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that $$\mu$$ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in $$[-1,1]$$. The same problem with a finite number of mass points off $$[-1,1]$$ was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they considered the constants $$M_{k,i}$$ to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I. A. Rocha for the Jacobi measure and mass points in $$\mathbb{R}\setminus [-1,1]$$.

##### MSC:
 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 33C47 Other special orthogonal polynomials and functions
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##### References:
 [1] López, G.; Marcellán, F.; Van Assche, W., Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product, Constr. approx., 11, 107-137, (1995) · Zbl 0840.42017 [2] Marcellán, F.; Osilenker, B.; Rocha, I.A., On Fourier series of jacobi – sobolev orthogonal polynomials, J. inequal. appl., 7, 5, 673-699, (2002) · Zbl 1016.42014 [3] Marcellán, F.; Osilenker, B.; Rocha, I.A., On Fourier series of a discrete jacobi – sobolev inner product, J. approx. theory, 117, 1-22, (2002) · Zbl 1019.42014 [4] Marcellán, F.; Van Assche, W., Relative asymptotics for orthogonal polynomials with a Sobolev inner product, J. approx. theory, 72, 192-209, (1992) · Zbl 0771.42014 [5] Maté, A.; Nevai, P.; Totik, V., Asymptotics for the leading coefficients of orthogonal polynomials, Constr. approx., 1, 63-69, (1985) · Zbl 0582.42012 [6] Maté, A.; Nevai, P.; Totik, V., Strong and weak convergence of orthogonal polynomials on the unit circle, Amer. J. math., 109, 239-282, (1987) [7] P. Nevai, Orthogonal Polynomials, Memoirs of American Mathematical Society, Vol. 213, Amer. Math. Soc., Providence RI, 1979. · Zbl 0405.33009 [8] Osilenker, B.P., Fourier series in orthogonal polynomials, (1999), World Scientific Singapore · Zbl 1001.42013 [9] G. Szegő, Orthogonal Polynomials, American Mathematical Society Colloquium Publication, Vol. 23, 4th Edition, Amer. Math. Soc. Providence, RI, 1975.
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