### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 024, 11 pages      arXiv:1704.01597      https://doi.org/10.3842/SIGMA.2018.024
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

### Fourier Series of Gegenbauer-Sobolev Polynomials

Óscar Ciaurri and Judit Mínguez
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain

Received January 19, 2018, in final form March 13, 2018; Published online March 17, 2018

Abstract
We study the partial sum operator for a Sobolev-type inner product related to the classical Gegenbauer polynomials. A complete characterization of the partial sum operator in an appropriate Sobolev space is given. Moreover, we analyze the convergence of the partial sum operators.

Key words: Sobolev-type inner product; Sobolev polynomials; Gegenbauer polynomials; partial sum operator.

pdf (313 kb)   tex (14 kb)

References

1. Bavinck H., Meijer H.G., Orthogonal polynomials with respect to a symmetric inner product involving derivatives, Appl. Anal. 33 (1989), 103-117.
2. Fejzullahu B.X., Marcellán F., A Cohen type inequality for Gegenbauer-Sobolev expansions, Rocky Mountain J. Math. 43 (2013), 135-148.
3. Guadalupe J.J., Pérez M., Ruiz F.J., Varona J.L., Weighted $L^p$-boundedness of Fourier series with respect to generalized Jacobi weights, Publ. Mat. 35 (1991), 449-459.
4. Guadalupe J.J., Pérez M., Ruiz F.J., Varona J.L., Weighted norm inequalities for polynomial expansions associated to some measures with mass points, Constr. Approx. 12 (1996), 341-360, math.CA/9505214.
5. Marcellán F., Quintana Y., Urieles A., On the Pollard decomposition method applied to some Jacobi-Sobolev expansions, Turkish J. Math. 37 (2013), 934-948.
6. Marcellán F., Xu Y., On Sobolev orthogonal polynomials, Expo. Math. 33 (2015), 308-352, arXiv:1403.6249.
7. Moreno A.F., Marcellán F., Osilenker B.P., Estimates for polynomials orthogonal with respect to some Gegenbauer-Sobolev type inner product, J. Inequal. Appl. 3 (1999), 401-419.
8. Muckenhoupt B., Mean convergence of Jacobi series, Proc. Amer. Math. Soc. 23 (1969), 306-310.
9. Muckenhoupt B., Transplantation theorems and multiplier theorems for Jacobi series, Mem. Amer. Math. Soc. 64 (1986), iv+86 pages.
10. Nevai P., Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc. 282 (1984), 669-698.
11. Pollard H., The mean convergence of orthogonal series. II, Trans. Amer. Math. Soc. 63 (1948), 355-367.
12. Pollard H., The mean convergence of orthogonal series. III, Duke Math. J. 16 (1949), 189-191.
13. Rodríguez J.M., Romera E., Pestana D., Alvarez V., Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials. II, Approx. Theory Appl. 18 (2002), 1-32.
14. Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
15. Xu Y., Mean convergence of generalized Jacobi series and interpolating polynomials. I, J. Approx. Theory 72 (1993), 237-251.
16. Xu Y., Mean convergence of generalized Jacobi series and interpolating polynomials. II, J. Approx. Theory 76 (1994), 77-92.