Uniform asymptotic estimates of hypergeometric functions appearing in Potential Theory

19 pages, no figures.-- MSC1991 codes: 33C05, 33C55, 31B15.MR#: MR1393128 (98b:33007)Zbl#: Zbl 0864.33001The solution of a Dirichlet problem for the Laplace-Beltrami operator with Bergman metric in the unit ball in the complex $n$-dimensional space can be expressed in terms of integrals of which the kernel can be expanded in spherical harmonics. The coefficients in this expansion contain ratios of Gauss hypergeometric functions of the form $F(p,q;p+q+n;r^2)/ F(p,q;p+q+n;1)$. The paper studies the uniform asymptotic behaviour of $F(q,mq;q+mq+n;t)$ for large values of $q$. Several results are formulated as inequalities for certain integrals containing ratios of hypergeometric functions [Zentralblatt MATH].Research of the second author was supported by a grant of the CICYT, Ministerio de Educación y Ciencia, Spain.Publicad

Distortion of boundary sets under inner functions. II

33 pages, no figures.-- MSC2000 codes: 32A30, 30C85, 30D50.MR#: MR1379286 (97b:30035)Zbl#: Zbl 0847.32005We present a study of the metric transformation properties of inner functions of several complex variables. Along the way we obtain fractional dimensional ergodic properties of classical inner functions.Publicad

Isoperimetric inequalities in Riemann surfaces of infinite type

75 pages, 1 figure.-- MSC2000 code: 30F45.MR#: MR1715412 (2000j:30075)Zbl#: Zbl 0935.30028Research partially supported by a grant from Dirección General de Enseñanza Superior (Ministerio de Educación y Ciencia), Spain.Publicad

Quantitative mixing results and inner functions

19 pages, no figures.-- MSC2000 codes: 30D05, 30D50, 37A05, 37A25, 37F10, 28D05, 11K55.MR#: MR2262783 (2007j:37003)Zbl#: Zbl 1125.30019We study in this paper estimates on the size of the sets of points which are well approximated by orbits of other points under certain dynamical systems. We apply the results obtained to the particular case of the dynamical system generated by inner functions in the unit disk of the complex plane.D. Pestana was supported by Grants BFM2003-04780 and BFM-2003-06335-C03-02, Ministerio de Ciencia y Tecnología, Spain. J. L. Fernández and M. V. Melián were supported by Grant BFM2003-04780 from Ministerio de Ciencia y Tecnología, Spain.Publicad

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials, I

36 pages, no figures.-- MSC2000 codes: 41A10, 46E35, 46G10.-- Part II of this paper published in: Approx. Theory Appl. 18(2): 1-32 (2002), available at: http://e-archivo.uc3m.es/handle/10016/6483MR#: MR2047389 (2005k:42062)Zbl#: Zbl 1081.42024In this paper we present a definition of Sobolev spaces with respect to general measures, prove some useful technical results, some of them generalizations of classical results with Lebesgue measure and find general conditions under which these spaces are complete. These results have important consequences in approximation theory. We also find conditions under which the evaluation operator is bounded.Research by first (J.M.R.), third (E.R.) and fourth (D.P.) authors was partially supported by a grant from DGI (BFM 2000-0206-C04-01), Spain.Publicad

Approximation theory for weighted Sobolev spaces on curves

17 pages, no figures.-- MSC2000 codes: 41A10, 46E35, 46G10.MR#: MR1882649 (2003c:42002)In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete. We also prove the density of the polynomials in these spaces for non-closed compact curves and, finally, we find conditions under which the multiplication operator is bounded on the completion of polynomials. These results have applications to the study of zeroes and asymptotics of Sobolev orthogonal polynomials.Research of V. Álvarez, D. Pestana and J.M. Rodríguez partially supported by a grant from DGI, BFM2000-0206-C04-01, Spain.Publicad

Weighted Sobolev spaces on curves

45 pages, no figures.-- MSC1987 codes: 41A10, 46E35, 46G10.MR#: MR1934626 (2003j:46038)Zbl#: Zbl 1019.46026In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete for non-closed compact curves. We also prove the density of the polynomials in these spaces and, finally, we find conditions under which the multiplication operator is bounded in the space of polynomials.Research of second (D.P.), third (J.M.R.) and fourth (E.R.) authors was partially supported by a grant from DGI (BFM 2000-0206-C04-01), Spain.Publicad

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials, II

32 pages, no figures.-- MSC1987 codes: 41A10, 46E35, 46G10.-- Part I of this paper published in: Acta Appl. Math. 80(3): 273-308 (2004), available at: http://e-archivo.uc3m.es/handle/10016/6482MR#: MR1928169 (2003h:42034)Zbl#: Zbl 1095.42014^aWe present a definition of general Sobolev spaces with respect to arbitrary measures, $W^{k,p}(\Omega,\mu)$ for $1\leq p\leq\infty$. In Part I [Acta Appl. Math. 80(3): 273-308 (2004), http://e-archivo.uc3m.es/handle/10016/6482] we proved that these spaces are complete under very mild conditions. Now we prove that if we consider certain general types of measures, then $C^\infty_c({\bf R})$ is dense in these spaces. As an application to Sobolev orthogonal polynomials, we study the boundedness of the multiplication operator. This gives an estimation of the zeroes of Sobolev orthogonal polynomials.Research partially supported by a grant from DGES (MEC), Spain.Publicad

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials: a survey

6 pages, no figures.-- MSC2000 codes: 41A10, 46E35, 46G10.MR#: MR2219917 (2006k:42051)Zbl#: Zbl 1146.42005In this paper we present a definition of Sobolev spaces with respect to general measures, prove some useful technical results, some of them generalizations of classical results with Lebesgue measure and find general conditions under which these spaces are complete. These results have important consequences in approximation theory. We also find conditions under which the evaluation operator is bounded.Research by first author (J.M.R.) was partially supported by grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain. Research by second author (V.A.) was partially supported by grants from MCYT (MTM 2004-00078) and Junta de Andalucía (FQM-210), Spain. Research by third author (E.R.) was partially supported by a grant from DGI (BFM 2003-06335-C03-02), Spain. Research by fourth author (D.P.) was partially supported by grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain.Publicad

The effects of the use of organic solid wastes on the growth of citrus trees

Two types of organic solid wastes were selected to be applied in a citrus orchard: the sewage sludge (the most common on the settlements) and the manure (the most common on the agricultural fields). The sewage sludge is a residue originated from the wastewater treatment - the solid phase. Its application as an organic fertilizer may represent an alternative to the pollution effects in nature. In an orchard of orange-trees (Citrus sinensis [L.] Osbeck), the application of sludge was compared with the application of manure and with the control in order to observe the growth response of the trees. Biometric methods where used - number of leaves per tree, diameter of the trunk, leaf area, specific leaf areas and leaf chlorophyll content. In general, the citrus plants response to the application of sewage sludge and manure was positive for the plant growth, compared to the control. The application of the manure, and especially the sewage sludge (once this fertilizer / soil amendment is available in larger amounts) may be a profitable alternative application to the use of mineral fertilizers and to other soil amendments. Moreover, as the possible destinations of sewage sludge (sea, deposition in land fields, incineration) provoke environmental problems, its agricultural reuse is essential to avoid those problems, when correctly applied in relation to trace elements and to pathogenic parameters. The reuse of this solid waste may be a clean and a safe technique to preserve the environment contamination