zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. (English) Zbl 1057.41003
The paper is devoted to Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. The approximation results are of the form $$ \left\Vert Q_N v-v \right\Vert _{B_1} \leq d_{N,\alpha,\beta}\vert v \vert_{B_2}, $$ where $B_1$ and $B_2$ are non-uniformly Jacobi weighted Sobolev spaces, $Q_N$ is an orthogonal projection upon the set of polynomials of degree at most $N$, and $d_{N,\alpha,\beta}$ is an explicit function of $N$, $\alpha$ and $\beta$, independent of $v$. The results are general and all estimates are as sharp as possible. First some basic results on Jacobi approximation are dissussed. Then several orthogonal approximations in non-uniformly Jacobi weighted Sobolev spaces, and also Jacobi-Gauss-type interpolations are studied. These results are useful tools in the numerical solutions of differential and integral equations.

41A10Approximation by polynomials
41A25Rate of convergence, degree of approximation
Full Text: DOI
[1] Askey, R.: Orthogonal polynomials and special functions. Regional conference series in applied mathematics 21 (1975) · Zbl 0298.33008
[2] Ka, I. Babu S. \check{}; Guo, B. Q.: Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two-dimensions. Numer. math. 85, 219-255 (2000) · Zbl 0970.65117
[3] Ka, I. Babu S. \check{}; Guo, B. Q.: Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces, part iapproximability of functions in the weighted Besov spaces. SIAM J. Numer. anal. 39, 1512-1538 (2001) · Zbl 1008.65078
[4] Bergh, J.; Löfström, J.: Interpolation spaces, an introduction. (1976) · Zbl 0344.46071
[5] C. Bernardi, M. Dauge, Y. Maday, in: P.G. Ciarlet, P.L. Lions (Eds.), Spectral Methods for Axisymmetric Domains, Series in Applied Mathematics, Vol. 3, Gauhtier-Villars & North-Holland, Paris, 1999. · Zbl 0929.35001
[6] C. Bernardi, Y. Maday, Spectral methods, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, Vol. 5, Techniques of Scientific Computing, Elsevier, Amsterdam, 1997, pp. 209--486.
[7] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A.: Spectral methods in fluid dynamics. (1988) · Zbl 0658.76001
[8] Canuto, C.; Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. comp. 38, 67-86 (1982) · Zbl 0567.41008
[9] Dubiner, M.: Spectral methods on triangles and other domains. J. sci. Comput. 6, 345-390 (1991) · Zbl 0742.76059
[10] Ezzirani, A.; Guessab, A.: A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations. Math. comp. 225, 217-248 (1999) · Zbl 0916.65020
[11] Gottlieb, D.; Orszag, S. A.: Numerical analysis of spectral methods: theory and applications. (1977) · Zbl 0412.65058
[12] Gottlieb, D.; Shu, C. W.: On the Gibbs phenomenon ivrecovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic functions. Math. comp. 64, 1081-1095 (1995) · Zbl 0852.42018
[13] Guo, Ben-Yu: Spectral methods and their applications. (1998) · Zbl 0906.65110
[14] Guo, Ben-Yu: Gegenbauer approximation and its applications to differential equations on the whole line. J. math. Anal. appl. 226, 180-206 (1998) · Zbl 0913.41020
[15] Guo, Ben-Yu: Jacobi spectral approximation and its applications to differential equations on the half line. J. comput. Math. 18, 95-112 (2000) · Zbl 0948.65071
[16] Guo, Ben-Yu: Gegenbauer approximation in certain Hilbert spaces and its applications to singular differential equations. SIAM J. Numer. anal. 37, 621-645 (2000) · Zbl 0947.65112
[17] Guo, Ben-Yu: Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. J. math. Anal. appl. 243, 373-408 (2000) · Zbl 0951.41006
[18] Guo, Ben-Yu; Shen, Jie; Wang, Zhong-Qing: A rational approximation and its applications to differential equations on the half line. J. sci. Comp. 15, 117-148 (2000)
[19] Guo, Ben-Yu; Shen, Jie; Wang, Zhong-Qing: Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval. Internat. J. Numer. meth. Eng. 53, 65-84 (2002) · Zbl 1001.65129
[20] Guo, Ben-Yu; Wang, Li-Lian: Jacobi interpolation approximations and their applications to singular differential equations. Adv. comput. Math. 14, 227-276 (2001) · Zbl 0984.41004
[21] Ben-yu Guo, Li-lian Wang, Orthogonal approximation on a triangle, unpublished. · Zbl 1116.65122
[22] Junghanns, V. P.: Uniform convergence of approximate methods for Cauchy type singular equation over (-1,1). Wiss. Z. Tech. hocsch. Karl-Mars stadt 26, 250-256 (1984) · Zbl 0575.65136
[23] Karniadakis, G.; Sherwin, S. J.: Spectral/hp element methods for CFD. (1999) · Zbl 0954.76001
[24] Kufner, A.: Weighted Sobolev spaces. (1985) · Zbl 0567.46009
[25] Schmeisser, H. J.; Triebel, H.: Topics in topics in Fourier analysis and function spaces. (1987) · Zbl 0661.46024
[26] Stephan, E. P.; Suri, M.: On the convergence of the p-version of the boundary element Galerkin method. Math. comp. 52, 31-48 (1989) · Zbl 0661.65118
[27] Szegö, G.: Orthogonal polynomials. (1959) · Zbl 0089.27501
[28] Timan, A. F.: Theory of approximation of functions of a real variable. (1963) · Zbl 0117.29001