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## Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces (2004)

Venue: | JOURNAL OF APPROXIMATION THEORY |

Citations: | 26 - 14 self |

### Citations

743 |
Interpolation Spaces: An Introduction,
- Bergh, Lofstrom
- 1976
(Show Context)
Citation Context ...In particular, L 2 w ðLÞ H0 w ðLÞ; ðu; vÞ w ðu; vÞ 0;w and jjvjj w jjvjj 0;w : For any real number r rŠþy; 0oyo1; we define the interpolation space Hr wðLÞ HrŠþ1 w ðLÞ; H rŠ w ðLÞŠ1 y as in =-=[4]-=-. Moreover, the followingGagliardo– Nirenberg-type inequality holds (see [4] and (1.10) of [6]), jjvjjr;wpjjvjj y y rŠþ1;wjjvjj1rŠ;w ARTICLE IN PRESS B.-y. Guo, L.-l. Wang / Journal of Approximation... |

302 |
Numerical Analysis of Spectral Methods
- Gottlieb, Orszag
- 1977
(Show Context)
Citation Context ...ome results on Jacobi approximations, see [2,22,23,26]. In particular, the Legendre and Chebyshev approximations have been widely used for spectral methods of non-singular differential equations, see =-=[6,7,11,13]-=-. Recently, some authors applied Jacobi approximations directly to singular problems and differential equations on unbounded domains and axisymmetric domains, see [5,14–17,20]. Furthermore, Dubiner [9... |

259 |
Spectral/hp Element Methods for CFD,
- Karniadakis, Sherwin
- 1999
(Show Context)
Citation Context ...ctions at endpoints, and so we need to study certain Jacobi interpolations, see [10]. Thirdly, in the numerical analysis of finite element methods, one used some results on Jacobi approximations, see =-=[2,22,23,26]-=-. In particular, the Legendre and Chebyshev approximations have been widely used for spectral methods of non-singular differential equations, see [6,7,11,13]. Recently, some authors applied Jacobi app... |

148 |
Orthogonal polynomials and special functions.
- Askey
- 1975
(Show Context)
Citation Context ...bi approximations; Orthogonal projections; Interpolations 1. Introduction ARTICLE IN PRESS The Jacobi polynomials J ða;bÞ l ðxÞ play important roles in mathematical analysis and its applications, see =-=[1,27,28]-=-. In the early work, one only considered Jacobi Correspondingauthor. E-mail address: byguo@guomai.sh.cn (B.-y. Guo). 1 The work of this author is partially supported by E-Institute of Shanghai Municip... |

147 |
Topics in Fourier analysis and function spaces. A WileyInterscience Publication.
- Schmeisser, Triebel
- 1987
(Show Context)
Citation Context ...N;a;bv vjj y mŠþ1;wða;bÞ ;AjjPN;a;bv 1 y vjjmŠ;wða;bÞ ;A m r p cðNðN þ a þ bÞÞ 2 jvjr;wða;bÞ ;A : & We may consider Jacobi approximations for functions belonging to Jacobiweighted Besov spaces, see =-=[3,25]-=-, and follow the same line as in [3] to derive the correspondingresult. This generalizes Theorem 2.3 of [3], since m and r are real numbers and a could be different from b: But in this case, the norm ... |

134 |
Spectral methods on triangles and other domains,
- Dubiner
- 1991
(Show Context)
Citation Context ...3]. Recently, some authors applied Jacobi approximations directly to singular problems and differential equations on unbounded domains and axisymmetric domains, see [5,14–17,20]. Furthermore, Dubiner =-=[9]-=- investigated an orthogonal approximation on a triangle in which the base functions are the products of two Jacobi polynomials, also see [23]. Jacobi approximations were also used for the numerical an... |

114 |
Weighted Sobolev Spaces,”
- Kufner
- 1985
(Show Context)
Citation Context ...e [5]. So we cannot use Jacobi approximations in uniformly weighted spaces to deal with this problem properly. It is also difficult to use such approximations for singular differential equations, see =-=[24]-=-. In the past decade, Jacobi approximations developed again because of several reasons. Firstly, Gegenbauer approximations were successfully used for removing Gibbs phenomenon, see [12]. Next, the usu... |

114 |
Theory of Approximation of Functions of a Real Variable,
- Timan
- 1963
(Show Context)
Citation Context ...bi approximations; Orthogonal projections; Interpolations 1. Introduction ARTICLE IN PRESS The Jacobi polynomials J ða;bÞ l ðxÞ play important roles in mathematical analysis and its applications, see =-=[1,27,28]-=-. In the early work, one only considered Jacobi Correspondingauthor. E-mail address: byguo@guomai.sh.cn (B.-y. Guo). 1 The work of this author is partially supported by E-Institute of Shanghai Municip... |

69 |
Approximation results for orthogonal polynomials in Sobolev spaces.
- Canuto, Quarteroni
- 1982
(Show Context)
Citation Context ...e rational approximations, see [18,19]. As we know, the more precise the results on Jacobi approximations, the more accurate the error estimates of related numerical algorithms. Canuto and Quarteroni =-=[8]-=- first studied the Legendre and Chebyshev approximations in Sobolev spaces. Bernardi and Maday [6] developed symmetric Jacobi approximations ða bÞ in uniformly weighted Sobolev spaces. However in ma... |

30 | A rational approximation and its applications to differential equations on the half line,”
- Guo, Shen, et al.
- 2000
(Show Context)
Citation Context ...arts, we studied Jacobi approximations with the parameters a; b; g; d4 1: But in some practical problems, we also need to consider certain critical cases, in which some parameters are equal to 1; see =-=[5,14,15]-=-. Here, we consider the case with a 1; b 0; g 1 and d 0: The notations H1 0;1;0 1;0ðLÞ; P1;0 N;1;0 1;0 and a1;0 1;0ð ; Þ have the same meanings as in Theorem 3.3. Theorem 3.6. For any vAH1 0;1... |

25 |
Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations,”
- Guo
- 2000
(Show Context)
Citation Context ...s orthogonal projections in non-uniformly Jacobi-weighted Sobolev spaces, in which the weights for different derivatives appearingin the expressions of norms are different. Babus˘ ka and Guo [3], Guo =-=[16,17]-=-, and Guo and Wang [20] developed such approximations. But the results in [3] are valid only for symmetric Jacobi approximations in the standard Jacobi-weighted Sobolev spaces in which the weight for ... |

16 |
Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions
- Babuška, Guo
- 2000
(Show Context)
Citation Context ...ctions at endpoints, and so we need to study certain Jacobi interpolations, see [10]. Thirdly, in the numerical analysis of finite element methods, one used some results on Jacobi approximations, see =-=[2,22,23,26]-=-. In particular, the Legendre and Chebyshev approximations have been widely used for spectral methods of non-singular differential equations, see [6,7,11,13]. Recently, some authors applied Jacobi app... |

15 |
Gegenbauer approximation and its applications to differential equations on the whole line,”
- Guo
- 1998
(Show Context)
Citation Context ...arts, we studied Jacobi approximations with the parameters a; b; g; d4 1: But in some practical problems, we also need to consider certain critical cases, in which some parameters are equal to 1; see =-=[5,14,15]-=-. Here, we consider the case with a 1; b 0; g 1 and d 0: The notations H1 0;1;0 1;0ðLÞ; P1;0 N;1;0 1;0 and a1;0 1;0ð ; Þ have the same meanings as in Theorem 3.3. Theorem 3.6. For any vAH1 0;1... |

15 |
Jacobi interpolation approximations and their applications to singular differential equations,”
- Guo, Wang
- 2001
(Show Context)
Citation Context ...n non-uniformly Jacobi-weighted Sobolev spaces, in which the weights for different derivatives appearingin the expressions of norms are different. Babus˘ ka and Guo [3], Guo [16,17], and Guo and Wang =-=[20]-=- developed such approximations. But the results in [3] are valid only for symmetric Jacobi approximations in the standard Jacobi-weighted Sobolev spaces in which the weight for derivative of order k i... |

11 |
On the convergence of the p-version of the boundary element Galerkin method
- Stephan, Suri
- 1989
(Show Context)
Citation Context ...ctions at endpoints, and so we need to study certain Jacobi interpolations, see [10]. Thirdly, in the numerical analysis of finite element methods, one used some results on Jacobi approximations, see =-=[2,22,23,26]-=-. In particular, the Legendre and Chebyshev approximations have been widely used for spectral methods of non-singular differential equations, see [6,7,11,13]. Recently, some authors applied Jacobi app... |

8 | Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces
- Babuˇska, Guo
(Show Context)
Citation Context ...dy various orthogonal projections in non-uniformly Jacobi-weighted Sobolev spaces, in which the weights for different derivatives appearingin the expressions of norms are different. Babus˘ ka and Guo =-=[3]-=-, Guo [16,17], and Guo and Wang [20] developed such approximations. But the results in [3] are valid only for symmetric Jacobi approximations in the standard Jacobi-weighted Sobolev spaces in which th... |

4 |
A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations
- Ezzirani, Guessab
- 1999
(Show Context)
Citation Context ...menon, see [12]. Next, the usual Gauss-type interpolations are not applicable to quadratures involvingderivatives of functions at endpoints, and so we need to study certain Jacobi interpolations, see =-=[10]-=-. Thirdly, in the numerical analysis of finite element methods, one used some results on Jacobi approximations, see [2,22,23,26]. In particular, the Legendre and Chebyshev approximations have been wid... |

3 |
Spectral methods, in: P.G
- Bernardi, Maday
- 1997
(Show Context)
Citation Context ...ome results on Jacobi approximations, see [2,22,23,26]. In particular, the Legendre and Chebyshev approximations have been widely used for spectral methods of non-singular differential equations, see =-=[6,7,11,13]-=-. Recently, some authors applied Jacobi approximations directly to singular problems and differential equations on unbounded domains and axisymmetric domains, see [5,14–17,20]. Furthermore, Dubiner [9... |

2 |
Gegenbauer approximation in certain Hilbert spaces and its application to singular differential equations
- Guo
(Show Context)
Citation Context ...s orthogonal projections in non-uniformly Jacobi-weighted Sobolev spaces, in which the weights for different derivatives appearingin the expressions of norms are different. Babus˘ ka and Guo [3], Guo =-=[16,17]-=-, and Guo and Wang [20] developed such approximations. But the results in [3] are valid only for symmetric Jacobi approximations in the standard Jacobi-weighted Sobolev spaces in which the weight for ... |

2 |
Zhongqing Wang. A rational approximation and its applications to di erential equations on the half line
- Guo, Shen
(Show Context)
Citation Context ...tion may not possess the regularity required by validity of approximation results in [17]. This fact also simplifies the analysis of various rational approximations induced by Jacobi polynomials, see =-=[18,19]-=-. In this paper, we described the explicit dependance of approximation results on the parameters a; b; g and d precisely. It helps us to deal with more complicated problems. For example, the convergen... |

2 |
Uniform convergence of approximate methods for Cauchy type singular equation over ð 1; 1Þ; Wiss
- Junghanns
- 1984
(Show Context)
Citation Context |

1 |
On the Gibbs phenomenon IV: recoveringexponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic functions
- Gottlieb, Shu
- 1995
(Show Context)
Citation Context ...equations, see [24]. In the past decade, Jacobi approximations developed again because of several reasons. Firstly, Gegenbauer approximations were successfully used for removing Gibbs phenomenon, see =-=[12]-=-. Next, the usual Gauss-type interpolations are not applicable to quadratures involvingderivatives of functions at endpoints, and so we need to study certain Jacobi interpolations, see [10]. Thirdly, ... |

1 |
Zhong-qing Wang, Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval
- Guo, Shen
(Show Context)
Citation Context ... a triangle in which the base functions are the products of two Jacobi polynomials, also see [23]. Jacobi approximations were also used for the numerical analysis of some rational approximations, see =-=[18,19]-=-. As we know, the more precise the results on Jacobi approximations, the more accurate the error estimates of related numerical algorithms. Canuto and Quarteroni [8] first studied the Legendre and Che... |

1 |
Orthogonal approximation on a triangle, unpublished
- Guo, Wang
(Show Context)
Citation Context ...04) 1–41 39 jjx j y r k j 2ð1 yÞ þjjx s 2ð1 x yÞ s 2q s xvjj2L 2 !1 2 ðT Þ : r 2q j j xqky vjj2L 2ðT Þ By usingthe results in Sections 2 and 4, we can derive some important approximation results, see =-=[21]-=-. For instance, if M OðL 1þ s 2rÞ; then jjPL;Mv vjjT pcM 2rs 2rþsjjvjjHr;sðT Þ :s40 But we cannot use the results in [17] for such problem. Indeed, all results in [17] are for fixed a; b; g and d; w... |