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15
Bounds on Projections onto Bivariate Polynomial Spline Spaces with Stable Bases
 Constr. Approx
, 2002
"... . We derive L1 bounds for norms of projections onto bivariate polynomial spline spaces on regular triangulations with stable local bases. We then apply this result to derive error bounds for best L 2  and ` 2 approximation by splines on quasiuniform triangulations. x1. Introduction Let X ` L1(\O ..."
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Cited by 19 (3 self)
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. We derive L1 bounds for norms of projections onto bivariate polynomial spline spaces on regular triangulations with stable local bases. We then apply this result to derive error bounds for best L 2  and ` 2 approximation by splines on quasiuniform triangulations. x1. Introduction Let X ` L1(\Omega\Gamma be a linear space defined a set\Omega with polygonal boundary. Suppose h\Delta; \Deltai is a semidefinite innerproduct on X with associated seminorm k \Delta k. We assume that hf; gi = 0, whenever fg = 0 on \Omega\Gamma (1:1) kfk kgk, whenever jf(x)j jg(x)j for all x 2\Omega : (1:2) Suppose S ` X is a linear space of polynomial splines (bivariate piecewise polynomials) defined on a regular triangulation 4 of\Omega (two triangles intersect only at a common vertex or along a common edge). We assume that S is a Hilbert space with respect to h\Delta; \Deltai. Let P : X ! S be the projection of X onto S defined by the minimization problem kf \Gamma Pfk = min s2S kf \Gamm...
Macroelements and stable local bases for splines on PowellSabin triangulations, manuscript
 Math. Comp
, 1999
"... Abstract. Macroelements of arbitrary smoothness are constructed on PowellSabin triangle splits. These elements are useful for solving boundaryvalue problems and for interpolation of Hermite data. It is shown that they are optimal with respect to spline degree, and we believe they are also optimal ..."
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Cited by 19 (9 self)
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Abstract. Macroelements of arbitrary smoothness are constructed on PowellSabin triangle splits. These elements are useful for solving boundaryvalue problems and for interpolation of Hermite data. It is shown that they are optimal with respect to spline degree, and we believe they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over PowellSabin refinements. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power. 1.
Smooth MacroElements Based on PowellSabin Triangle Splits
 Adv. Comp. Math
, 2000
"... . Macroelements of smoothness C r on PowellSabin triangle splits are constructed for all r 0. These new elements are improvements on elements constructed in [10] in that certain unneeded degrees of freedom have been removed. x1. Introduction A bivariate macroelement defined on a triangle T co ..."
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Cited by 15 (4 self)
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. Macroelements of smoothness C r on PowellSabin triangle splits are constructed for all r 0. These new elements are improvements on elements constructed in [10] in that certain unneeded degrees of freedom have been removed. x1. Introduction A bivariate macroelement defined on a triangle T consists of a finite dimensional linear space S defined on T , and a set of linear functionals forming a basis for the dual of S. It is common to choose the space S to be a space of polynomials or a space of piecewise polynomials defined on some subtriangulation of T . The members of , called degrees of freedom, are usually taken to be point evaluations of derivatives. A macroelement defines a local interpolation scheme. In particular, if f is a sufficiently smooth function, then we can define the corresponding interpolant as the unique function s 2 S such that s = f for all 2 . We say that a macroelement has smoothness C r provided that if the element is used to construct an interpolati...
On the Approximation Order of Splines on Spherical Triangulations
 Adv. in Comp. Math
, 2004
"... Bounds are provided on how well functions in Sobolev spaces on the sphere can be approximated by spherical splines, where a spherical spline of degree d is a C r function whose pieces are the restrictions of homogoneous polynomials of degree d to the sphere. The bounds are expressed in terms of ap ..."
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Cited by 14 (2 self)
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Bounds are provided on how well functions in Sobolev spaces on the sphere can be approximated by spherical splines, where a spherical spline of degree d is a C r function whose pieces are the restrictions of homogoneous polynomials of degree d to the sphere. The bounds are expressed in terms of appropriate seminorms defined with the help of radial projection, and are obtained using appropriate quasiinterpolation operators. x1.
Nonlinear approximation from differentiable piecewise polynomials
 SIAM J. Math. Anal
"... piecewise polynomials ..."
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MacroElements and Stable Local Bases for Splines on CloughTocher Triangulations
 Math. Comp
, 1999
"... Macroelements of arbitrary smoothness are constructed on CloughTocher triangle splits. These elements can be used for solving boundaryvalue problems or for interpolation of Hermite data, and are shown to be optimal with respect to spline degree. We conjecture they are also optimal with respect to ..."
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Cited by 10 (8 self)
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Macroelements of arbitrary smoothness are constructed on CloughTocher triangle splits. These elements can be used for solving boundaryvalue problems or for interpolation of Hermite data, and are shown to be optimal with respect to spline degree. We conjecture they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over CloughTocher refinements of arbitrary triangulations. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power. x1.
Error Bounds for minimal energy interpolatory spherical splines
 NUMERISHE MATH
, 2005
"... The convergence of the minimal energy interpolatory splines on the unit sphere is studied in this paper. An upper bound on the difference between a sufficiently smooth function and its interpolatory spherical spline in the infinity norm is given. The error bound is expressed in terms of a second or ..."
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Cited by 7 (5 self)
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The convergence of the minimal energy interpolatory splines on the unit sphere is studied in this paper. An upper bound on the difference between a sufficiently smooth function and its interpolatory spherical spline in the infinity norm is given. The error bound is expressed in terms of a second order spherical Sobolevtype seminorm of the original function.
Penalized least squares fitting
 Serdica Math. J
, 2002
"... Abstract. Bounds on the error of certain penalized least squares data setting, more detailed results are included for several particularly interesting special cases, including splines in both one and several variables. 1. Introduction. We ..."
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Cited by 4 (0 self)
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Abstract. Bounds on the error of certain penalized least squares data setting, more detailed results are included for several particularly interesting special cases, including splines in both one and several variables. 1. Introduction. We
Energy minimization method for scattered data Hermite interpolation
 APPLIED NUMERICAL MATHEMATICS
, 2007
"... ..."
Locally Linearly Independent Bases for Bivariate Polynomial Spline Spaces
, 1999
"... Locally linearly independent bases are constructed for the spaces S r d (4) of polynomial splines of degree d 3r + 2 and smoothness r defined on triangulations, as well as for their superspline subspaces. ..."
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Cited by 3 (3 self)
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Locally linearly independent bases are constructed for the spaces S r d (4) of polynomial splines of degree d 3r + 2 and smoothness r defined on triangulations, as well as for their superspline subspaces.