For a prescribed set of lacunary data{(x_(ν),f_(ν),f^(″)_(ν)):ν=0,1,...,N}with equally spaced knot sequence in the unit interval,we show the existence of a fam-ily of fractal splines S^(α)_(b)∈C 3[0,1]satisfyin...For a prescribed set of lacunary data{(x_(ν),f_(ν),f^(″)_(ν)):ν=0,1,...,N}with equally spaced knot sequence in the unit interval,we show the existence of a fam-ily of fractal splines S^(α)_(b)∈C 3[0,1]satisfying S^(α)_(b)(x_(ν))=f_(ν),(S^(α)_(b))^(2)(x_(ν))=f^(″)_(ν)forν=0,1,...,N and suitable boundary conditions.To this end,the unique quintic spline introduced by A.Meir and A.Sharma[SIAM J.Numer.Anal.10(3)1973,pp.433-442]is generalized by using fractal functions with variable scaling pa-rameters.The presence of scaling parameters that add extra“degrees of freedom”,self-referentiality of the interpolant,and“fractality”of the third derivative of the in-terpolant are additional features in the fractal version,which may be advantageous in applications.If the lacunary data is generated from a functionΦsatisfying certain smoothness condition,then for suitable choices of scaling factors,the corresponding fractal spline S^(α)_(b)satisfies||Φ^(r)−(S^(α)_(b))(r)||∞→0 for 0≤r≤3,as the number of partition points increases.展开更多
文摘For a prescribed set of lacunary data{(x_(ν),f_(ν),f^(″)_(ν)):ν=0,1,...,N}with equally spaced knot sequence in the unit interval,we show the existence of a fam-ily of fractal splines S^(α)_(b)∈C 3[0,1]satisfying S^(α)_(b)(x_(ν))=f_(ν),(S^(α)_(b))^(2)(x_(ν))=f^(″)_(ν)forν=0,1,...,N and suitable boundary conditions.To this end,the unique quintic spline introduced by A.Meir and A.Sharma[SIAM J.Numer.Anal.10(3)1973,pp.433-442]is generalized by using fractal functions with variable scaling pa-rameters.The presence of scaling parameters that add extra“degrees of freedom”,self-referentiality of the interpolant,and“fractality”of the third derivative of the in-terpolant are additional features in the fractal version,which may be advantageous in applications.If the lacunary data is generated from a functionΦsatisfying certain smoothness condition,then for suitable choices of scaling factors,the corresponding fractal spline S^(α)_(b)satisfies||Φ^(r)−(S^(α)_(b))(r)||∞→0 for 0≤r≤3,as the number of partition points increases.
