Quasi-interpolation operators on hexagonal grids with high approximation orders in spaces of polyharmonic splines

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• 作者：Milvia Rossini ^{milvia.rossini@unimib.it" class="auth_mail" title="E-mail the corresponding author} ; Elena Volontè ; ; ^{e.volonte2@campus.unimib.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Polyharmonic splines ; Bees-splines ; Quasi-interpolation operators ; High degree polynomial reproduction ; Scaling functions ; Subdivision
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：272
• 期：part_P1
• 页码：223-234
• 全文大小：1609 K

文摘

In this paper we discuss the problem of approximating data given on hexagonal lattices. The construction of continuous representation from sampled data is an essential element of many applications as, for instance, image resampling, numerical solution of PDE boundary problems, etc. A useful tool is quasi-interpolation that doesn’t need the solution of a linear system. It is then important to have quasi-interpolation operators with high approximation orders, and capable to provide an efficient computation of the quasi interpolant function. To this end, we show that the idea proposed by Bozzini et al. [1] for the construction of quasi-interpolation operators in spaces of m>m  m>-harmonic splines with knots in mmlsi37" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315010413&_mathId=si37.gif&_user=111111111&_pii=S0096300315010413&_rdoc=1&_issn=00963003&md5=69e77c46e652e8ed489d0a0293b1aa9b" title="Click to view the MathML source">Z²mathContainer hidden">mathCode"><math altimg="si37.gif" overflow="scroll"><msup><mi mathvariant="double-struck">Zmi><mn>2mn>msup>math> which reproduce polynomials of high degree, can be generalized to any spaces of m>mm>-harmonic splines with knots on a lattice m>Γ  m> of mmlsi38" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315010413&_mathId=si38.gif&_user=111111111&_pii=S0096300315010413&_rdoc=1&_issn=00963003&md5=a65dd3becbe56e1d022c3c208242da28" title="Click to view the MathML source">R²mathContainer hidden">mathCode"><math altimg="si38.gif" overflow="scroll"><msup><mi mathvariant="double-struck">Rmi><mn>2mn>msup>math> and in particular on hexagonal grids. Then by a simple procedure which starts from a generator mmlsi39" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315010413&_mathId=si39.gif&_user=111111111&_pii=S0096300315010413&_rdoc=1&_issn=00963003&md5=244fb05f330e07d4d482abd31e3694b6">mg class="imgLazyJSB inlineImage" height="21" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300315010413-si39.gif">mathContainer hidden">mathCode"><math altimg="si39.gif" overflow="scroll"><msubsup><mi>ϕmi><mn>0mn><mi>Γmi>msubsup>math> with corresponding quasi-interpolation operator reproducing only linear polynomials, it is possible to define recursively generators mmlsi40" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315010413&_mathId=si40.gif&_user=111111111&_pii=S0096300315010413&_rdoc=1&_issn=00963003&md5=851aadb77bb66ebbcafa41aa576f3bac">mg class="imgLazyJSB inlineImage" height="21" width="104" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300315010413-si40.gif">mathContainer hidden">mathCode"><math altimg="si40.gif" overflow="scroll"><mrow><msubsup><mi>ϕmi><mn>1mn><mi>Γmi>msubsup><mo>,mo><mo>⋯mo><mo>,mo><msubsup><mi>ϕmi><mrow><mi>mmi><mo>&minus;mo><mn>1mn>mrow><mi>Γmi>msubsup>mrow>math> with corresponding quasi-interpolation operators reproducing polynomials up to degree mmlsi41" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315010413&_mathId=si41.gif&_user=111111111&_pii=S0096300315010413&_rdoc=1&_issn=00963003&md5=7d9b7270383e07d37bf02160b8c73449" title="Click to view the MathML source">3,5,⋯,2m&minus;1.mathContainer hidden">mathCode"><math altimg="si41.gif" overflow="scroll"><mrow><mn>3mn><mo>,mo><mn>5mn><mo>,mo><mo>⋯mo><mo>,mo><mn>2mn><mi>mmi><mo>&minus;mo><mn>1mn><mo>.mo>mrow>math> We show that this new generators of quasi-interpolation operators on a general lattice are positive definite functions, and are scaling functions whenever mmlsi39" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315010413&_mathId=si39.gif&_user=111111111&_pii=S0096300315010413&_rdoc=1&_issn=00963003&md5=244fb05f330e07d4d482abd31e3694b6">mg class="imgLazyJSB inlineImage" height="21" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300315010413-si39.gif">mathContainer hidden">mathCode"><math altimg="si39.gif" overflow="scroll"><msubsup><mi>ϕmi><mn>0mn><mi>Γmi>msubsup>math> has those properties. Moreover we are able to associate with mmlsi42" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315010413&_mathId=si42.gif&_user=111111111&_pii=S0096300315010413&_rdoc=1&_issn=00963003&md5=2279e3a1dc8f6caa3b39e5033436dd9d">mg class="imgLazyJSB inlineImage" height="24" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0096300315010413-si42.gif">mathContainer hidden">mathCode"><math altimg="si42.gif" overflow="scroll"><msubsup><mi>ϕmi><mi>jmi><mi>Γmi>msubsup>math> a dyadic convergent subdivision scheme that allows a fast computation of the quasi-interpolant.