Some Problem of Spline's Exactness

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1 Ê ³ Î ² Ê 28 ÊÊ 5 May, 2006 MATHEMATICA NUMERICA SINICA Vol.28, No.2 ÏÎ 1 ÍÌË ÊÉÈ Û ±Í± ± Ã, Á»Ó ÍÅÀ Á Schoenberg Á 1946 [20] ÝÇ ÀÅ. Schoenberg Û Å, ÝÔ»ÓÅÀ Ó¼Å, Ý Ô É»Ó ÅÀ Ó Å. ³, ÝÔ È»ÓÅ Ù À ÈÅ, ½ ½ ÛÅ ÈÅÛÈ Ë, ÆÞ ÉÅ È box»óåà ӼŠŠÃ. :»ÓÀ,»Ó, box»ó MR 2000 Æ ¾: 65D07 SOME PROBLEM OF SPLINE S EXACTNESS Zhao Yanxiang Wang Renhong Institute of Mathematical Science, Dalian University of Technology, Dalian , China Abstract Spline s exactness is firstly mentioned in [20] by Schoenberg in In this paper, we further investigate the relationship between the spline exactness and its span, and construct a special spline with minimal span when its exactness reach the maximum. Then, we consider how to describe the above problem with multivariate mathematical tool when we extend the univariate case to multivariate one so that we can continuously investigate certain multivariate box splines with their exactness and span. Keywords: Spline s Exactness, Spline s Span, Box Spline 2000 Mathematics Subject Classification: 65D07 1. À ¹ ²½ 1.1 ³ Schoneberg Ï [20] Ü ºÒ Ó ß ÅÄ Â. À ºÒÄ ÄÚ ºÒ ÌÄ Â É Ë 28 ¹ÅÁ. 1 ³ : No , No , No ÅĐÎ ³ : No Å.

2 134 ² Í ± 2006 Ï[20]Ü, ºÒÄÍ ¼«: 1.1. Í Ï Ä Ì Fx ¼ ß ¼«Ä Û: 1 Õ Ì µ k 1 Ä «; 2Fx C k 2,+ ; 3 k ß Ì, Fx ÌËß Ë; k ß Ì, Fx Ë x = n + 1 2,n = 0, ±1, ±2, ß Ë. ß k ºÒ, Æ Π k x., Π k x Ç Ì k 1, k 2. Í Ñ, Π 1 x À Ö Ì, Ï x = n Π 2x À ÖÔ Ä Ì, ÌËÏ Ë x = n. ºÒ ÌÄм ÄÏÀ ºÒÄ «Ã Äß Í Ì Ä ÄºÒ. ÄĐ¼Ú ¾ÄÀà µ ºÒÄ Í «, Ü ¼ ºÒ Ä ÈÄ Û. ¼ Đ, «ÂĺÒÌÖ ßƻɻÉß 1 Ä À ÌÄºÒ ¼«ÊÄ Ìß k B- ºÒ: M k x := 1 2π u/2 k e iux du k = 1,2, ; < x < +. 1 Ï[20]Ü, Schoenberg ¼Ó B- ºÒij Õ, M k x = 1 2π u/2 k e iux du = B- ºÒÀÇ Ð Ä k ºÒ Ì. «ÄÚ ÄÀ k B- ºÒ 1 Ì g k u = M k xe iux dx = 1 k 1! δk x k M k xcos uxdx 3 ßÚÇ Â ºÒ Ì ÛÄ. ÜÉ ÖÆ À ß M k x ß Ì. Ʋ u/2 k. g k u = 1.3. ¼ Px = n= PnFx n 4 ÀÙ k Ì, ºÒ Fx Ç k, ÖÒ¾ ƺÀ¼«Ä x ν = n ν Fx n ν = 0,1,,k 5 n= ºÒ Fx Ç s, ÀÖ Fx ÄÐ Ä Lebesgue ß s., Ì Fx x > s/2 ß 0, À x > s /2s < s, Fx Û ß 0.

3 2 ¹ Ç:»Ó ÍÀ Å Ã 135 Ù, ¼Ó ÒºÒ Fx Ä 4 ÖÖ : Ì md m, µc µ, ke k, s. Schoenberg Ï[20]ÄÚ Đ¼ÀÜ ÈĺÒ, ß ÍÄ Ì,,,. Ï ÄĐ¼Ú Ò, Í ÖºÒ Ä ÍÍ, ¾À «Đ ¼Ú Ä Đ¼Ä Ü Æ ÙºÒ Fx Ä Ìß gu, ºÒ Fx Ç k 1, ¼ «Ñ ÖÒ¾Ö : 1 u = 0 ß gu 1 Ä k ÐØ, 2 u = 2πnn 0 ß gu Ä k ÐØ. 1.2 B-  ³ Å À À k B- ºÒ 1 Ì 3 ßÚÇÄ, ¼Æ «B- ºÒ ÌÄ Ò Û k B- ºÒ 1 Ç Ì k 1D k 1, k 2C k 2, 1E 1, k. k B- ºÒ 1 Ä Ìß g k u = Laurent Í, Æ» Ã Ä Ü ÄºÒ Π k x Ì ÆÞ ¼«, Π k x = n= u/2 k, Í 1.1, ν Ì y n M k x n, 6 Ü y n À ÍÄ Ì. Ä À M k x ÄÐ, ²ÀÄÐÄ. Π k x ÀÐ Ä {y n } n= Ö Ç. 1.3 ±¼ ĵ Schoneberg Ï[20]ÜÜ Ó ±Ö ÈĺÒ, ß ÍÄ., Ò»Ä ²Ä ßÓ ¼ÏÚÄĐ. ½ Ò, Π k x Ä? Ä, Π k x Ä? À «Ä Ü Ó ÖĐ k ºÒ Π k x Ä ÔÀ k 1, Ä Ô 2k 2 s = 2k 1 k À Ì, k À Ì. Ã. Æ ÏÚ k ºÒÜÇ Ä B- ºÒ M k x, M k x = 1 2π k 2 7 u/2 k e iux du, 8

4 136 ² Í ± 2006 À M k x Ä Ìß g k u, g k u = Ð Ä ÜÑ sinu/2 k = u/2 = M k x e iux dx = u/2 k 9 [ 1 u/2 1 u/22 u 2 u/23 + u/25 3! 5! 3! + u/24 5! k = 1 + u 2 À u Ä Ö Ì. ] k sinu/2 k 1 = u 2 À u Ä Ö Ì. 10 u/2 ß g k u 1 Ï u = 0 À 2 ÐØ. ØÚ,» Ñ sinu/2 u/2 k Ï u = 2πnn 0 À k ÐØ, M k x Ä ß 1, k. ÏÚ k ºÒÜÇ Ä Ì Lx, Í 1.3, Ä À k + 2, Lx Ç ¼«Ä : Lx = y 1 M k x y 0 M k x + y 1 M k x 1, 11 Ü y 1 = y 1. ÏÚ Ì y 1, y 0, y 1 ÍÄÔ, Lx Ä. ²ÏÚ Lx Ä Ì g k u, g k u = Lxe iux dx = y 1 e iu + y 0 + y 1 e iu sinu/2 u/2 Laurent Í, g k u = 1 y u y iu2 + iu4 + 2! 4! = 1 y u y 1 + 2y 1 iu2 + 2! À y 0, y 1 À Ò, Í ÈÄ y 0, y 1, Ã Ä Laurent Í ß : a Ì ß 1, b u 2 Ìß 0. Ä k, g k u 1 = u 4 À u Ä Ö Ì. 12

5 2 ¹ Ç:»Ó ÍÀ Å Ã 137 ß, g k u 1 Ï u = 0 À 4 ÐØ. ØÚ,» Ñ g k u Ï u = 2πnn 0 À k Ð Ø. s = k + 2 Ä, k ºÒÄ 3. ÖºÄ, s = k + 4, Ä k ºÒÄ 5. g k u Ï u = 2πnn 0 À k ÐØ, Í 1.1 Ñ, k ºÒÄ µ k 1. ß, Ä Ø, s = 2k 2k ß Ì, k 1, s Î, Î ; s = 2k 1k ß Ì, k 1, s Î, Î. Ó Ú, Í ÜÄĐ¼ Ü. ¹Í ß k 1 k ß Ì, Ä Ôß 2k 4, «Ä Ü, Ù k 3, Ã ß k 1 ÀÐ Ä. À[20]¼ Ä», ±Í À ÏÚ Đ, È ÓºÒ Ä µ, Prof. Carl de Boor, Prof. Amos Ron Æ, ÀÌ ¼ ±Ä Ð. 2. À box ¹ ²½ 2.1 ± 2.1. À Ä Ð X = {x 1,...,x n } R s \{0}, X = spanx = R s, box ºÒ Bx X ¼«Í : fxbx Xdx = ft 1 x t n x n dt 1... dt n, f DR s 13 R s [ 1 2, 1 2 ]n Ü DR s Í Ï R s ÝÄÐ Ì Ð. Ï Ô Ä Ç : α,β Z s, α = α 1,...,α s, α = α α s, x α = x α xαs s, α! = α 1!... α s!, α β = α!/β!α β!. P ß ÐÜ Ä», P k ß Ì µ k Ä ÐÜ Ä», P k = P k R s = c α x α : c α R. D i,d α ßÝ, α k D i fx = x i fx, D α fx = x α 1 1 α... xαs s fx. ÜÄ box ºÒß X Z s, ÛÏ[8], [9], [14], [15], [16] Ü ÅÄ. À Carl de Boor Ï[8],[9] Ä, Ï ÄÀ W. Dahmen C. A.Micchelli Ï[14], [15], [16] Ü Ä. ßÓà ÇÄ Åº Ú, Ä Åß 13 ÜÄ [ 1 2, 1 2 ]n, À Carl de Boor, W. Dahmen C. A.Micchelli Æ Ä [0,1] n.

6 138 ² Í ± 2006 box ºÒÜ ÖÐ¼Ä Â À» lx = span{bx α X : α Z s }. 14 ßÓ Ù Ä Ê» 14 ÜÄ, «Í ±Ö À Box ºÒÄ, Ŷ X = {Y X : X\Y R s } 15 fx Ä Ì D y fx, y R s Í ß D y fx = y i D i fx 16 } DX = {f : f C R s,d Y f = 0,Y Ŷ X 17 D Y = y Y D y,, 2.2 º box DX = À Ç box ºÒÏÚÄÀÏ» Y Ŷ X ker D Y. 18 lx = span{bx α X : α Z s } 19 Ü, ºÒÄ Ä, Ä ß. À ÄÀºÒ Ì Ð Ä, supp. Ï Ç «ºÒ, «[15],[16]Ü Ä ÑºÒÄ : S φ fx = α Z s fαφx α 20 ±ÍÄ À φx Ä ß Schoenberg. ºÒ Ï ÇÄ «ÄÀÖÏ Schoenberg «Åº Ä Ä Ó Ì. ÄĐ¼ ÄÀÏ lx ܵ φx, à φx Ä Schoenberg S φ DX ź, supp φ. S φ DX ź, ÀÖ Ä fx DX, Ï S φ fx = fx. ÖĐ Á Ä W.Dahmen C.A.Micchelli Ï[14]Ü Ë, ¼Ó¼«ÄÍ : 2.1. Ù X Z s \{0} ß Ð X = R s. À Ä y R s \cx, ÏÌÕ {d α : α bx X} à Schoenberg S φ DX ź. φx = β bx X d βbx + β X.

7 2 ¹ Ç:»Ó ÍÀ Å Ã 139 Ʋ Ü Ä φx ß supp φx =Vol s α by X X[0,1]s + α. supp φx ¼Ä Ä«. ßÓ Ë, ÎÅ Ä Bx X. s = 2, e 1 = 1,0, e 2 = 0,1, X = {e 1,...,e 1,e 2,...,e 2 }{{}}{{} m 1 Bx X À Ò Ä box ºÒ, ¾ ß «Æ» Ã Í 1.4 ļ«ØÐ: 2.2. s = 2 X = {e 1,...,e 1 }{{} m 2 } Bx X = M m1 x 1 M m2 x 2. 21,e 2,...,e 2 }, Ï φx lx, à Schoen- }{{} m 1 berg S φ Dx ź, Ö supp Bx X Ä m m 1 2 m 2 m m 2 2 Ä ÃÍ 1.4 Û, ¼ Ç. 22 À Ä Bx X, Í 2.1 Å lx ÜºÒ Ì DX ıÔÌ, À¼«±Í φx lx à supp φx À Ö ÐÍÄĐ. Ö ¼ÄÀ, Í 1.4 Í 2.2 ĵ Ð Ä À Ä Bx X Â. À È ÄºÒ Ì Ä È. Ï ÇÄ «À DX Ñ ÇÄºÒ ÒºÒ ÌÄ±Ô ÛÄ ÖÄ. Ï [16] Ü W. Dahmen C.A. Micchelli ¾Ó P φ, ÜÓ P Bx X DX Ò»Ä, Ö¼Ó DX À P Bx X Ä Ö». Px Ä»ÀÖ Ä P ß a P À Ä»; b Ä p P, pax + y P, Ü a C,y R s. P Bx X Ä Ö» P, W. Dahmen C.A. Micchelli Ó P lx, Schoenberg S φ À P ÄÏ. Æ ²ÄÏÚ P Bx X Ä» P,»ÄÀÖ¾Ó, Đ Ä» P. Ï «P à DX Æ? À P Ñ DX Ò ÇºÒÄ±Ô Û? ¾ÄÀ À Ï φx lx à Schoenberg S φ P ź? ¼ Ï ºÄ φx, ÏÚ ºÄ φx À Þ, ¼ Þ, ¼«µ Ç Ð Ä φx?

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