Applied Mathematics Letters

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1 Appled Matheatcs Letters 2 (2) 46 5 Contents lsts avalable at ScenceDrect Appled Matheatcs Letters journal hoepage: wwwelseverco/locate/al Calculaton of coeffcents of a cardnal B-splne Gradr V Mlovanovć a,, Zlatko Udovčć b a Faculty of Coputer Scences, Megatrend Unversty, Bulevar uetnost 29, 7 Nov Beograd, Serba b Faculty of Scences, Departent of Matheatcs, Zaja od Bosne 5, 7 Sarajevo, Bosna and Herzegovna a r t c l e n f o a b s t r a c t Artcle hstory: Receved 5 Aprl 2 Receved n revsed for 6 June 2 Accepted 2 June 2 It s well known that a cardnal B-splne of order, N, s a pecewse polynoal functon In ths note we propose an effectve ethod for calculatng the coeffcents of polynoals whch consttute a cardnal B-splne 2 Elsever Ltd All rghts reserved Keywords: Cardnal B-splne Polynoal Coeffcents Introducton Cardnal B-splnes play a very portant role n the approxaton theory (dfferent ethods for solvng ntal and boundary value probles, splne nterpolaton, ult-resoluton approxaton, etc) Roughly speakng, a cardnal B-splne of order, N, s a real functon wth the followng propertes: ts support s the nterval, ; t belongs to the class C,, 2; at each nterval k, k +, k, t s a polynoal of degree In ths short note we gve an effectve, sple and useful algorth for calculatng the coeffcents of the entoned polynoals The paper s organzed as follows In ths secton we gve the defnton of the cardnal B-splne and a lst of ts basc propertes The an result s gven n the second part where we deduce recurrence relatons for calculatng the coeffcents Also, we descrbe an algorth for realzaton of the obtaned relatons Fnally, n the last secton we gve nuercal results, whch explctly descrbe cardnal B-splnes of orders 2,,, 7 Defnton A cardnal B-splne of frst order, denoted by ϕ ( ), s a characterstc functon of the nterval, ), e, {, x, ), ϕ (x), otherwse A cardnal B-splne of order, N, denoted by ϕ ( ), s defned as a convoluton ϕ (x) (ϕ ϕ ) (x) ϕ (x t)ϕ (t) dt ϕ (x t) dt R The frst author was supported n part by the Serban Mnstry of Scence and Technologcal Developent (Project: Orthogonal Systes and Applcatons, grant nuber #444) Correspondng author E-al addresses: gv@egatrendedurs (GV Mlovanovć), zzlatko@pfunsaba (Z Udovčć) /$ see front atter 2 Elsever Ltd All rghts reserved do:6/jal2629

2 GV Mlovanovć, Z Udovčć / Appled Matheatcs Letters 2 (2) Theore 2 A cardnal B-splne of order, N, has the followng propertes: supp ϕ ( ), ; 2 ϕ ( ) C, ; At each nterval k, k +, k, the cardnal B-splne of order s a polynoal of degree ; 4 For each t, ϕ (t) t ϕ (t) + t ϕ (t ), 2, () ϕ (t) ϕ (t) ϕ (t ), 2; (2) 5 A cardnal B-splne s syetrc on the nterval,, e, for each t,, ϕ (t) ϕ ( t) The proof of ths theore, as well as uch ore detals on cardnal B-splnes, can be found n or 2 2 Man result Equaltes () and (2), after soe splfcaton, gve the followng dfferental equaton ( x)ϕ (x) + ( )ϕ (x) ϕ (x) We wll look for a soluton of ths equaton (obvously t s a cardnal B-splne of order ) n a polynoal for, assung that the coeffcents of the cardnal B-splne of order are known Hence, let x k, k +, k N, k and let ϕ (x) x Then ϕ (x) ( + )a(,k) + x, and so a(,k) ( x)ϕ (x) + ( )ϕ (x) ( x) whle on the other hand we have ϕ (x) a (,k) x ( + )a (,k) ( + )a,k + + x + ( ) a (,k) x + ( )a(,k) After dentfyng the correspondng coeffcents, we obtan the followng relatons ( + ) + a(,k) + a (,k) + a(,k), 2 (2) These relatons are a syste of 2 lnear equatons wth unknown coeffcents Ths syste can be extended by the followng equaton ( ) k a (,k) + a(,k) ( ) k a(,k ) + a (,k ), whch s a consequence of the contnuty of the dervatve of order 2 of the cardnal B-splne at the pont x k The last two equatons are a subsyste of second order and one soluton of ths subsyste s gven by a (,k) a (,k) k( )a (,k ) a (,k ) ( )( k) The reanng coeffcents are obtaned fro (2) n the followng way a (,k) ( + )a (,k) + a (,k), 2 + If k, t s not possble to apply the descrbed procedure, so n ths case we use specal relatons Hence, let x, Fro the fact that s a root of the polynoal x wth ultplcty 2, t edately follows that a (,), 2 To deterne the coeffcent a (,) a (,) a(,) a(,) we use the relaton (2) (whch also holds n the case k ) and obtan,

3 48 GV Mlovanovć, Z Udovčć / Appled Matheatcs Letters 2 (2) 46 5 The obtaned relatons should be used to calculate the coeffcents of the cardnal B-splne at the left hand sde of the nterval, To reduce the calculatng process, for calculatng the coeffcents on the rght hand sde of the nterval, we use a syetry of the cardnal B-splne More precsely, we use a syetry of ts dervatves of the correspondng order Fro the fact that ϕ (x) ϕ ( x), t follows that ϕ (j) (x) ( )j ϕ (j) ( x), for all x, and all j {,,, } Now, let x k, k Then j (x) ( + )( + 2) ( + j)a (, k ) ϕ (j) ( ) j ϕ (j) ( x) j ( ) j ( + )( + 2) ( + j)a (,k) ( x) +j +j j ( ) j ( + )( + 2) ( + j)a (,k) +j x l j ( ) j ( + )( + 2) ( + j)a (,k) + The coeffcents of x have to be equal, fro whch follows a (, k ) j ( )j j! j +j ( ) l ( ) l x l l ( + )( + 2) ( + j)a (,k) +j, j 2 It s easy to verfy that the last relaton can be extended to the case j, and therefore, the leadng coeffcent can be calculated by a (, k ) ( ) a (,k) Accordng to the prevous consderaton we are able to forulate the followng result Theore 2 Let N, k, x k, k + and let ϕ (x) a (,k) x Furtherore, let the coeffcents of the cardnal B-splne of order be known The coeffcents of the cardnal B-splne of order satsfy the followng relatons: a (,) a(,) ; a (,), 2 (24) (25) If s odd and k /2 (a denotes the largest nteger not greater than the real nuber a), e, f s even and k /2, then a (,k) a (,k) ( )( k) + a (,k) k( )a (,k ) ( + )a (,k) + a (,k) 2 If s odd and /2 + k, e, f s even and /2 k, then a (, k ) a (, k ) j ( ) a (,k), ( )j j! j a (,k ), (26), 2 ; (27) (28) ( + )( + 2) ( + j)a (,k) +j, j 2 (29)

4 GV Mlovanovć, Z Udovčć / Appled Matheatcs Letters 2 (2) and a (, ), becoes (24) Furtherore, under the sae assuptons, relaton (27) becoes (25) Regardng the prevous result we can forulate the followng algorth for calculatng the coeffcents of the cardnal B-splne The nput data n the algorth s the order of the cardnal B-splne, postve nteger M The output data are coeffcents of the cardnal B-splnes of orders, 2,, M The obtaned coeffcents are eleents of the three densonal array a, where It s easy to check that the relaton (26), n the case k, under assuptons a (, ) we used tag a(,, k) a (,k) The algorth s realzed n such a way that the coeffcents of the polynoals defned on the syetrc ntervals are calculated n the sae loop If the nuber of ntervals s odd, the coeffcents of the polynoal defned on the ddle nterval are calculated n the last step The descrbed procedure repeats M tes Algorth Set a(,, ) for 2 to M do Set a(,, ), a( 2,, ) and g 2 for k to g do to calculate coeffcent a(,, k) use equalty (26) to calculate coeffcent a(,, k ) use equalty (28) for 2 to do to calculate coeffcent a(,, k) use equalty (27) to calculate coeffcent a(,, k ), set j, renae ndex of suaton and use equalty (29) f s odd then to calculate coeffcent a(,, g + ) set k g + and use equalty (26) for 2 to do to calculate coeffcent a(,, g + ) set k g + and use equalty (27) Nuercal results By usng the descrbed algorth, we calculate the coeffcents of the cardnal B-splnes of orders 2,,, 7 The correspondng coeffcents are gven n the atrx for snce that cardnal B-splne s copletely deterned n that way Naely, the atrx product a (,) a (,) a (,) a (,) a (,) a (,) a (,) a (,) a (,) x x s equal to the colun whose coponents are the correspondng polynoals The cardnal B-splne of order s equal to the frst polynoal at the nterval,, to the second polynoal at the nterval, 2, etc The cardnal B-splne of order s equal to the last polynoal at the nterval, Now we lst results for 2,,, 7 Case 2 For the cardnal B-splne of second order, the correspondng atrx s 2 Case For the cardnal B-splne of thrd order, the correspondng atrx s Case 4 For the cardnal B-splne of order 4, the correspondng atrx s

5 5 GV Mlovanovć, Z Udovčć / Appled Matheatcs Letters 2 (2) 46 5 Case 5 For the cardnal B-splne of order 5, the correspondng atrx s Case 6 For the cardnal B-splne of order 6, the correspondng atrx s Case 7 For the cardnal B-splne of order 7, the correspondng atrx s References C Chu, An Introducton to Wavelets, Acadec Press, Boston, C Chu, Wavelets: A Matheatcal Tool for Sgnal Analyss, SIAM, Phladelpha, 997