On Bivariate Haar Functions and Interpolation Polynomial

Transcription

1 Joural of atheatics ad coputer sciece (24), -2 O Bivariate Haar Fuctios ad Iterpolatio Polyoial R. Dehgha, K. Rahsepar Fard Departet of Matheatics, Islaic Azad Uiversity, Masjed Soleia brach, Masjed Soleia, Ira. Departet of Coputer Egieerig, Uiversity of Qo, Qo, Ira Article history: Received February 24 Accepted March 24 Available olie March 24 E-ail: rezadehghaaoli@gail.co Abstract I this paper we cosider bivariate Haar series i geeral case, where bivariate Haar fuctios are defied o the plae. Here we defie a ew bivariate Haar fuctio that is icluded two idepedet variables. Ideed we preseted the ew fuctio that is ot i previous researches. Matheaticias have applied bivariate Haar fuctio based o tesor product that is a special case of bivariate case. I this research we defie the Haar fuctios by applyig aother way. Therefore, we defie the Haar fuctio differetly. Ad also, the iterpolatio polyoial with two variables is explaied. The we copare two ethods for calculatig the approxiatig fuctio. Naely, we cosider a uerical exaple for coparig the ew approxiatio to bivariate iterpolatio polyoial. I this exaple we copute iterpolatio polyoial by poits with Newto lattice for. The calculatios idicate that the accuracy of the obtaied solutios is acceptable whe the uber of calculatio poits is sall. Keywords: Bivariate Haar fuctio, Bivariate iterpolatio polyoial, Haar Fourier coefficiet, Haar series. Itroductio The wavelets techique allows the creatio of very fast algoriths whe copared to algoriths which are ordiarily used. Various wavelet basis are applied, we ca see soe of wavelet applicatios i [- 3]. Oe of the wavelet basis is Haar wavelet where we use of this id of base o Bivariate iterpolatio polyoial.before defiig the Haar syste, we itroduce the stadard otatio for biary itervals, which will be used throughout the rest of the paper. A biary iterval is a iterval of the for i 2, i 2, where i =, 2,, 2, =,,.

2 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), MATERIALS AND METHODS 2.. Mai properties of the Haar Syste The Haar fuctios are the ost eleetary wavelets. They still illustrate i the ost direct way soe of the ai features of wavelet decopositios. Here, we shall cosider i soe detail the properties that ae the suitable for uerical applicatios. Let be the set of (, ) such that 2,,,..., : (,). Deote (,), (, 2 2 [,], ), [, 2 2 ], (, ). Such itervals are aed dyadic. Clearly, if two dyadic itervals itersect, the oe of the cotais the other. The iclusio q p is equivalet to coditios p p p, 2 ( ) q 2. Put ( ). t If (, ), the χ x = 2, t Δ + 2, t Δ +, t Δ. The value of () t i a discotiuity poit t is defied as ( t ) li ( ( t ) ( t )). 2 If = or = 2, the the value () t i ad is defied so that () t is cotiuous i ad. The set of fuctios ( t ),(, ) is called the Haar syste (for coveiece show by H.s.). The oe-to-oe appig of o the set of itegers is realized by the followig: (,),(2, ) 2, (, ).

3 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - 2 The syste 2 x ( t ), (, ) is orthooral. Let 2, (, ). Deote by A the set of itervals: Deote by 2 A (,) ; () A {,,...,,,..., }. F the σ-algebra geerated by partitioig A. Clearly, each fuctio { i } ( i ) is easurable with respect to j F ad x ( t ) c ( t ), j j F is the sallest σ -algebra. Give, the polyoial with respect to the H.s. is a step fuctio. It is a costat o each iterval fro A. It satisfies the coditio x ( t ) 2 li ( x ( t ) x ( t )) t (,), ad x(t) is cotiuous i ad. Deote by D the set of such fuctios. Sice D is dece i L p, p, the the Haar syste is a coplete oe i L p. Give x L, the Fourier - Haar coefficiets are defied by c ( ) ( ) ; x x s ds ( ) ( ) 2 ( ) c x c x x s ( s) ds, x 2 ( 2 x ( s ) ds 2 x ( s ) ds ) 2 ( ( ) ( 2 2 x s x s )) ds, where 2, (, ). Usig stadard arguets we ca obtai the represetatio of the partial su of Fourier - Haar series S x ( t ): c ( x ) X ( t ) K ( t, s ) x ( s ) ds, i i i (3)

4 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - where K ( t, s ) X ( t ) X ( s ). X L i i i i Oe of the ai properties of the H.s. is that it fors a basis i C, L ( p ). Ay fuctio χ p (t) ( > ) is discotiuous. Therefore if x C[,], the the covergece S (x) to x is eat i. Theore 2... If x C[,], the the correspodig Haar series is covergece to the fuctio f. For proof of this Theore see [7]. 2.. Bivariate Haar systes i a special case (Tesor product i diesio two) Frequetly, the coefficiets of a everywhere coverget series a f ( x ) with respect to soe orthooral syste of fuctios { f ( x)} are recostructed by its su S( x) with the help of the usual Fourier forulas a S ( x ) f ( x ) ds. (4) Sice the fuctio S( x) is ot ecessarily su able, we assue that the itegral i forula (4) is ot the Lebesgue oe. For exaple, the coefficiets of a everywhere coverget trigooetric series are recostructed with the help of the so-called MT itegral ([9]). I this sectio, we cosider the proble of recostructio of coefficiets of the bivariate Haar series. Sice there are differet defiitios of the (oe-diesioal) Haar fuctios ([]), we ote that we use the stadard defiitio ([6]) which iplies that the Haar syste is coplete i C [,], i.e., we assue that X ( x ) o [,]; if 2 i,, i 2,, the L χ x = 2 2, x ( 2i 2 2i, ) 2 2, x ( 2i 2 +, 2i 2 +), x ot i 2i 2 2 +, 2i 2 +. We assue that at poits ad the fuctio X (x) equals the right ad left liits, respectively, ad at other poits of the seget [,] it equals the arithetic average of the right ad left liits. I [5] the p-regular covergece is cosidered, i. e., a covergece of the Haar series a (, ) ( ) ( ),,, x y a, x y 3

5 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - such that the sequece of the rectagular partial sus S ( x, y ) a ( x, y ) N, M,, verges to S ( x, y ) as i( M, N ) ad i( N / M, N / N ) p Mai properties of the ultivariate iterpolatio proble This sectio is iteded to itroduce ultivariate polyoial iterpolatio. First cosider iterpolatio proble i uivariate case. I this case, this proble has a well developed theory, see [4] ad [5] for coditios esurig its solvability Itroductio to the ultivariate case Defiitio The Lagrage iterpolatio proble (Π, X s ) is called correct (poised), if for ay values {c,, c s } there exists a uique polyoial p Π, satisfyig the coditios p(x (i) ) = c i, i =,, s. I other words, the Lagrage iterpolatio proble is to fid a uique polyoial p(x) = γ a γ x γ Π such that p(x (i) ) = γ a γ (x (i) ) γ = c i, i =,, s. (5) Thus, the correctess of iterpolatio eas that the liear syste (5) has a uique solutio for arbitrary right had side values. A ecessary coditio for this is that the uber of uows be equal to the uber of equatios: i.e., s = N. We ow that i this case the liear syste (5) has a uique solutio for arbitrary values {c,, c s }, if ad oly if the correspodig hoogeeous syste has oly trivial solutio. I other words we have Propositio The Lagrage poitwise iterpolatio proble (Π, X N ) is correct if ad oly if p Π ad p x i =, i =,, N p =. Equivaletly: The iterpolatio proble (Π, X N ) is ot correct if ad oly if 4

6 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - p Π, p such that p x i =, i =,, N. (6) Whe the Lagrage iterpolatio proble for ay N distict poits i R is poised i V Π, the V is called a Haar space of order N. Haar spaces exist i abudace for =. The situatio for > is draatically differet. I this case there are o Haar spaces of diesio greater tha oe. For refieets of this iportat result see [6] For >, the Haar space of order N of least diesio is yet to be deteried ad oly be ow for a few special cases Costructio of sets of iterpolatio poits Sice the poisedess of ultivariate polyoial iterpolatio depeds o the geoetric structure of the poits at which oe iterpolates, there has bee iterested i idetifyig poits ad polyoial subspaces, for exaple Π, for which iterpolatio is poised Regular grids ad atural lattices: Here we will focus o various ethods to choose poits x,, x N i R such that the iterpolatio proble with respect to these poits is poised i R ad oreover the Lagrage forula ca be easily costructed. Clearly, this requires the followig N = N = diπ = +. The first ad ost atural approach to choose such iterpolatio odes is the triagular grid of the uit siplex fored by the poits i N,. I the bivariate case, this cofiguratio has bee discussed i classical textboos o uerical aalysis, for exaple (Gasca ad Sauer, 2). This also deals with the ore geeral case of arrays fored by poits (x i, y j ), i + j, where x i, y j, i, j =,,, are two sets of + distict poits. A Newto forula with bivariate (tesor product) divided differeces is provided for this case. The bivariate array is triagular whe x i ad y j are ordered ad uiforly spaced. It was this subject which apparetly otivated the costructio i the paper [], writte by Chug ad Yao. Accordig to [], a set of N poits X = {x,, x N } i R satisfies the GC coditio (Geoetric Characterizatio) if for each poit x i there exist hyperplaes G il, l =, 2,,, such that x i is ot o ay of these hyperplaes, ad all poits of X lies o at least oe of the, i.e., we have Theore let ad be give. Let a set of N odes x,, x N be give i R. If there exists a hyperplae H ij i R, with i N ad j, such that 5

7 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - x j v= the arbitrary data o the ode set with Π is correct. H iv j i, ( i N) For exaple, let r, r,, r + be + 2 straight lies i R 2 such that ay two of the r i, r j itersect at exactly oe poit x ij ad these poits have the property that x ij x l {i, j} {, l}. The the set X = {x ij : i < j + } satisfies the GC coditio ad forula (3.3.2.) reads as p = i= + j =i+ f(x ij ) + =, i,j r. (7) r (x ij ) The set X is called a atural lattice of order The Newto lattice Let a uber Z + ad arbitrary poits a,, a R be give such that vol (S), where S is the siplex [a,, a ]. Figure : The Newto lattice Λ N (5) iside the triagle [a, a, a 2 ]. It is coveiet for us to itroduce the Newto lattice for Π iside the siplex S as follows: Λ N () = {x γ = a + i= γ i(a i a ): γ Γ}, 6

8 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - where Γ = {γ: γ = (γ,, γ ) Z +, γ }. 3. RESULTS AND DISCUSSION Now we cosider two itervals sae as the above iterval o the plae. Without loss of geerality, let us Λ =, : Z +, = 2 + i, = 2 + j, i, j =,, 2, =,,. (8) We oly cosider the proble where, Λ. Deote by χ, the Haar fuctio of two variables i diesio two, where, are positive itegers. Thus, by the above etioed we have the followig defiitios χ, x, y =, x, y,,, x, y 2, x χ 2,2 x, y = 2, 2 y, 2 x, y 2, x, y. 2 Accordig to (9) the fuctios χ,2, χ 2,, χ 2,3, χ 3,2 ca ot be defied. I iterval, [, ] we also defie: χ 3,3 (x, y) - - χ 3,4 (x, y) 7

9 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), χ 4,3 (x, y) - - χ 4,4 (x, y) - - By this way we ca defie the other fuctios. 8

10 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - Now, suppose that the fuctio f is defied o [,]. A approxiatio of a fuctio f is based o Haar fuctio is as follows: M N f x, y = = a, χ,, (2) where M, N are big possitive ubers ad a, s are Haar Fourier coefficiet. Naely, a, = f x, y, [,]. χ, x, y da. Note that each bracet is a collectio i a sae power of uber two. Based o chapter two we coclude that Theore 3. Suppose f C(, 2 ) the li, a, χ, x, y = f(x, y), x, y,,. It is easily see that the syste of {χ, },= is orthooral, i.e., where δ,,p,q is Kroecer delta. χ, x, y. χ p,q x, y dxdy = δ,,p,q, [,] 2 Accordig to (2) the fuctio f is approxiated by sixth first ters, aely f x, y [a, χ, x, y ] + [a 2,2 χ 2,2 x, y ] + a 3,3 χ 3,3 x, y + a 3,4 χ 3,4 x, y + a 4,3 χ 4,3 x, y + a4,4 χ4,4x,y (2) Now we brig a exaple of this fuctio ad approxiated also by bivariate iterpolatio polyoial. Exaple 3.2 Let f(x, y) = iterpolatio polyoial. +x 2.(+y 2 ) be a fuctio. The we decide to approxiate f by Haar ad Usig Matheatica progra the Haar Fourier coefficiets are as follows: a = π2 6, a 22 = π π 8ArcCot 2, 6 a 33 = ArcTa ArcTa , a 34 = 9 ArcTa ArcTa ,

11 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - a 43 = ArcTa[ ]ArcTa[ ], a 44 = ArcTa[ ]ArcTa[4 ] By approxiatig the above coefficiets ad forula (2) we have p x, y χ x, y χ 22 x, y χ 33 x, y χ 3,4 x, y Therefore +.727χ 4,3 x, y χ 4,4 x, y. p 4, The error is = This exaple shows that the proble is uerically stable. Now, for istace, cosider six poits,, a,, a +,,, a, a, a,, a +, 2 2 where < a < 2. The above poits have Newto lattice for. Thus, the iterpolatio polyoial coicide with these poits is uique. Agai by applyig Matheatica progra the polyoial is p(x, y) = + ( 5 4a 7a2 + 6a 3 + 8a 4 a 3 (5 + 4a + 4a 2 ) )x + ( 5 4a 7a2 + 6a 3 + 8a 4 a 3 (5 + 4a + 4a 2 )y ) + 2( 5 4a + a2 + 2a 3 ) a 3 (5 + 4a + 4a 2 x 2 + ( 2 + 2a2 a 6 ) a 4 ( + a 2 ) 2 )x y + 2( 5 4a + a2 + 2a 3 ) a 3 (5 + 4a + 4a 2 ) y 2 If a = 4, the p a, a + 2 = + 8a + 26a2 + 8a 3 + 6a 4 + 2a 5 + 3a 6 4a 7 4a 8 2a 3 ( + a 2 ) 2 (5 + 4a + 4a 2. ) p 4, 3 4 =

12 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - Note that this is far fro to exact solutio, i.e., = This eas that the proble is ot stable. 4. CONCLUSION I this wor, the Haar wavelet approach for uerical solutio of the syste of iterpolatio proble ad the syste of Haar Fourier series are preseted, Illustrative exaples are icluded to deostrate the validity ad applicability of the techique. The calculatios idicate that the accuracy of the obtaied solutios is quite high eve whe the uber of calculatio poits is sall. ACKNOWLEDGEMENTS This study was partially supported by Islaic Azad Uiversity, Masjed Soleya Brach. The authors are grateful for this fiacial support. Refereces [] K. C. Chug, ad T.H. Yao, " lattices adittig uique Lagrage iterpolatio," SIAM J. Nu. Aal 4: ,977. [2] R. Devore, ad B. Lucier, " wavelet, " Acta Nuerica, :-56,992. [3] M. Gasca, ad T. Sauer, " O the history of ultivariate polyoial iterpolatio, " Joural of Coputatioal ad Applied Matheatics, 22, 23-35,2. [4] M. Gasca, ad T. Sauer, " Polyoial iterpolatio i several variables, " Advaces i Coputatioal Matheatics, 2 :377-4,2. [5] L.S. Lee, ad G.M. Phillips, " Polyoial iterpolatio at poits of a geoetric esh o a triagle, " Proc. Roy. Soc. Edi. 8(A): 75-87,988. [6] J.R. McLaughli, J.J Price " Copariso of Haar series with gaps with trigooetric series, " Padf. J. Math., 28( 3): ,969. [7] M.G. Plotiov, " Recostructio of coefficiets of the bivariate Haar series, " Russia Matheatics (Iz. VUZ), 4.9(2): 42-5,25. [8] M.G. Plotiov, " The uiqueess of the everywhere coverget ultiple Haar series, " Vesti

13 R. Dehgha, K. Rahsepar Fard / J. Math. Coputer Sci. ( ), - Mos. Uiv., Ser. Mate., : 23-28,2. [9] V.A. Svortsov, " Differetiatio with respect to eshes ad the Haar series, " zaeti, 4( ):33-4,986. [] P.L. Ul'yaov, " Series by the Haar syste, Mate. sbori, "63( 3): ,964. [] M. Bahapour, ad M.A. Fariborzi Araghi, " Nuerical Solutio of Fredhol ad Volterra Itegral Equatios of the First Kid Usig Wavelets bases" The Joural of Matheatics ad Coputer Sciece Vol.5 No.4 (22) [2] S. Ahava, " Nuerical solutio of sigular Fredhol itegro differetial equatios of the secod id via Petrov Galeri ethod by usig Legedre ultiwavelet " Joural of atheatics ad coputer sciece 9 (24), [3] A. Neaaty, B. Agheli, ad R. Darzi, " Solvig Fractioal Partial Differetial Equatio by Usig Wavelet Operatioal Method " Joural of atheatics ad coputer Sciece 7 (23) [4] B. Bojaov,H. Haopia H, ad A.Sahaia, " Splie fuctios ad ultivariate iterpolatio, " Kluwer Acadeic Publishers, 993. [5] A. Cohe, " Aalysis of Wavelet ethods, " Elsevier, 23. [6] l. Noviov, ad E.Seeov E, " Haar series ad Liear operator, "Kluwuer Acadeic Publishers,996. [7] B.S. Kashi, ad A.A. Sahaya, " orthogoal series, " Aerica Matheatical Society,996. [8] K. Rahsepar Fard, "Orthogoal series, " Ph. D. Thesis, Yereva State Uiversity, 2. [9] A. Zigud, " Trigooetric Series, " Mir, Moscow (Russ. trasl.),965. 2