TWO DIMENSIONAL INTERPOLATION USING TENSOR PRODUCT OF CHEBYSHEV SYSTEMS

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1 Proceedings of the Third Interntionl Conference on Mthetics nd Nturl Sciences (ICMNS ) TWO DIMENSIONAL INTERPOLATION USING TENSOR PRODUCT OF CHEYSHEV SYSTEMS Lukit Abrwti, nd Hendr Gunwn Anlsis nd Geoetr Group Institut Teknologi ndung, West Jv, Indonesi Mthetics Deprtent Universits Negeri Jkrt, West Jv, Indonesi Abstrct. In this pper we will discuss bout n interpoltion proble of set of dt on n grid on A A. We will use tensor product of two Chebshev sstes on A nd A s the spce of interpolnts. The procedure will be eplined nd soe eples will be presented. Kewords: interpol tion, Chebshev sste. Introduction The double Fourier e series (), ) : ( k )( l ) kl k l cn be used to interpolte set of points {( i,, ci ) : i,,..., ;,,..., } on n grid on [,] (see [],[]). substituting the given points, we obtin the sste of liner equtions AX, where A ( k ) ( l ) is block tri of size. Since det( A) det ( k ) ( l ) () i det ( k i) det ( l ), i 97

2 Two Diensionl Interpoltion ug Tensor Product of Chebshev Sstes AX hs unique solution. In this pper we will generlize the bove result. Let A nd A be copct Husdorff topologicl spces. Let {( i,, ci ) : i,,..., ;,,..., } be set of points on A A, where or. Let {,,..., } nd {,,..., } be the set of functions on A nd A respectivel. We will show tht the functions, ) n n ( ) n ( ) () cn interpolte the given points if ( ) nd ( ) i re nongulr. The set of functions {,,..., } nd {,,..., } which stisf this condition re known s Chebshev sste. Furtherore, the function () cn be used to interpolte n set points tht is contined in n grid on A A. Two Diensionl Interpoltion. Proble : Full Grid i Definition (Chebshev Sste) [] Let A be copct Husdorff topologicl spce tht contin t lest n points. A set of continuous, cople or rel vlue, functions {,..., n } on A is clled Chebshev sste on A if it stisfies the following condition: For rbitrr n distinct points,,..., n, on A then deterinnt D (,..., n ) det( ( i )) n n where row nd [ is tri of size n n (with ) ( i )] n n th -colun). ( i being the eleent on th i - Let A nd A be copct Husdorff topologicl spces. Let {( i,, ci ) : i,,..., ;,,..., } be set of points on grid on A {,,..., A, where or. Let {,,..., } } nd be Chebshev sstes on A nd A respectivel. We will show tht 97

3 LUKITA AMARWATI nd HENDRA GUNAWAN, ) ( ) ( ) n n n cn interpolte the given points. Now, substituting the given points to (), we obtin the sste of liner equtions AX, where () ( ) ( ) ( ) ( )... ( ) ( ) i i i ( ) ( i ) ( ) ( i )... ( ) ( i ) A. (5) ( ) ( ) ( ) ( )... ( ) ( ) i i i The eistence of the coefficient n on () is deterined b tri A. Definition (Kronecker Product) [] k respectivel. The Kronecker Let A nd be tries of size n nd l product of A nd, A is defined b block tri of size k nl, given b A :.... n n n (6) k nl Theore [] Let A nd be squre tries of size nd n n respectivel. Then n det( A ) det( A) det( ). (7) Theore Let {,,..., } nd {,,..., } be Chebshev sstes on A nd A respectivel. Let {(,, ci ) : i,,..., ;,,..., } A points on i be set of grid on A. Then the sste of liner equtions 97

4 Two Diensionl Interpoltion ug Tensor Product of Chebshev Sstes n n hs unique solution. ( ) ( ) c, i,,..., ;,,..., i n i (8) Proof. The sste of liner equtions (8) cn be written s AX, where ( ) ( ) ( ) ( )... ( ) ( ) i i i ( ) ( i ) ( ) ( i )... ( ) ( i ) A. ( ) ( ) ( ) ( )... ( ) ( ) i i i (9) ecuse nd re Chebshev sstes, we hve det( A) det ( ) ( ) i i det ( ) det ( ) (). i i This iplies tht (8) hs unique solution. Eple. Suppose we wnt to interpolte,,,,,,,,,,,,,,,,,,,,,,,,,, ug the product of two Chebshev sstes on [,]. The points re given s grid: 97

5 LUKITA AMARWATI nd HENDRA GUNAWAN Since we wnt to interpolte points on. grid on [,] Chebshev sstes tht consist of functions on [,]. [,], we use two. If we use {,, } nd {,, } s Chebshev sstes, then the generl interpolnt hs the for, ). () Substituting the vlues fro the given points nd reducing the tri into row echelon for, we get, so 97

6 Two Diensionl Interpoltion ug Tensor Product of Chebshev Sstes, ) () interpoltes the given points. b. If we use {,, } nd {,, } s Chebshev sstes, then the generl interpolnt hs the for, ) (). Substituting the vlues fro the given points nd solving the sste of liner equtions,, ) interpoltes the given points. () 6 c. Let us use {,, } nd {,, } s Chebshev sstes. Siilr to the bove, we get U (, ) (5)

7 LUKITA AMARWATI nd HENDRA GUNAWAN s n interpolnt. The grph of the functions tht interpolte the given points cn be seen in Figure. Figure : The grph of the interpolnts ug two Chebshev sstes:, } {,, (left); {, nd } {,, } nd {,, } (center); {,, } {,, } (right). nd Eple. Suppose we wnt to interpolte,,,,,,,,,,,, product of two Chebshev sstes on [,].,,,,, ug the 976

8 Two Diensionl Interpoltion ug Tensor Product of Chebshev Sstes The points re given s grid: Since we wnt to interpolte points on grid on [,] [,], we use two Chebshev sstes tht consist of functions nd functions. Siilr to the bove eple, we hve, ) (6) s n interpolnt if we use {,, } Chebshev sstes. If we use {,, } (7) s n interpolnt. Menwhile, if we use {,, } then we get nd {, } s nd {, } s Chebshev sstes, then we get, ) nd {, } s Chebshev sstes, 5 U (, ) 7 (8) s n interpolnt. The grph of the functions tht interpolte the given points cn be seen in Figure. 977

9 LUKITA AMARWATI nd HENDRA GUNAWAN Figure : The grph of the interpolnts ug the product of two Chebshev sstes: {,, } nd {, } (left), {,, } nd {, } (center) nd {,, } {, } (right). nd. Proble : Prt of Grid Let G l, l, cl : l,,..., k be n set of points on is set i i A A. So there H,, c : i,,..., ;,,..., (hs grid for), such tht H is inil grid tht contins G. This iplies k. Let {,,..., },,..., A nd } be Chebshev sstes on on A nd respectivel. We cn use, ) n n ( ) n ( ) s n interpolnt of G. Substituting the points on G to (9), we obtin the sste of liner equtions AX. It hs k equtions nd vribles. Since k, AX (9) hs n solutions. This iplies there re n sets of interpolte the given points. n such tht (9) Eple. Suppose we wnt to interpolte,,,,,,,,,,, 978

10 Two Diensionl Interpoltion ug Tensor Product of Chebshev Sstes,,,,,,,, ug the product of two Chebshev sstes on [,]. The points re given on subset of grid: Since the inil grid tht contins the given points is grid on [,] [,], we use two Chebshev sstes tht consist of functions.. If we use {,, } nd {,, } s the Chebshev sstes, then the generl interpolnt hs the for U (, ). () Substituting the vlues fro the given points nd reducing the tri into row echelon for, we get. 979

11 LUKITA AMARWATI nd HENDRA GUNAWAN There re n functions tht interpolte the given points, one of the is: U (, ).5. () b. If we use {,, } nd {,, } s the Chebshev sstes, then the generl interpolnt hs the for, ) () The se process pplied to the bove gives., ) 6 6 () s one of the functions tht interpolte the given points. c. If we use {,, } nd {,, } s the Chebshev sstes, then one of the functions tht interpolte the given points is, ) () The grph of the functions tht interpolte the given points cn be seen in Figure. 98

12 Two Diensionl Interpoltion ug Tensor Product of Chebshev Sstes Figure : The grph of the interpolnts ug the product of two Chebshev sstes: {,, } nd {,, } (left); {,, } nd {,, } (center); {,, } {,, } (right). nd Acknowledgeent L. Abrwti nd H. Gunwn re supported b PRI Reserch Grnt /. References [] H. Gunwn, E. Rusn, L.Abrwti (9), Surfces with prescribed nodes nd iniu energ integrl of frctionl order, subitted. [] G.. Lorentz (966), Approition of Function, AMS Chelse Publishing, USA. [] C.R. Ro nd M.. Ro (998), Mtri Algebr nd Its Appliction to Sttistics nd Econoetric, World Scientific, Singpore. [] E. Rusn, H. Gunwn, A.K. Supritn, R.E. Siregr (), Eksistensi interpol n usoid berdiensi du, to pper in JMS. 98

13 LUKITA AMARWATI nd HENDRA GUNAWAN Detils of uthor(s) LUKITA AMARWATI Anlsis nd Geoetr Group Institut Teknologi ndung, West Jv, Indonesi Mthetics Deprtent Universits Negeri Jkrt Jkrt, Indonesi. eil: HENDRA GUNAWAN Mthetics Deprtent Universits Negeri Jkrt Jkrt, Indonesi. eil: 98