In this present paper we study the general even degree spline  i.e. the spline of degree 2m, where m is the positive integer, matches derivatives upto the order of m at the knots of uniform partition. Tarazi and Sallam[6], have been constructed an interpolating quartic spline with matching first and second derivative of a given function at the knots. A similar study was made by Tarazi and Karaballi [5], for even degree splines upto degree 10. Further, it was conjectured by Tarazi and Karaballi [5], that higher degree splines can be obtained in a similar way. They also raised a question for getting a proof for general degree splines. We provide a proof for general degree spline of degree 2m. Explicit formula for these splines are obtained. Error estimation to these splines in terms of Chebyshev norm is also represented by using the result due to Cirlet, Schults and Varga [2]. On combining the result of this paper and the result obtained by Kumar and Jha [4] with some modification we get deficient spline of general degree for approximation. The deficient splines are found useful because of the fact that, in this case we require less continuity requirements (see De Boor [3], P. 125). The restrictions of smoothness are compensated by considering additional interpolatory conditions

Jha, V. N.

Kumar, Arun

Nisar, K. S.

American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS)

  41   American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers  http://asrjetsjournal.org/   Even Degree Deficient Spline Interpolation V. N. Jhaa*, K. S. Nisarb, Arun Kumarc a,bDepartment of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Aldwasir, Kingdom of Saudi Arabia cDepartment of Mathematics & Computer Science, RD University, Jabalpur, India   Abstract In this present paper we study the general even degree spline [0,1],mS C∈  i.e. the spline of degree 2m, where m is the positive integer, matches derivatives upto the order of m at the knots of uniform partition. Reference[6], have been constructed an interpolating quartic spline with matching first and second derivative of a given function at the knots. A similar study was made by [5], for even degree splines upto degree 10. Further, it was conjectured by [5], that higher degree splines can be obtained in a similar way. They also raised a question for getting a proof for general degree splines. We provide a proof for general degree spline of degree 2m. Explicit formula for these splines are obtained. Error estimation to these splines in terms of Chebyshev norm is also represented by using the result due to [2]. On combining the result of this paper and the result obtained by [4] with some modification we get deficient spline of general degree for approximation. The deficient splines are found useful because of the fact that, in this case we require less continuity requirements (see De Boor [3], P. 125). The restrictions of smoothness are compensated by considering additional interpolatory conditions. Keywords: Even degree; Deficient; Spline interpolation.  1.  Introduction Let 0 1: 0 ... 1,nx x x∆ = < < < =  with 1 , 1, 2,..., .i ix x h i n−− = =  If mπ   denotes the polynomials of degree not greater then m. We define the class of all 2m degree deficient splines over ∆ with deficiency of order r  2 2 1(2 , , ) { : ( ) [0,1] , [ , ], 1, 2,..., 1}.m rm i iS m r S S x C x x x i nπ−+∆ = ∈ ∩ ∈ = −   Here we consider deficient splines with deficiency m. We claim the following: ------------------------------------------------------------------------ * Corresponding author.  American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 32, No  1, pp 41-48 42  2. Existence and Uniqueness We write ( ) / ,ix x h q− =  for {0,1,..., },i n∈  0 1;q≤ ≤  the value of s(x) as is  at ix  and ( ) ( )( ) .j ji is x f=   Theorem 2.1 There exists a function of unique splines ( ) [0,1],ms x C∈ of degree 2m, with matching conditions,         ( ) ( ) , 1, 2,..., ,r ri is f r m= =                                                                   (1) For i = 0,1,…,n and the boundary condition 0 0.s f=  It has the following representation in the sub interval 1[ , ], 0,1,..., 1,i ix x i n+ = −   1( ) ( ) ( ),1 1 ,2 ,2 1 1 ,2 2 ,2 11( ) ( ) ( ) { ( ) ( )} ( ),mj j j m mi m i m i m j i m j i m mjs x s T q s T q h f T q f T q h f T q−+ + + + +== + + + +∑        (2) where  ,2 10 011( ) ( 1) , 0,1,..., ,!m j mi j k im jk im k mT q q j mk ij−+ ++= =− +     = − =        ∑ ∑               ( )11,2 20 021( ) 1 ( 1) , 0,1,..., 1.1!m j jk i m k im jk im k m j jT q q j mk m j k ij− −+ + ++= =+ −      = − − = −     − − −     ∑ ∑  We observe that the functions ,1 ,2 ,2 2, ,..., ,m m m mT T T +  are linearly independent polynomials of 2 .mπ   We need the following lemma for proof of the theorem. Lemma. We have (i) ( )011 0,r jkkm k mk r j k−=− +  − =  − −  ∑    (ii)  ( )1021 1,10, .km jkm k m rfor j rk m j kfor j r− −=+ −  − = =  − − −  = <∑   (iii) 01 2,m jkm k m jk m j−=− + −   =   −   ∑  (iv) ( ) ( )01 1 1, for .m i jm j i kkm m km i jm i j k k− −− −=+  − − = ≥ +  − − −  ∑  American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 32, No  1, pp 41-48 43  Proof of Lemma. We use the notation that EB(1-x)q is the binomial expansion.  For (i) we consider EB(1-x)-m EB(1-x)m and collecting the coefficients for xr-j , we get the result. We get part (ii) if we compare the coefficients of xm-i-j in the expansion EB(1-x)-m-1 EB(1-x)2m-r . For (iii) we compare the coefficients of xm-j in the identity,                              1 1 2(1 ) (1 ) (1 ... ) (1 ) .m m m mx EB x x x x EB x+ − − −− − = + + + + −   The last result follows when we compare the coefficients of xm-1-j in the expansion of                                               1 1(1 ) (1 ) (1 ) .m mEB x x EB x− − −− = − = −  Proof of Theorem 2.1 Let D denote the differential operator with respect to x, 1 0( ), 0, ( 1).r rD D D r D−= < =  We observe that D(q) = 1/h. We have for r = 1,2,…,m-1,                               ,2 10 011( ) ( 1) .!m j mr i r j k im jk im k mD T q D qk ij−+ ++= =− +   = −      ∑ ∑  In case r < j, at q = 0,       ,2 1( ) 0.rm jD T q+ =                                                                  (3) For the case r = j, the term 1,r j kD q + + at q = 0 is not zero if and only if k = i = 0, we get   ,2 1( ) ,r rm jD T q h−+ = at q = 0.                                                     (4) For the case r > j, and i = r - j - k. But i ≥ 0, i.e. k ≤ r - j we obtain ( ) ( ),2 101!( ) 1 1 0!r jr j krm j rkm k mrD T qk r j kj h−−+=− +  = − − =  − −  ∑                    (5) by the result (i) of the lemma. It is direct to see that                                              ,2 2 ( ) 0,rm jD T q+ = at q = 0.                                                   (6) From the foregoing observations (2), yields American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 32, No  1, pp 41-48 44                                              ( ) ( )( ) ( ), 1, 2,..., 1, 0,1,..., ,r ri is x f x r m i n= = − =  from the definition of s(x) in 1[ , ].i ix x +  Now we claim  ,2 1( ) 0,rm jD T q+ = at q = 1.                                                                          (7) We can write                                       ,2 1011( ) (1 ) .!m jr j k mm jkm kD T q q qkj−++=− + = −  ∑  Using Leibnitz’s formula for successive differentiation, we get                                     ,2 10 011( ) (1 ) ,!m j rr r l m l j km jk lm k rD T q D q D qk lj−− ++= =− +   = −      ∑ ∑  which establishes the claim. Writing                                ( )01 (1 ) ,ji i jijq qi= − = −  ∑  we get              ( )11,2 20 021( ) 1 ( 1) .1!m j rkr r l j l m km jk lm k m j rD T q D q D qk m j k lj− −− + ++= =+ −    = − −    − − −    ∑ ∑  If j > r, the above expression is zero at q = 1. Now let j≤ r the term ( 1)r l jD q− −  at q = 1 is not zero if j = r – l i.e. l  = r – j. We have                   ( )1,2 202(2 )!( ) 1 ,1(2 )!m jkrm j rkr m k m rm jD T qr j k m j km r h− −+=+ −    −= −    − − − −−    ∑  1 ,rh= for j = r, at q = 1;                                                                          (8) and  ,2 2 ( ) 0,rm mD T q+ = for j < r by lemma (ii). Whence ( ) ( )1 1( ) ( ),r ri is x f x+ +=  from the definition of s(x) in 1[ , ].i ix x +   American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 32, No  1, pp 41-48 45  Hence we get 1( ) [0,1].ms x C −∈      Now we proceed to find the recurrence relation for explicit computation of is  , by using the condition that 1( ) [0,1].ms x C −∈  The m-th differential coefficient of  ,2 1( )m jT q+  and ,2 2 ( )m jT q+ at q = 0, j = 0,1,…,m, are given below:                           ,2 10 011( ) ( 1) ,!m j mr i r j k im jk im k mD T q D qk ij−+ ++= =− +   = −      ∑ ∑   1 ,mh=  for j = m.                                                                             (9) and  ,2 2 ( ) 0,m jT q+ = at q = 0.                                                                                                         (10) The above differential coefficients at q=1, i.e.                           ( ),2 1011 !( )!m m jrm j mkm kmD T qkj h−+=− +−  =   ∑ ( )1 (2 )!, 1.!( )!mmm jat qj m j h− −= =−                                                            (11) By lemma (iii), and at q = 1.                  ( ) ( )11,2 20 021( ) 1 1 .1!m j mk jm m l m k lm jk lm k m j mD T q D q D qk m j k lj− −− + ++= =+ −    = − −    − − −    ∑ ∑  The term ( 1)l jD q − is not zero only if l  = j; hence                   ( )1,2 20(2 )!( ) 1 ,1!( )!m jkmm j mkm k mm jD T qk m j kj m j h− −+=+  −= −   − − −−   ∑  1( 1) (2 )!,!( )!m jmm jj m j h− −− −=−                                                                            (12) by lemma (iv). American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 32, No  1, pp 41-48 46  Thus the condition get ( ) [0,1],ms x C∈  that is ( ) ( )1( ) ( ), 1, 2,...,m mi i i is x s x i n−= =  gives                          ( ) 1 ( )111{ ( 1) }! (2 )! ,(2 )! !( )!j j jmj i ii ijf fms s m j hm j m j+−−=+ −− = −−∑           ,m iF=   (say)                                                                                            (13) Using the boundary condition 0 0 ,s f=  we obtain                                 , 01, 1, 2,..., .ii m rrs F f i n== + =∑         3.  Error Bounds Now we shall obtain error bound for the above splines and its derivatives. For this we use the following result due to Cirlet, Schultz and Varga [2]. Let 2 [0, ]mf C h∈  be given. Let 2 1mQ − be the unique Hermite interpolant polynomial of degree 2m-1 that matches g and its first m-1 derivatives ( )rg  at 0 and h. Then the error ( ) ( ) ( )2 1 ,r r rme g Q −= −  for 0 x h≤ ≤  ([5], P. 161) ( ) [ ( )]( ) , 0,1,..., ,!(2 2 )!r m rr h x h x Fe x r mr m r−−= =−                                                          (14) where (2 )0max ( ) , 0 .mx hF g x x h≤ ≤= ≤ ≤        The bounds in (14) are best possible for r = 0 only. For some values of m (m=2 and m = 3) optimal error bounds on the derivatives ( ) ( )re x do exist due to [1], (see also Varma and Howell [7]). Let ( )s x′ be Hermite interpolation polynomial of degree 2m-1 matching ( ) , 1, 2,...,rf r m=  at xI and xi+1, therefore for any 1[ , ],i ix x x +∈  we have by (14) , with 2 1, ,mg f Q s−′ ′= =                                            ( 1) ( 1) (2 1)1[( )( )] , 0,1,..., .!( 2 )!r m rr r mi ih x x x xs f f r mr m r−+ + ++∞− −− ≤ =−  By the simple calculation we get American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 32, No  1, pp 41-48 47                         2( 1) ( 1) (2 1)[ (1 )] , 0,1,..., .!( 2 )!r m rr r mh h q qs f f r mr m r−+ + +∞−− ≤ =− Where x = xi + qh. Then for 0 1,x≤ ≤ we have for q=1/2, 2( 1) ( 1) (2 1) , 0,1,..., .4 !(2 )!m rr r mm rhs f f r mr m r−+ + +− ∞− ≤ =−                                    (15) Provided 2 1[0,1],mf C +∈  using 0 0 ,s f=  and integrating equation (15) over [0, x] for r = 0, we obtain                                2(2 1)( ) ( ) .4 (2 )!mmmhs x f x fm+∞− ≤  Thus we have proved the following: Theorem 3.1 If [0,1]ms C∈  is the Hermite interpolating polynomial of degree 2m-1, matching the derivatives  ( ) , 0,1,...,rf r m=  at the knots xi and xi+1. Then for 2 1[0,1],mf C +∈  and 1[ , ],i ix x x +∈                                   ( 1) ( 1) 2 (2 1)( ) ,r r m r ms f K r h f+ + − +∞− ≤  Where 1( ) [4 !(2 )!] , 0,1,...,m rK r r m r r m− −= − =  is constant depending on r. 4. Conclusion  In this paper we studied the existence and uniqueness of a class of general even degree deﬁcient spline the matching derivatives at the knots to a given order. We have also obtained explicit formula with error estimation of approximation. In study of statistical distributions we are getting such type of integrals                                  ( ) ( ) , for [ , ]xaf x f t dt x a b′= ∈∫  which allow us to approximate and the above obtained spline establish a new class of numerical quadrature rules. References [1]. G. Birkhoff and A. Priver, Hermite Interpolation Errors for Derivatives, J. Math. Phys., 46(1967), 440-447. [2]. P. C. Cirlet, M. H. Schultz and R. S. Varga, Numerical Methods of High-Order Accuracy. Numer. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 32, No  1, pp 41-48 48  Math, 9(1967), 394-430. [3]. C. De. Boor, A Practical Guide to Spline, Springer - Verlag, New York, 1978. [4]. A. Kumar and V. N. Jha, Odd Degree Deficient Splines, Proc. On Wavelet Analysis and Applications, Narosa Pub. 2002, 131-140. [5]. M. N. El. Tarazi and A. A. Karaballi, On Even Degree Splines with Application to Quadratures, J. Approx. Theory, 60(1990), 157-167. [6]. M. N. El. Tarazi and S. Sallam, On Quartic Spline with Application to Quadratures, Computing, 38(1987), 355-361. [7].  A. K. Varma and G. Howell, Best Error Bounds for Derivatives in Two point Birkhoff Interpolation Problems, J. Approx. Theory, 38(1983), 258-268. 

Even Degree Deficient Spline Interpolation

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Even Degree Deficient Spline Interpolation

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