An Interpolation Process on Laguerre Polynomial

Transcription

1 Global Joural of Pure ad Applied Mathematics. ISSN Volume 13, Number 10 (2017), pp Research Idia Publicatios A Iterpolatio Process o Laguerre Polyomial R. Srivastava 1 ad Geeta Vishwakarma 2 Departmet of Mathematics ad Astroomy, Luckow Uiversity, Luckow , Idia. Abstract I the preset paper,we have cosidered the problem i which {ξ i } i=1 ad {ξ i } i=1 the two sets of iterscaled odal poits o the iterval [0, ). Here we deal with the problem i which fuctio values are prescribed at the zeros of L k (x) ad the first derivative values are prescribed o the zeros L k 1 (x). We ivestigate the existece, uiqueess explicit represetatio of iterpolatory polyomial. Estimatio of the fudametal polyomials leadig to a covergece theorem have also bee obtaied. Keywords: Lacuary iterpolatio, Pál - Type iterpolatio, Laguerre Polyomial MSC 2000: 41 A D INTRODUCTION Pál [10], Mathur P. ad Datta S. [8] ad may other authors [1][2][6][7] [12] [14] have discussed about various kid of iterpolatio problems.. I 1975 Pál [10] proved that whe the fuctio values are prescribed o oe set of poits ad derivative values o other set of -1 poits, the there exist o uique polyomial of degree 2-2, but prescribig fuctio value at oe more poit ot belogig to former set of poits there exists a uique polyomial of degree 2-1. Léárd M. [5] ivestigated the Pál type iterpolatio problem o the odes of Laguerre abscissas. I this paper we study Pál type iterpolatioal polyomial with ω +k (x) = x k L (k) (x).we have examied the existece, uiqueess,explicit

2 7090 R. Srivastava ad Geeta Vishwakarma represetatio ad estimatio of fudametal polyomials of such special kid of mixed type of iterpolatio o iterval [0, ). For this we have cosidered the problem i which {ξ i } i=1 ad {ξ i } i=1 the two sets of iterscaled odal poits. (1.1) 0 ξ 0 < ξ 1 < ξ 1 < < ξ 1 < ξ < ξ < o the iterval [0, ). We seek to determie a polyomial R (x) of miimal possible degree 3+k satisfyig the iterpolatory coditios : (1.2) R (ξ i ) = g i, R (ξ i ) = g i, R (ξ i ) = g i, for i = 1(1) (1.3) R (j) (ξ 0 ) = g 0 (j), j = 0,1,, k where g i, g i, g i ad g 0 (j) L (k) (x) ad L (k 1) (x) have zeroes {ξ i } i=1 are arbitrary real umbers. Here Laguerre polyomials ad {ξ i } i=1 respectively ad x 0 = 0. We prove existece, uiqueess, explicit represetatio ad estimatio of fudametal polyomials. 2. PRELIMINARIES I this sectio we shall give some well-kow results which are as follws : As we kow that the Laguerre polyomial is a costat multiple of a cofluet hypergeometric fuctio so the differetial equatio is give by (2.1) xd 2 L k (x) + (1 + k x)dl k (x) + L k (x) = 0 (k 1) (2.2) L (x) = L (k) 1 (x) Also usig the idetities (2.3) L (k) (k+1) (k+1) (x) = L (x) L 1 (x) (k) (2.4) xl (k) (k) (x) = L (x) ( + k)l 1(x) We ca easily fid a relatio d (2.5) dx [xk L k (x)] = ( + k)x k 1 (k 1) L (x) By the followig coditios of orthogoality ad ormalizatio we defie Laguerre polyomial L (k) (x), for k > 1

3 A Iterpolatio Process o Laguerre Polyomial (2.6) e x x k L (k) (x)l (k) m (x)dx = Γk + 1 ( +k )δ m, m = 0,1,2,.. (2.7) L (k) (x) = ( +k μ=0 ) ( x)μ μ! The fudametal polyomials of Lagrage iterpolatio are give by L (k) (x) (2.8) l j (x) = (k) L = δ i,j (x j )(x x j ) L (k 1) (x) (2.9) l j (x) = (k 1) L = δ i,j (x j )(x x j ) (2.10) l j (y j ) = { L (k 1) (y i ) L (k 1) (y j )(y i y j ) (k y j) 2y j i j i = j i, j = 1(1) (2.12) l j (y j ) = 1 (k) (k) (y j x j ) [L (y j ) (k) L L (yj ) (k) (x j ) L ] (x j )(y j x j ), j = 1(1) For the roots of L (k) (x) we have (2.13) x k 2 ~ k2 (2.14) η(x) S (l) (x) = О(1) where η(x) is the weight fuctio (2.15) L (k) (x j ) ~j k 3 2 k+1, (0 < x j Ω, = 1,2,3,. ) k (2.16) L k (x j ) = { x 2 1 4О ( k 2 1 4), c 1 x Ω О( k ), 0 x c 1 3 (2.17) l j (x) = О ( jk+ 2 ) (k+), for 0 x c 1 1 k = О ( jk+ 2x k (k+) ), c 1 x Ω

4 7092 R. Srivastava ad Geeta Vishwakarma 1 (2.18) l j (x) = О ( jk+ 2 ) (k+ 1), for 0 x c 1 1 k = О ( jk+ 2x k (k+ 1) ), for c 1 x Ω 3. NEW RESULT Theorem 1 : For >1 fixed iteger let {g i } i=1, {g i } i=1, {g i } i=1 ad, {g (j) k 0 } j=0 arbitrary real umbers the there exists a uique polyomial R (x) of miimal possible degree 3+k o the odal poits (1.1) satisfyig the coditio (1.2) ad (1.3). The polyomial R (x) ca be writte i the form (3.1) R (x) = U j (x)g j + V j (x)g j + W j (x)g k (j) j + C j (x)g 0 where U j (x), V j (x ), W j (x) ad C j (x) are fudametal polyomials of degree 3+k give by (3.2) U j (x) = x(k+1) l j (x)[l (k 1) (x)] 2 x j (k+1) [L (k 1) (xj )] 2 (3.3) V j (x) = xk+1 l j (x)l k (x) [L (k) (k+1) (k) y j [L (yj )] 2 (x) + y j 3k+2 L (k 1) (x)] 2y j j=0 are (3.4) W j (x) = xk+1 l j (x)l (k) (x)l (k 1) (x) y j k+1 L k (y j )L (k 1) (y j ) (3.5) C j (x) = p j (x)x j [L k (x)] 2 L (k 1) (x)+x k L (k) (x)l (k 1) (x)[cj L k (x)p j (x)+q j (x)l (k 1) (x) x k j ] vc, j = 0,1,, k 1 (3.6) C k (x) = 1 ( +k k )k!l (k 1) (0) x k L (k) (x)[l (k 1) (x)] 2 where p j (x) ad q j (x) are polyomials of degree at most k-j-1. c j is defied i (4.14)

5 A Iterpolatio Process o Laguerre Polyomial 7093 Theorem 2 Let the iterpolatory fuctio f: R R be cotiuously differetiable such that, C(m) = {f(x): f is cotiuous i[0, ), f(x) = О(x m )as x ; m 0 is a iteger} For every f C(m) ad α 0, The (3.7) R (x) = U j (x)g j + V j (x)g j + W j (x)g k (j) j + j=0 C j (x)g 0 satisfies the relatios 3k (3.8) R (x) f(x) = О ( jj (3.9) R (x) f(x) = О ( j k 2 1 j 2 (k+) 2 +1 x k k 2 (k+)(k+ 1) where ω is the modulus of cotiuity. ) ω (f, log ), for 0 x c 1 log ) ω (f, ), for c 1 x Ω 4. PROOF OF THEOREM 1 Let U j (x), V j (x), W j (x) ad C j (x) are polyomials of degree 3+k satisfyig coditios (4.1), (4.2), (4.3) ad (4.4) respectively. (4.1) U j (x i ) = 0 whe i j 1 whe i=j U j(x i ) = 0, U j (x i ) = 0 { U (l) j (0) = 0, i = 1(1) ad l = 0,1,, k (4.2) 0 whe i j V j (y i ) = 0 V j (y i ) = 1 whe i=j, V j (y i ) = 0 { V (l) j (0) = 0, i = 1(1) ad l = 0,1,, k (4.3) 0 whe i j 1 whe i=j W j (y i ) = 0 W j (y i ) = 0, W j (y i ) = { W (l) j (0) = 0, i = 1(1) ad l = 0,1,, k

6 7094 R. Srivastava ad Geeta Vishwakarma (4.4) { C k(x i ) = 0, C k (y i ) = 0, C k (y i ) = 0 C (l) k (0) = 0, i = 1(1) ad l = 0,1,, k To determie U j (x) let (4.5) U j (x) = C 1 x k+1 l j (x)[l (k 1) (x)] 2 where C 1 is a costat. l j (x) is defied i (2.8). U j (x) is a polyomial of degree 3+k By usig (2.8) we determie (4.6) C 1 = 1 x j (k+1) L (k 1) (xj ) Hece we fid the first fudametal polyomial U j (x)of degree 3+k To fid secod fudametal polyomial let ( 4.7) V j (x) = C 2 x k+1 l j (x)[l (k) (x)] 2 + C 3 x k+1 l j (x)l (k) (x)l (k 1) (x) where C 2 ad C 3 are arbitrary costats. By usig (2.9) ad (4.2) we determie (4.8) C 2 = 1 y j (k+1) [L k (y j )] 2 ad (4.9) C 3 = y j 3k+2 (k+1) (k) 2y j y j [L (yj )] 2 Hece we fid the first fudametal polyomial V j (x)of degree 3+k Agai let (4.10) W j (x) = C 4 x k+1 l j (x)l (k) (x)l (k 1) (x) where C 4 is a costat, l j (x) is defied i (2.8).W j (x) is polyomial of degree 3+k satisfyig the coditios (4.3) by which we obtai (4.11) C 4 = 1 (k+1) y j )L k (k 1) (y j )L (y j ) Hece we fid the third fudametal polyomial W j (x) of degree 3+k To fid C j (x), we assume C j (x) for fixed j ε {0,1,.., k 1} i the form

7 A Iterpolatio Process o Laguerre Polyomial 7095 (4.12) C j (x) = p j (x)x j [L k (x)] 2 L (k 1) (x) + x k L (k) (x)l (k 1) (x)g (x) Where p j (x) ad g (x) are polyomials of degree k-j-1 ad respectively. Now it is clear that C j (l) (0) = 0 for (l = 0,, j 1) ad sice L (k) (x i ) = 0 ad L (k 1) (y i ) = 0 we get C j (x i ) = 0 ad C j (y i ) = 0 for i = 1(1). The coefficiet of the polyomial p j (x) are calculated by the system (4.13) C j (l) (0) = dl dx l [p j(x)x j [L k (x)] 2 L (k 1) (x)]x=0 = δ i,j (l = j,, k 1) ow from the equatio C j (k) (0) = 0 we get (4.14) c j = g (0) = 1 d k ( +k k )k!l (k 1) (0) Now usig the coditio C j (y i ) = 0 of (4.7), we get dx k [p j(x)x j [L k (x)] 2 L (k 1) (x)]x=0 (4.15) g (y i ) = (y i ) j k L k (y i )p j (y i ) Which implies g (x) as follows (4.16) g (x) = L k (x)p j (x)+q j (x)l (k 1) (x) x k j Where q j (x) is a polyomial of degree k-j ad fuctio g (x) will be a polyomial iff for r = 0,1,, k j 1 (4.17) d r dx r [L k (x)p j (x) + q j (x)l (k 1) (x)]x=0 = 0 The coefficiets of q j (x) are uiquely calculated by this system (4.13) Usig (4.12) ad (4.14) we obtai C j (x) of degree 3+k satisfyig the coditios (4.4) 5. ESTIMATION OF THE FUNDAMENTAL POLYNOMIALS Lemma 5.1. For j = 1(1) ad [0, ), we have 3 2 ) (k+) (5.1) U j (x) О ( jk+, for 0 x c 1 k (5.2) U j (x) О ( j 2 +1 k x 2 1 k 42 ), for c 1 x Ω (k+)

8 7096 R. Srivastava ad Geeta Vishwakarma Wher U j (x) is give i equqtio (3.2) Proof : from (3.2) we have x(k+1) (k 1) 2 l j (x) L i=1 (k+1) (k 1) x j L (xj ) 2 (5.3) U j (x) where U j (x) is give i equqtio (3.2) usig equatios (4.1) (2.13) ad (2.17) we yield the result. Lemma 5.2. For j = 1(1) ad [0, ), we have 1 2 ) (k+ 1) ( 5.4) V j (x) О ( jk+, for 0 x c 1 5k (5.5) V j (x) О ( j2k+1 x k ), for c 1 x Ω (k+ 1) Where V j (x) is give i equqtio (3.3) Proof : from (3.3) we have (5.6) V j (x) x k+1 l j (x) L k (x) y j (k+1) L (k) (yj ) 2 [ L (k) (x) + y j 3k+2 L (k 1) 2y (x) ] j usig equatios (2.16) ad (2.18) we yield the result. Lemma 5.3. For j = 1(1) ad [0, ), we have 3k (5.7) W j (x) О ( j , for 0 x c 1 (k+ 1) ) 7 3k 4x 2 1 k (5.8) W j (x) О ( j2k+ ), for c 1 x Ω (k+ 1) where W j (x) is give i equqtio (3.4) Proof : from (3.3) we have (5.9) W j (x) = xk+1 l j (x) L (k) (x) L (k 1) (x) y j k+1 L k (y j ) L (k 1) (y j )

9 A Iterpolatio Process o Laguerre Polyomial 7097 By equatios (2.15), (2.16) ad (2.17) we yield the result. Lemma 5.4. For j = 0,1,2,, k ad [0, ), we have k (5.8) j=0 C j (x) O (j j 3k 2 j 2 ), for 0 x c 1 k (5.9) j=0 C j (x) O (j j x 3k (3k j) 7 4), for c 1 x Ω Proof : by usig equatio (3.5) dealig with similar method we get the result. 6. PROOF OF THEOREM 2 We prove theorem 2 with the help of certai theorem metioed as below Theorem (5.5): Let C(m) = {f(x): f is cotiuous i[0, ), f(x) = О(x m )as x ; m 0 is a iteger} The by Szego[12] is lim f(x) H (α) (f, x) = 0 I For every f C(s) ad (0, )for α 0, or I (0, )for 1 < α < 0. furthermore there exists a fuctio i C(m) such that {H (α) (f, x)} diverges for α 0 at x=0.as for thr rate of covergece the followig result is due to Vertesi (5.10) f(x) H (α) (f, x) I = О(ω(f, 1 α )); 1 < α < 0 { log О (ω (f, )) ; α 1 2 Proof of mai theorem 2 : Sice R (x) give by equatio (3.1) is exact for all polyomial S (x) of degree 3+k, we have (5.11) S (x) = S (x j )U j (x) + S (y j )V j (x) + S (y j )W j (x) + S (y i )C j (x) k j=0

10 7098 R. Srivastava ad Geeta Vishwakarma From equatio (5.11) ad (3.1) we get R (x) (x) = S (x) f(x) + f(x j ) S (x j ) U j (x) + f(y j ) S (y j ) V j (x) k + f (y j ) S (y j ) W j (x) + f l (y j ) S l (y j ) C j (x) l=1 Owig to equatios (2.14) ad lemmas (5.1), (5.2), (5.3), (5.4) we get the result. REFERENCES [1] Balázs, J. ad Turá.P., Notes o iterpolatio, II. Acta Math. Acad. Sci. Huger., 8 (1957), pp [2] Balázs, J., Weighted (0; 2)-iterpolatio o the roots of the ultraspherical polyomials, (i Hugaria: Súlyozott (0; 2)-iterpoláció ultraszférikus poliom gyökei), Mat. Fiz. Tud. Oszt. Közl., 11 (1961), [3] Balázs, J..P., Modified weighted (0,2) iterpolatio, Approx. Theory, Marcel Dekker Ic., New York, 1998, 1-73 [4] Chak, A.M. ad Szabados, J. : O (0,2) iterpolatio for Laguerre abscissas, Acta Math. Acad. Hug. 49 (1987) [5] Léárd, M. : Pál type- iterpolatio ad quadrature formulae o Laguerre abscissas, Mathematica Paoica 15/2 (2004), [6] Léárd, M., O weighted (0; 2)-type iterpolatio, Electro. Tras. Numer. Aal., 25 (2006), [7] Léárd, M., O weighted lacuary iterpolatio, Electro. Tras. Numer. Aal., 37 (2010), [8] Mathur, P. ad Datta S., O Pál -type weighted lacuary (0; 2; 0)- iterpolatio o ifiite iterval (-,+ ), Approx. Theor. Appl., 17 (4) (2001), [9] Mathur, K.K.ad Srivastava, R.: Pál type Hermite iterpolatio o ifiite iterval, J.Math. Aal.ad App. 192, (1995) [10] Pál L. G., A ew modificatio of the Hermite-Fejér iterpolatio, Aal.

11 A Iterpolatio Process o Laguerre Polyomial 7099 Math., 1 (1975), [11] Prasad, J. ad Jhujhuwala,N. : Some lacuary iterpolatio problems for the Laguerre abscissas, Demostratio Math. 32 (1999), o. 4, [12] Szegö G., Orthogoal polyomials,amer. Math. Soc.,Coll. Publ., 23, New York, 1939 (4th ed.1975). [13] Szili, L.,Weighted (0,2) iterpolatio o the roots of classical orthogoal polyomials, Bull. Of Allahabad Math. Soc., 8-9 ( ), [14] Srivastava, R.ad Mathur, K.K.:weighted (0;0,2)- iterpolatio o the roots of Hermite polyomials, Acta Math. Huger. 70 (1-2)(1996), [15] Xie,T.F., O ] Pál s problem, Chiese Quart. J. Math. 7 (1992), 48-2.

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