(0, 1, 2, 4) Interpolation by G -splines

Perio&.a

Mathenmtica

Hungarica

VoZ. 21 (4), (1990),

(0, 1, 2, 4) INTERPOLiiTION A.

SAXENA

pp. 261-271

BY G-SPLINES

(Lucknow)

1. Introduction Saxena and Tripathi [3], [4] h ave introduced spline methods for solving the (0, 1, 3) interpolation problem. They have used spline interpolants aA c L$$\ *t3) of degree six for functions / c C6 to solve the problems ad 8: C fin,6 @(a$

= yp), q = 0, 1, 3, k = O(1) n, $&J

= 9;;

and

are given reals. Their Let / c (Y(I).

convergence

results are as follows:

Then for q = o(l)& /j@’ - fq’ ~~~-~x~,x~+~~ < &h5-’

%@h

II sZq) - fq) lL.-[x~,x~+J < w5-q(%m~ where 14whenk=O 60 when k = l(l)n

-

1,

and I&= Here ti6( -) is the modulus

591 when k = 0 60 when k=l(l)m--2 27 when k = n - 1. of continuity

of f@.

AMS (MOS) Subject du-v~+cutkns (1980/85). Key words ati phmwx. Deficient Chpline.

Primary

4lA.

262

SAXENA:

(0, 1, 2, 4) INTERPOLATION

BY

G-SPLINES

Fawzy [2], in one of his very recent, papers, has solved the (0, 1,3) interpolation problem by constructing a spline method using piecewise polynomials such that for functions f c C4, the method converges faster than the method given in [5] for solving the same interpolation problem, and for functions f c C5, the order of approximation is the same as the best order of approximation using quintic splines. For this purpose he has used special kind of g-splines, to which he refers as lacunary g-splines. Motivated by the method and results of Fawzy [2], we solve here a different interpolation problem, namely, the (0, I, 2,4) interpolation problem, which is formulated as follows: Given a uniform

partition

A: of the interval

q

1 = [0, 1] and real numbers

find 8 in a suitable (1.1)

o=~o