A class of generalized cardinal splines
1979; Elsevier BV; Volume: 27; Issue: 2 Linguagem: Inglês
10.1016/0021-9045(79)90112-6
ISSN1096-0430
Autores Tópico(s)Mathematical Analysis and Transform Methods
Resumoal. 121. iMore recently Tzimbalario [9] and Mohapatra and Sharma [4] have derived analogous results for certain classes of cardinal discrete splines. In this paper we derive results for a broad class of generalized cardinal splines which both unify and generalize the results of all the above papers. In particular we extend the results of [9] to cardinal Mermite interpolation. Let P, denote the space of all real-valued polynomials of degree not exceeding 12, and for 1 < s 9 n, let r = (yO , y1 ,.~., y8-J denote a set of linearly independent linear functionals on P, . We define Yn(r) to be the class of all functions S from [w to itself such that for v = 0, +l, &2,..., S 1 IV, u + 1) = S, E P, and y[S,-,(x + v)] = y[S,(x + Y)], Vy E F. If y&7) = p’i’(O), i = o,..., n - s, then Y,Z(r) is the class of cardinaji splines with integer nodes of multiplicity s studied ir, [2, 31. If Y&P) = @“j(O), i = o,..., r? - s - 1, and ~+~(p) = ~(‘L-s+l)(0), then Y.(S) consists of the cardinal g-splines studied by Lee and Sharma [I]. The cardinal discrete splines of [4, 9] are obtained by putting s = 1, y,(p) = p(iil2) and yi(p) = pci)(ih), i = 0, I,..., n - 1 (0 < h < 1 jiz). We note that our above definition of 5QF) is the analogy for cardinal po- lynomial splines of Schumakers’ classes of generalized splines as defined in [S]. Now if .cP = (pl , pz ,..., pJ is a basis for (p E P,: y(p) = 0, tl'y E f'), then it is easily seen that 5?Jr) comprises all functions S of the form S(x) = P(x) + i cyp,<x - l)+ + i $p,(x - 2),
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