Cardinal Spline Interpolation by I. J. Schoenberg

By I. J. Schoenberg

As this monograph exhibits, the aim of cardinal spline interpolation is to bridge the distance among the linear spline and the cardinal sequence. the writer explains cardinal spline features, the fundamental houses of B-splines, together with B- splines with equidistant knots and cardinal splines represented by way of B-splines, and exponential Euler splines, resulting in crucial case and principal challenge of the publication - cardinal spline interpolation, with major effects, proofs, and a few purposes. different themes mentioned comprise cardinal Hermite interpolation, semi-cardinal interpolation, finite spline interpolation difficulties, extremum and restrict houses, equidistant spline interpolation utilized to approximations of Fourier transforms, and the smoothing of histograms.

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3. An extremum property of the exponential Euler polynomial. 6), it satisfies the conditions: 1. P(0) = 1. 2. P

Remark 1. Theorem 5 shows that if Assumption 1 is satisfied, we get functions Ns(x) which really deserve the name of B-splines. 8) for the case when m = 3 and r = 2. Their role in the finite quintic Hermite interpolation problem will be the subject of § 5 of Lecture 7. f. 17158. 8). Remark 2. The developments of this lecture will leave the reader, as they leave the author, with the feeling that the last word on this subject has not been said. Assumption 1 concerns too deep an arithmetic problem in comparison with the linear algebra nature of the interpolation problem.

THEOREM 4. The pn(t) defined by the generating Junction are monic reciprocal polynomials of degree n having integer coefficients. In terms of the B-spline Qn+i(x) we may write 2. pn(t) can be defined independently by the expansion 3. pn(t) satisfies the recurrence relation 4. The zeros ofpn(t) are simple and negative. 1) shows that O*(x; t) is a shifted version of B(x; t). However, it is clear that O*(x, t) can also be constructed in (-1/2, 1/2) in terms of Bn(x; t), afterwards extending its definition by the functional equation 26 LECTURE 3 5.

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