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 elemental houses of B-splines, together with B- splines with equidistant knots and cardinal splines represented when it comes to B-splines, and exponential Euler splines, resulting in an important case and principal challenge of the e-book - cardinal spline interpolation, with major effects, proofs, and a few functions. different issues mentioned contain 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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1). 13) we conclude that the curve F traced out by is entirely contained within the circle |z| ^ 1, only touching it at the points tv. Morever, the relation shows that all arches of the curve are obtained from the arc joining z = 1 to z = eiu by rotations through angles which are multiples of u. However, these remarks are already contained in the case \t\ = 1 of Theorem 4 of Lecture 2. Actually much more follows from that Theorem 4 (Lecture 2), in particular the following. THEOREM 6. Let — ii
V ) (l)^ r/(v)(0), v = 0, • - . , w - 1. The following theorem holds. THEOREM 3. 1), is the unique element of 2Fn that minimizes the norm giving it its least value For details see [4, § 6]. 4. The exponential Euler polynomials of the class & *. In § 1 of Lecture 1 we have defined the class &* * of cardinal midpoint splines. The analogues for this class of the results of §§ 1 and 2 require no new ideas. ; t). However, it is of interest to obtain its expansion in powers of x. 3), for x = 0, the expansion It is convenient to define a new sequence of polynomials pn(i) by setting The analogue of Theorem 2 is as follows.
35]). 2. The construction of the exponential splines. 5) of Lecture 2 is well suited for the evaluation of O n (x; t) because the defining series has at most n + 1 nonvanishing and consecutive terms for each value of x. 1) and the use of the exponential Euler polynomial An(x;t). 3). 3) by t — ez and comparing the coefficients of the powers of x, we obtain the relations giving the an(t) recursively. 1) becomes showing that An(x; 2) has integer coefficients. 2), the very definition of An(x; t) yields Finally, we extend this function to all real x by the functional equation of Lemma 2 of Lecture 2.