Titlebook: Spline Functions and Multivariate Interpolations; B. D. Bojanov,H. A. Hakopian,A. A. Sahakian Book 1993 Springer Science+Business Media B.

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Multivariate Polynomial Interpolations Arising by Hyperplanes,given traces on the .-dimensional hyperplanes (0 ⩽ . ⩽ . - 1) which are intersections of some hyperplanes from .. We start with the case . = 0, i.e., pointwise interpolation with a node set arising by the hyperplanes ...

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The Space of Splines,ration [.] into sufficiently small subintervals . and then uses a low degree polynomials .. for approximation over .,..., .. This procedure produces a piecewise polynomial approximating function ., ..

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ion curve through given n + k points that has a minimal norm of its k-th derivative. Isolated facts about the functions, now called splines, can be found in the papers of L. Euler, A. Lebesgue, G. Birkhoff, J. Favard, L. Tschakaloff. However, the Theory of Spline Functions has developed in the last

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,-Splines,We introduce here another basis for the spline space ..(.., ... ,..) which consists of functions that have a finite support (i.e., which vanish outside a certain finite interval). The new basis functions possess some remarkable properties which make them a widely used tool in calculating with splines as well as in other theoretical studies.

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Perfect Splines,A . of degree . with knots ξ.,..., ξ. is any expression of the form . where {α.}. and γ are real numbers.

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Periodic Splines,The function . on ℝ is said to be . with a period . (or in brief, . if . for each . ∈ ℝ. For the sake of convenience, we shall consider here 2π-periodic functions. In this case, one may think of . as a function defined on the unit circle.

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Multivariate ,-Splines and Truncated Powers,In the following, it is convenient to accept slightly a different normalization for .-splines. Namely, a .-spline with a knot set θ = {.., ..., ..}, . ⩽ .. ⩽ ... ⩽ .. ⩽ . is defined by the rule . and therefore (see 3.2.1) ..

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Multivariate Spline Functions and Divided Differences,In this section, we consider linear combinations of .-splines. Let ..,..., ..} be an arbitrary finite set of knots in ℝ. (not necessary distinct) with vol .[.] ≠ 0.

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