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An integral method for constructing bivariate spline functions. (English) Zbl 0698.65006

Let \(\Delta_ 1\) and \(\Delta_ 2\) respectively denote the three direction and four direction meshes of a region in \(R^ 2\) given by: \(\Delta_ 1:\) \(x=i\), \(y=j\), \(x-y=k\), and \(\Delta_ 2:\) \(x=i\), \(y=j\), \(x-y=k\) and \(x+y=k\), (i,j,k integers) [cf. W. Dahmen and C. A. Micchelli, Approximation theory IV, Proc. int. conf., Tex. A.&M. Univ. 1983, 27-121 (1983; Zbl 0559.41011)]. Let \(S_ k^{\mu}(\Delta_ i)\), \(i=1,2\) denote the space of bivariate splines of degree at most k over \(\Delta_ i\) which belong to \(C^{\mu}\), a locally supported spline \(s\in S_ k^{\mu}(\Delta_ i)\) with minimal support will be called a B-spline. A necessary condition for the existence of a locally supported nontrivial B-spline in \(S^{\mu}_ k(\Delta_ i)\) has been established by C. K. Chui and the first author [J. Math. Anal. Appl. 94, 197- 221 (1983; Zbl 0526.41027)]. In the present paper the authors show that the condition is also sufficient. Using an integral method a recurrence relation is also obtained.
Reviewer: H.P.Dikshit

MSC:

65D07 Numerical computation using splines
41A15 Spline approximation
41A63 Multidimensional problems
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