In this note we state some problems on approximation by univariate splines with free knots, bivariate segment approximation and tensor product splines with variable knot lines. There is a vast literature on approximation and interpolation by univariate splines with fixed knots (see e.g. the books of de Boor [1], Braess [2], DeVore & Lorentz [4], Powell [20], Schumaker [21], Nürnberger [13] and the book of Chui [3] on multivariate splines). On the other hand, numerical examples show that in general, the error is much smaller if variable knots are used for the approximation of functions instead of fixed knots. This is true for univariate splines as well as for bivariate splines. But approximation by splines with free knots leads to rather difficult nonlinear problems.[...]
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