Best approximations from Hilbert submanifolds

Learning Best Concept Approximations from Examples

This paper addresses the problem of learning the best approximation of a concept from examples, when the concept cannot be expressed in the learner’s representation language. It presents a method that determines the version space of the best approximations and demonstrates that for any given approximation of the target concept there is a better approximation in this version space. The method do...

Best subspace tensor approximations

In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition which allows to compute the best rank k approximations. For t-tensors with t > 2 many generalizations of the singular val...

A Splitting Theorem for Isoparametric Submanifolds in Hilbert Space

The best Diophantine approximations: the

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear sub-spaces generated by the best Diophantine approximations. Originally most of these results have been established by the author in [14, 15, 16, 17, 18]. Here we collect all of them togeth...

Best Approximations by Smooth Functions

THEOREM 1.1 (U. Sattes). Let r > 2 and g E C[O, l]\B$,‘. Then f”EB$’ is a best approximation to g, in L” (such a best approximation necessari/J) exisrs) if and only if there exists a subinterual (a, /?) c IO. 1 I and a positilse integer M > r + 1 for which the following conditions hold (i) f”l,n.ll, is a Perfect spline of degree r with exactly) M ~ r -1 knots arzd I.f”““(s)l = I a. e. on [u,pI....