On linearity of spline algorithms (Q1823607)

The main result of this paper can be stated in a simple form as follows: Let F and Z be Banach spaces. Then, if the dimension of F is greater than 2, statements (a) through (e) are equivalent: (a) F is a Hilbert space. For every linear problem S with domain F and information operator N there exists (b) a linear spline algorithm. (c) a linear interpolatory algorithm, (d) a linear strongly optimal algorithm, (e) a linear almost strongly optimal algorithm. In the context of information-based complexity, this provides a converse to the well-known result that a Hilbert structure is sufficient for such linearity properties.