On the closedness of the linear image of a set with applications to generalized spline functions^†

^†This work was supported in part by contract number N 00014-67-A-0128-004 at the University of Wisconsin, from which the first author is one leave, and by grant USAFOSR 69-1812B at the Center for Numerical Analysis at the University of Texas

Abstract

In many problems of approximation theory, especially in the field of spline functions and their generalizations, one seeks a u minimizing ||Tu-f|| for u in some constraint set U; it is usually easy to prove existence of such a u if IT/is closed. We consider hypotheses that will guarantee the closedness of TU, and, more generally, the closedness of the sum of U and a closed linear subspace

^†This work was supported in part by contract number N 00014-67-A-0128-004 at the University of Wisconsin, from which the first author is one leave, and by grant USAFOSR 69-1812B at the Center for Numerical Analysis at the University of Texas

^†This work was supported in part by contract number N 00014-67-A-0128-004 at the University of Wisconsin, from which the first author is one leave, and by grant USAFOSR 69-1812B at the Center for Numerical Analysis at the University of Texas

Notes

^†This work was supported in part by contract number N 00014-67-A-0128-004 at the University of Wisconsin, from which the first author is one leave, and by grant USAFOSR 69-1812B at the Center for Numerical Analysis at the University of Texas