Construction of functions from their values at the binary-rational points of an m-dimensional cube
This paper contains the results published by the author in a preprint (No.92/1986) edited by M. V. Keldysh Institute of Applied Mathematics of the Academy of Sciences of the USSR, entitled ``A basis of the space \(C(I^ m)''\). (I denotes the segment [0,1].) The author looks for a basis of the space \(C(I^ m)\), the space of continuous functions on \(I^ m\) in order to study the multidimensional variants of the following problems formulated for the one-dimensional case. ``Let D be the set of binary rational numbers of the segment \(I=[0,1]\). A continuous function f: \(I\to {\mathbb{R}}\) is uniquely defined by its values on the set D. But how do we find out if a function, defined on D, is the trace of a continuous function f: \(I\to {\mathbb{R}}?\) What can be said in this case about the smoothness of the function f? Sufficient conditions for the continuity and the Lipschitz property of a function f (constructively verifiable in concrete situations) can be obtained by making use of the expansion of the function with respect to the Faber-Schauder basis.''
scientific article; zbMATH DE number 4159365