ABSTRACT
Approximation Theory is an old and rich branch of Analysis and Applied Mathematics, but still full of formidable problems. For example, there are aspects of that theory that were understood and elucidated for algebras and modules of continuous functions, at least to a reasonable degree of our present needs. However, the counterpart of such a basic development is still drastically lacking in the continuously differentiable situation. Our purpose here is to enumerate a few open outstanding problems when continuous differentiability is called for. We precede them with the exact results involving only continuity that serve as their motivation. In this exposition, we shall try to be simple, by avoiding a sort of completeness in enumerating some differentiability flavored problems in Approximation Theory. As a matter of fact, we strongly believe that a kind of progress has to be accomplished along the lines laid down in the following description, before we engage ourselves in other fundamental problems lined up in our mind as their natural sequel, for example by using weights. We find it always stimulating to formulate new 370basic problems, both from the viewpoint of the theory and the applications as well. While these problems are for what is typically referred to as hard Analysis, there is clearly potential for applying them in concrete situations, as they are natural problems. As past history has vividly demonstrated, supposedly abstract investigations end up being brought out to help in the solution of applied problems. We have indeed seen the deep and abstract prospective theorems envisaged here come up in very special forms in concrete problems of the applied fields. Thus we need a plausible unifica of them, and the resulting new tools at hand to apply wherever ion needed. May we refer also to our previous survey lectures [11], [15], as they are not completely included in the present exposition. We restrict ourselves here to the real valued case.
