Three Conjectures on Shepard Interpolatory Operators

After establishing direct and converse approximation theorems for the Shepard interpolatory operators, J. Szabados (Approx. Theory Appl.7, No. 3, 1991, 63–76) left some open saturation problems (“the most intriguing questions” as he said), which he raised as three conjectures. The present paper proves the second parts of some conjectures, but constructs counterexamples to show that the first parts of three conjectures are not true. The constructive procedure uses some novel ideas and techniques.