Uniqueness of Tchebycheff Spaces and their Ideal Relatives

In the first part of the paper we show that the space of polynomials of degree n−1 is the unique n-dimensional Tchebycheff subspace of polynomials. We also define a generalization of Tchebycheff spaces: ”Ideal complements” and demonstrate their uniqueness. In the second part we discuss various analogues of Tchebycheff spaces (minimal interpolating systems) in several variables. Preface: I first met Professor Sharma twenty seven years ago. I was a young graduate student, my head was filled with “Bourbakisms”, my Ph.D. thesis was about interpolation (in Banach spaces, of course) and I was looking forward to learn more from the renown expert in the field. To my surprise Sharma told me that he didn’t understand what a functional was and the only theorems worth knowing in analysis was the Taylor formula and maybe “integration by parts” although he had his doubts about the latter. With typical modesty, he told me that he wasn’t bright enough for the abstractions. The best he could do, was to compute a few “right” examples and hope to get lucky. That send me for a spin, that lasted awhile. I tried to “compute” with Sharma only to learn that there is no way for me to keep up with his speed and accuracy. I believe that this was a lesson learned by many of my colleagues. Fortunately “Maple” came about and like “Colt 45”, equalized the playing field. This paper is about solvability of various interpolation problems and its generalizations, the topic that benefited greatly by many contributions of Sharma and his collaborators (cf. [4-7], [10], [12], [16-19]). The paper is divided into two parts. The aim of the first part is to investigate the general form and uniqueness of Tchebycheff and Extended Tchebycheff subspaces as well as “ideal complements” in the spaces of polynomials. In particular