Interpolation of medians

Take n > 0 and n + 1 disjoint, bounded intervals I0, . . . , In, possible single points. In [2], motivated by applications in signal processing, it was conjectured that for any numbers a0, . . . , an, there is a unique polynomial of degree n whose median value on Ij equals aj for j = 0, . . . ,n, and this was confirmed for n = 2. We shall verify this conjecture in section 2. Indeed we prove a much more general result which includes median interpolation by spline functions. Firstly, in this section, we define precisely what is meant by the median value of any continuous function on a bounded interval and show that this varies continuously both with the function and with the interval. The concept of median is well known in statistics but we include elementary proofs of propositions 1.1 and 1.3 in order to make the paper self-contained. We also mention a connection between the median and best L1 approximation. Finally, section 3 studies the derivative of the median for the case when the function is a polynomial. We denote Lebesgue measure on R by m.