\input zb-basic \input zb-ioport \iteman{io-port 05222865} \itemau{Berzins, M.} \itemti{Adaptive polynomial interpolation on evenly spaced meshes.} \itemso{SIAM Rev. 49, No. 4, 604-627 (2007).} \itemab When using polynomial interpolation on equispaced data, it is well known that the interpolant may have unbounded oscillations as the degree increases. This paper removes this unwanted effect by defining different polynomials in each subinterval, obtained by combining three ideas that have been proposed earlier. These are (1) Change the stencil (i.e., the set of chosen interpolation points) to get smaller divided differences. That is the so called ENO (essential nonoscillatory) method of {\it A. Harten, B. Engquist, S. Osher, S. Chakravarthy} [J. Comput. Phys. 71, 231--303 (1987; Zbl 0652.65067)], (2) change the degree of the polynomial to remove the effect of higher derivatives of the given function [{\it A. Harten}, Commun. Pure Appl. Math. 48, No. 12, 1305--1342 (1995; Zbl 0860.65078)], (3) modify the polynomials so that the differences between them do not change drastically [{\it M. Berzins}, Appl. Numer. Math. 52, No. 2--3, 197--217 (2005; Zbl 1064.65109)]. The main idea of this paper is that a certain ratio appearing in a recursive formulation of the divided differences should be bounded. Therefore, it is replaced by a bounded function of that ratio. This, together with an adaptive stencil, gives good numerical results on Runge type problems. It is shown that the interpolated values are monotone in between two surrounding data points (and hence bounded by these), and that all the computed differences have the same sign. \itemrv{Adhemar Bultheel (Leuven)} \itemcc{} \itemut{adaptive polynomial interpolation; data-bounded polynomials; Runge's function; divided differences; numerical results} \itemli{doi:10.1137/050625667} \end