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The problem of oscillatory polynomial interpolants arising from equally spaced mesh points is considered. It is shown that by making use of adaptive approaches the oscillations may be contained and the resulting polynomials are data-bounded and monotone on each interval. This is achieved at the cost of using a different polynomial on each subinterval. Computational results for a number of challenging functions including a number of problems similar to Runge's function with as many as 511 points per interval are shown.