  We present a method for constructing superoscillatory functions the
superoscillatory part of which approximates a given polynomial with arbitrarily
small error in a fixed interval. These functions are obtained as the product of
the polynomial with a sufficiently flat, bandlimited envelope function whose
Fourier transform has at least N-1 continuous derivatives and an N-th
derivative of bounded variation, N being the order of the polynomial.
Polynomials of arbitrarily high order can be approximated if the Fourier
transform of the envelope is smooth, i.e. a bump function.
