  We give a general strategy to construct superoscillating/growing functions
using an orthogonal polynomial expansion of a bandlimited function. The degree
of superoscillation/growth is controlled by an anomalous expectation value of a
pseudodistribution that exceeds the band limit. The function is specified via
the rest of its cumulants of the pseudodistribution. We give an explicit
construction using Legendre polynomials in the Fourier space, which leads to an
expansion in terms of spherical Bessel functions in the real space. The other
expansion coefficients may be chosen to optimize other desirable features, such
as the range of super behavior. We provide a prescription to generate
bandlimited functions that mimic an arbitrary behavior in a finite interval. As
target behaviors, we give examples of a superoscillating function, a
supergrowing function, and even a discontinuous step function. We also look at
the energy content in a superoscillating/supergrowing region and provide a
bound that depends on the minimum value of the logarithmic derivative in that
interval. Our work offers a new approach to analyzing
superoscillations/supergrowth and is relevant to the optical field spot
generation endeavors for far-field superresolution imaging.
