  This paper considers the problem of approximating a Boolean function $f$
using another Boolean function from a specified class. Two classes of
approximating functions are considered: $k$-juntas, and linear Boolean
functions. The $n$ input bits of the function are assumed to be independently
drawn from a distribution that may be biased. The quality of approximation is
measured by the mismatch probability between $f$ and the approximating function
$g$. For each class, the optimal approximation and the associated mismatch
probability is characterized in terms of the biased Fourier expansion of $f$.
The technique used to analyze the mismatch probability also yields an
expression for the noise sensitivity of $f$ in terms of the biased Fourier
coefficients, under a general i.i.d. input perturbation model.
