The spectral norm of a Boolean function f:{0,1}nβ{β1,1} is the sum
of the absolute values of its Fourier coefficients. This quantity provides
useful upper and lower bounds on the complexity of a function in areas such as
learning theory, circuit complexity, and communication complexity. In this
paper, we give a combinatorial characterization for the spectral norm of
symmetric functions. We show that the logarithm of the spectral norm is of the
same order of magnitude as r(f)log(n/r(f)) where r(f)=max{r0β,r1β},
and r0β and r1β are the smallest integers less than n/2 such that f(x)
or f(x)β parity(x) is constant for all x with βxiββ[r0β,nβr1β]. We mention some applications to the decision tree and communication
complexity of symmetric functions