On the degree of polynomials that approximate symmetric boolean functions

Abstract

In this paper, we provide matching (up to a constant factor) upper and lower bounds on the degree of polynomials that represent symmetric boolean functions within an error 1/3. Let Γ(f) = min{|2κr-n+ 1: fκ ^ fκ+l and 0 ≤ κ ≤ n - 1} where fi is the value of f on inputs with exactly i l's. We prove that the minimum degree over all the approximating polynomials of f is ⊕(√n(n - Γ(f))). We apply the techniques and tools from approximation theory to derive this result.