2017/07/28 by Defant, Andreas, Mastyło, Mieczysław, Pérez, Antonio · 1 citation
#06E30 #42A16 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1707.09186
We study the asymptotic decay of the Fourier spectrum of real functions f\colon \-1,1\N → ℝ in the spirit of Bohr's phenomenon from complex analysis. Every such function admits a canonical representation through its Fourier-Walsh expansion f(x) = ∑_S⊂ \1,…,N\\widehatf(S) xS , where xS = ∏k ∈ S xk. Given a class F of functions on the Boolean cube \-1, 1\N , the Boolean radius of F is defined to be the largest ρ≥ 0 such that ∑S|\widehatf(S)| ρ|S| ≤ ‖f‖∞ for every f ∈ F. We give the precise asymptotic behaviour of the Boolean radius of several natural subclasses of functions on finite Boolean cubes, as e.g. the class of all real functions on \-1, 1\N, the subclass made of all homogeneous functions or certain threshold functions. Compared with the classical complex situation subtle differences as well as striking parallels occur.