Real Analysis II · Power Series & Capstones
Abel regularization of an absolutely convergent Fourier series
Problem
Let (c_n)_{n∈ℤ} satisfy Σ_{n∈ℤ}|c_n|<∞, and define g(x)=Σ_{n∈ℤ}c_ne^{inx}. For 0≤r<1 define g_r(x)=Σ_{n∈ℤ}c_nr^{|n|}e^{inx}. Prove both series converge uniformly on ℝ, g and every g_r are continuous and 2π-periodic, and g_r→g uniformly as r↑1.
Hint
For every x and 0≤r≤1, |c_nr^{|n|}e^{inx}|≤|c_n|; the same majorant works for g and g_r.
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