Extensions of Otto Toeplitz’ Combinatorial Construction of Almost Periodic Functions on the Real Line
Hans Haller, Konrad Jacobs · Operator theory · 1982
In 1928 Otto Toeplitz constructed a very interesting class of real Bohr almost periodic functions on the line. His construction was based a) on an arbitrary choice of a uniformly convergent sequence f 1 , f 2 ,... of continuous real functions on the unit interval [O,1] such that f n (O) = O = f n (1) (n = 1,2,...) and b) a binary “filling scheme” which can shortly be described as follows: fill every second unit interval [k - 1, k] with translates of f 1 and leave the remaining intervals as “holes”; ect fill every second hole with translates of f 2 and leave the remaining unit intervals as “holes” etc. Toeplitz 1 contains formulas for the spectrum and Fourier coefficients of the resulting continuous function f: ℝ → ℝ, which is always Bohr almost periodic.