Step 2. The deduction in Step 1 can be iterated repeatedly due to the symmetry of

The universal approximation Theorem 3.1 immediately implies another universal approximation Thus y (t) solves the ODE (2.6), with initial condition y (0) = y (0) = 0 . Reconstruction of a continuous signal from its sine transform. Step 0: (Equicontinuity) We recall the following fact from topology. F (τ):= null f (τ), for τ 0, f ( τ), for τ 0. Since F is odd, the Fourier transform of F is given by We provide the details below. The next step in the proof of the fundamental Lemma 3.5 needs the following preliminary result in By (B.3), this implies that It follows from Lemma 3.4 that for any input By the sine transform reconstruction Lemma B.1, there exists It follows from Lemma 3.6, that there exists Indeed, Lemma 3.7 shows that time-delays of any given input signal can be approximated with any Step 1: By the Fundamental Lemma 3.5, there exist It follows from Lemma 3.6, that there exists an oscillator Step 3: Finally, by Lemma 3.8, there exists an oscillator network,

In this supplementary material, we provide the proof of all theoretical results stated in the paper. We complete the proof in 8 steps by showing statements 1 - 8 above. Markov's inequality the first term on the RHS is also O This follows from Step 8 and the fact that by Chebyshev's inequality, The reasoning here is similar to Step 1. Since the number of splits K is bounded, we only need to verify for any k { 1, 2,...,K }, null null null null 1 n null Below we'll prove 1 n null Combining the above, we obtain (30). In the last inequality we utilize (32).