Approximation of Fractals via Lagrange-type Superoscillations

Abstract

We study the approximation of the Weierstrass function by means of superoscillating sequences. Superoscillatory functions are band-limited functions whose local oscillation rate can exceed the highest frequency contained in their Fourier spectrum. Starting from Lagrange-type interpolation at nodes in $[-1,1]$, we construct a double-indexed family ${\mathcal{W}}_{N,n}(x)$ that approximates the truncated Weierstrass function $W_N(x)$ for each fixed truncation order~$N$. We prove that if the number of interpolation nodes $n_N$ grows sufficiently fast relative to the highest frequency $b^N\pi$, namely $b^N\pi /n_N\to 0$, then ${\mathcal{W}}_{N,n_N}$ converges uniformly to the full Weierstrass function on every compact set. We also show that the two limits in $N$ and $n$ do \emph{not} commute: for any fixed~$n$ the series ${\lim }_{N\to \infty }{\mathcal{W}}_{N,n}(x)$ diverges for every $x\ne 0$, a phenomenon called the Divergence Wall.