GIBBS OVERSHOOT ON A FRACTAL

Gibbs' overshoot refers to the persistent discrepancy between a Fourie r series' approximation and actual values near a functional discontinu ity. Here this phenomenon is generalized to a fractal discontinuity on a trial function. Analytic results support the conclusion that fracta l dimension can parameterize the Gibbs' overshoot on an example sawtoo th with a fractal distribution of discontinuous depths. Simulations co nfirm this finding on a fractal Brownian walk and the devil's staircas e. Results indicate that the overshoot arises on a fractal not as much from any general ruggedness, but more from the fractal discontinuity directly.