Oscillation Space Embedding and the Estimation of Box Dimension: A Correction on Previous Attempts

Based on an embedding on a Besov space, many fractal dimension estimations have been proposed using wavelet transformation. Many of them were diminished by the counter-example on the embedding but corrected by dening a bounded variational seminormed space. Thus, we correct the fractal function estimation approaches which have employed the wrong embedding. The results are given for the problem of estimation of multi-dimensional functions instead of simple univariate case. We also obtain the consistency of the proposed estimator in the probability space induced by random functions. The results are examined through a simulation study on the index- family of incremental stationary Gaussian elds. Moreover, under the laboratory situations, the procedure of equilibration of heap in a liquid is studied in view of this estimator.