Dimensions of Graphs of Functions and Lacunary Decompositions of Spline Approximations

Abstract

A construction is put forward which shows how to decompose an arbitrary continuous function $f\in C[0,1]$ into a sum of two continuous functions each having a graph of Hausdorff dimension one. An example is given where the terms in the decomposition have graphs of Hausdorff dimension 1 and packing dimension 2.