Compression and Restoration of Square Integrable Functions

We consider classes of smooth functions on [0, 1] with mean square norm. We present a wavelet-based method for obtaining approximate pointwise reconstruction of every function with nearly minimal cost without substantially increasing the amount of data stored. In more detail: each function f of a class is supplied with a binary code of minimal (up to a constant factor) length, where the minimal length equals the ε-entropy of the class, ε > 0. Given that code of f we can calculate f , ε-precisely in L2, at any specific N,N ≥ 1, points of [0, 1] consuming O(1+ log∗((1/ε)(1/α)/N)) operations per point. If the quantity N of points is a constant, then we consume O(log∗ 1/ε) operations per point. If N goes up to the ε-entropy, then the per-point time consumption goes down to a constant, which is less than the corresponding constant in the fast algorithm of Mallat [11]. Since the iterated logarithm log∗ is a very slowly increasing function, we can say that our calculation method is nearly optimal.