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 $\varepsilon$-entropy of the class, $\varepsilon > 0$. Given that code of $f$ we can calculate $f$, $\varepsilon$-precisely in $L_2$, at any specific $N, N\geq 1,$ points of $[0,1]$ consuming $O(1+\log ^*((1/\varepsilon )^{(1/\alpha )}/N))$ operations per point. If the quantity $N$ of points is a constant, then we consume $O(\log ^*1/\varepsilon )$ operations per point. If $N$ goes up to the $\varepsilon$-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.