On the Almost Everywhere Convergence of Wavelet Summation Methods

Let ψ be a rapidly decreasing one-dimensional wavelet. We show that the wavelet expansion of anyL^pfunction converges pointwise almost everywhere under the wavelet projection, hard sampling, and soft sampling summation methods, for 1 <p< ∞. In fact, the partial sums are uniformly dominated by the Hardy–Littlewood maximal function.