In this note we give a simple proof of the endpoint regularity for the uncentred Hardy–Littlewood maximal function on $\mathbb{R}$. Our proof is based on identities for the local maximum points of the corresponding maximal functions, which are of interest in their own right.

Most existing source search algorithms suffer from a high travel cost, and few of them have been analyzed in performance in noisy environments where local basins are presented. In this paper, the theseus gradient search (TGS) is proposed to effectively overcome local basins in search. Analytical performances of TGS and the gradient ascend with correlated random walk (GACRW), which is a variant of correlated random walk, are derived and compared. A gradient field model is proposed as an analytical tool that makes it feasible to analyze the performances. The analytical average searching costs of GACRW and TGS are obtained for the first time for this class of algorithms in the environments with local basins. The costs, expressed as functions of searching space size, local basin size, and local basin number are confirmed by simulation results. The performances of GACRW, TGS, and two chemotaxis algorithms are compared in the gradient field and a scenario of indoor radio source search in a hallway driven by real data of signal strengths. The results illustrate that GACRW and TGS are robust to noisy gradients and are more competitive than the chemotaxis-based algorithms in real applications. Both analytical and simulation results indicate that in the presence of local basins, TGS almost always costs the lowest.

We give explicit expressions for the Slepian model process of non-stationary Gaussian processes following level crossings and local maxima. We also include a detailed analysis of the high-level case.