We propose the Rescaled Pure Greedy Learning Algorithm (RPGLA) for solving the kernel-based regression problem. The computational complexity of the RPGLA is less than the Orthogonal Greedy Learning Algorithm (OGLA) and Relaxed Greedy Learning Algorithm (RGLA). We obtain the convergence rates of the RPGLA for continuous kernels. When the kernel is infinitely smooth, we derive a convergence rate that can be arbitrarily close to the best rate [Formula: see text] under a mild assumption of the regression function.
We construct the blending-type modified Bernstein–Durrmeyer operators and investigate their approximation properties. First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. Finally, we derive the rate of convergence for functions with a derivative of bounded variation. The results show that the new operators have good approximation properties.
We propose the Weak Rescaled Pure Super Greedy Algorithm (WRPSGA) for approximation with respect to a dictionary D in Hilbert space. The WRPSGA is simpler than some popular greedy algorithms. We show that the convergence rate of the RPSGA on the closure of the convex hull of the μ-coherent dictionary D is optimal. Then, we design the Rescaled Pure Super Greedy Learning Algorithm (RPSGLA) for kernel-based supervised learning. We prove that the convergence rate of the RPSGLA can be arbitrarily close to the best rate O(m−1) under some mild assumptions.
We study the approximation capability of the orthogonal super greedy algorithm (OSGA) with respect to μ-coherent dictionaries in Hilbert spaces. We establish the Lebesgue-type inequalities for OSGA, which show that the OSGA provides an almost optimal approximation on the first [1/(18μs)] steps. Moreover, we improve the asymptotic constant in the Lebesgue-type inequality of OGA obtained by Livshitz E D.
This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed [Formula: see text] and the basic reproduction number [Formula: see text] are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder’s fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when [Formula: see text], for every speed [Formula: see text] there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any [Formula: see text] when [Formula: see text] or [Formula: see text] when [Formula: see text]. Finally, we analyze the effects of nonlocal time delay on the minimum wave speed.
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