Analysis and Applications

An elementary proof is given for the positivity of the heat kernel associated with the sub-Laplacian on Heisenberg group. Starting from an integral representation of the heat kernel, the positivity is shown by choosing a simple, explicit path of integration. The positivity of the heat kernel associated with the Grušin operator then follows.

We study the asymptotical properties of indefinite kernel network with coefficient regularization and dependent sampling. The framework under investigation is different from classical kernel learning. Positive definiteness is not required by the kernel function and the samples are allowed to be weakly dependent with the dependence measured by a strong mixing condition. By a new kernel decomposition technique introduced in [27], two reproducing kernel Hilbert spaces and their associated kernel integral operators are used to characterize the properties and learnability of the hypothesis function class. Capacity independent error bounds and learning rates are deduced.

Let θ ∈ (0, 1), s₀, s₁ ∈ ℝ, τ₀, τ₁ ∈ [0, ∞), p₀, p₁ ∈ (0, ∞), q₀, q₁ ∈ (0, ∞], s = s₀(1 - θ) + s₁θ, τ = τ₀(1-θ) + τ₁θ, and . In this paper, under the restriction , the authors establish the complex interpolation, on Triebel–Lizorkin-type spaces, that , where denotes the closure of the Schwartz functions in . Similar results on Besov-type spaces and Besov–Morrey spaces are also presented. As a corollary, the authors obtain the complex interpolation for Morrey spaces that, for all 1 < p₀ ≤ u₀ < ∞, 1 < p₁ ≤ u₁ < ∞ and 1 < p ≤ u < ∞ such that , and p₀u₁ = p₁u₀, , where denotes the closure of the Schwartz space in . It is known that, if p₀u₁ ≠ p₁u₀, these conclusions on Morrey spaces may not be true.

We prove the generalized Hyers–Ulam stability of mapping on normed spaces for the following Jensen functional equation:

Asymptotic formulas are derived for the Stieltjes–Wigert polynomials S_n(z; q) in the complex plane as the degree n grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the q-Airy function A_q(z) is used as the approximant. For real x > 1/4, a limiting relation is also established between the q-Airy function A_q(x) and the ordinary Airy function Ai(x) as q → 1.

In this article, we continue the analysis of a class of singularly perturbed parabolic equations with alternating boundary layer type solutions. For such problems, the degenerate (reduced) equations obtained by setting a small parameter equal to zero correspond to algebraic equations that have several isolated roots. As time increases, solutions of these equations periodically go through two comparatively long lasting stages with fast transitions between these stages. During one of these stages, the solution outside the boundary layer (i.e. the regular part of the asymptotic solution) is close to one of the roots of the degenerate equation. During the other stage, the regular part of the asymptotic solution is close to the other root. Here we discuss some specific features of the solutions' behavior for such problems in certain two-dimensional spatial domains.