A Riemann--Lebesgue lemma for Jacobi expansions

Fatou's Lemma for Weakly Converging Measures under the Uniform Integrability Condition

This note describes Fatou's lemma and Lebesgue's dominated convergence theorem for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to these measures. The note also provides new formulations of uniform Fatou's lemma, uniform Lebesgue convergence theorem, the Dunford-Pettis theorem, and the fundamental theorem for Young measures based on the equivalence of uniform integrability and the apparently weaker property of asymptotic uniform integrability for sequences of functions and finite measures…

加权Jacobi-Radau函数的等相单调性与Jacobi Lebesgue常数

Equal-Phase Monotonicity for Weighted Jacobi-Radau Functions and Jacobi Lebesgue Constants

针对加权Jacobi-Radau函数，证明等相单调性定理，恢复并简化了Wong-Zhang与Qu-Wong关于Lebesgue常数的结论。

关于零阶贝塞尔–傅里叶级数的黎曼–勒贝格引理的部分逆命题

A partial converse to the Riemann--Lebesgue lemma for Bessel--Fourier series of order zero

本文证明零阶贝塞尔–傅里叶系数衰减的黎曼–勒贝格引理在 $0\leq α<1/2$ 时存在部分逆命题，并猜想该结论可推广到 $α=1/2$。

On Direct Integral Expansion for Periodic Block-Operator Jacobi Matrices and Applications

We construct a functional model (direct integral expansion) and study the spectra of certain periodic block-operator Jacobi matrices, in particular, of general 2D partial difference operators of the second order. We obtain the upper bound, optimal in a sense, for the Lebesgue measure of their spectra. The examples of the operators for which there are several gaps in the spectrum are given.

Rapidly growing Fourier integrals

The Riemann-Lebesgue Lemma says that the Fourier transform of an absolutely integrable function on the real line tends to zero as the transform parameter tends to infinity. When the integral is allowed to converge conditionally, the transform can have arbitrarily rapid pointwise growth as the transform parameter tends to infinity. Smoothness of the function to be transformed need not decrease growth of the transform.