We introduce an ap-Henstock-Kurzweil type integral with a non-atomic Radon measure and prove the Saks-Henstock type lemma. The monotone convergence theorem, -Henstock-Kurzweil equi-integrability, and uniformly strong Lusin condition are discussed.
We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.
We prove that if for certain values of , then
It is shown that a uniform version of Sklyarenko's integrability condition for Perron integrals together with pointwise convergence of a sequence of integrable functions are sufficient for a convergence theorem for Perron integrals.
We give a definition of uniform PU-integrability for a sequence of -measurable real functions defined on an abstract metric space and prove that it is not equivalent to the uniform -integrability.
The purpose of this note is to provide characterizations of operator convexity and give an alternative proof of a two-dimensional analogue of a theorem of Löwner concerning operator monotonicity.