We work with a fixed N-tuple of quasi-arithmetic means $M₁,...,M_N$ generated by an N-tuple of continuous monotone functions $f₁,...,f_N:I\to ℝ$ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^N\ni b\mapsto (M₁\left(b\right),...,M_N\left(b\right))$ tend pointwise to a mapping having values on the diagonal of $I^N$. Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M₁,...,M_N$ taken on b. We effectively...

A function f: ℝⁿ → ℝ satisfies the condition $Q_i\left(x\right)$ (resp. $Q_s\left(x\right)$, $Q_o\left(x\right)$) at a point x if for each real r > 0 and for each set U ∋ x open in the Euclidean topology of ℝⁿ (resp. strong density topology, ordinary density topology) there is an open set I such that I ∩ U ≠ ∅ and $|(1/\mu \left(U\cap I\right)){\int }_{U\cap I}f\left(t\right)dt-f\left(x\right)|<r$. Kempisty’s theorem concerning the product quasicontinuity is investigated for the above notions.

We study the integrability of Banach valued strongly measurable functions defined on $[0,1]$. In case of functions $f$ given by ${\sum }_{n=1}^{\infty }{x}_n{\chi }_{E_n}$, where $x_n$ belong to a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for the Bochner and for the Pettis integrability of $f$ (cf Musial (1991)). In this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions.

A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case.

For a merely continuous partition of unity the PU integral is the Lebesgue integral.

This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s $L^2$-estimates for the $\overline{\partial }$-equation is used as motivation. We also use the method to prove $L^2$-estimates for the $d$-equation with a weight $e^{-t\phi }$ where $\phi$ is a nondegenerate Morse function.