In a little known paper, Haslam-Jones defined a collection of higher order derivatives in terms of an integral and Legendre polynomials. One member of this collection is equivalent to the higher order Peano derivatives. The purpose of this paper is to present a more direct proof of this equivalence.

Kostyrko and Šalát showed that if a linear space of bounded functions has an element that is discontinuous almost everywhere, then a typical element in the space is discontinuous almost everywhere. We give a topological analogue of this theorem and provide some examples.

Weighted Hardy-type inequalities between suitable amalgams $\ell^q(L^{p},u)$ and $\ell^{\bar q}(L^{\bar p},v)$ are characterized. The Hardy-type operator involved in the inequalities involves functions which are not necessarily non-negative.

A \textquoteleft pressure' functional $\Phi^s (T_1,\ldots ,T_N)$, defined as the limit of sums of singular value functions of products of linear mappings $(T_1,\ldots ,T_N)$, is central in analysing fractal dimensions of self-affine sets. We investigate the continuity of $\Phi^s$ with respect to the linear mappings $(T_1,\ldots ,T_N)$ which underlie the self-affine sets.

Let $\{\varphi_n(x)$, $n=1,2,\dots\}$ be an arbitrary complete orthonormal system (ONS) on the interval $I:=[0,1)$ that consists of a.e. bounded functions. Then there exists a rearrangement $\{ \varphi_{\sigma_1(n)}$, $n=1,2, \dots\}$ of the system $\{\varphi_n(x)$, $n=1,2,\dots\}$ that has the following property: for arbitrary nonnegative, continuous and nondecreasing on $[0,\infty)$ function $\phi(u)$ such that $u\phi (u)$ is a convex function on $[0,\infty)$ and $\phi (u) = o(\ln u)$, $u \to \infty$, there exists a function $f \in L(I^2)$ such that $\int_{I^2} | f(x,y) |$ $\phi( | f(x,y) | )\;dx\; dy \infty$ and the sequence of the square partial sums of the Fourier series of $f$ with respect to the double system $\{ \varphi_{\sigma_1 (m)}(x)\varphi_{\sigma_1 (n)}(y)$, $m,n \in\N \}$ on $I^2$ is essentially unbounded in measure on $I^2$.

Let $\tau$ be a topology on the real numbers which is finer than the usual topology such that $\R_\tau$ is a weak $P$-space. In this paper, conditional completeness and conditional $\sigma$-completeness of $C(X)$ and $C(X,\R_\tau)$ are compared. In particular, it is shown for zero-dimensional spaces that $C(X,\R_\tau)$ is conditionally $\sigma$-complete if and only if $X$ is a $P$-space and that $C(X,\R_\tau)$ is conditionally complete if and only if $X$ is an extremally disconnected $P$-space.

A regulated function on the real line is a real valued function whose left-hand and right-hand limits exist at all points. In this paper we examine a generalization of regulated functions to functions defined on Davison Spaces, which are topological spaces with a little extra structure. Properties of such functions are discussed. Our main result concerns the set of discontinuities of these functions. We also prove that regulated functions defined on the natural numbers, with the cofinite topology, coincide with convergent sequences.

In this paper, we show that if an R-integrable (or BV integrable function) defined over a compact interval $[a,b]\times [c,d]\subset \mathbb R^2$ satisfies certain conditions, then over any subinterval of $[a,b]\times [c,d],$ the iterated integral exists and equals the double integral. We also present some examples relevant to our theory.