We give conditions in order to approximate a real function by $\phi$-strongly modulus $c$ convex functions.

We consider finitely many interval maps simultaneously acting on the unit interval $I = [0, 1]$ in the real line $\mathbb{R}$; each with utmost finitely many jump discontinuities and study certain important statistical properties. Even though we use the symbolic space on $N$ letters to reduce the case of simultaneous dynamics to maps on an appropriate space, our aim in this paper remains to resolve ergodicity, rates of recurrence, decay of correlations and invariance principles leading to the central limit theorem for the dynamics that evolves through simultaneous action. In order to achieve our ends, we define various Ruelle operators, normalise them by various means and exploit their spectra.

This paper proves some results concerning the lower Hewitt-Stromberg dimension of projections. We give non-trivial lower bounds on the lower Hewitt-Stromberg dimension of the projections onto almost all $m$-dimensional subspaces. In addition, we estimate the dimension of the exceptional sets of $m$-dimensional subspaces where the dimensions of projection are smaller than the typical value. The condition where the quasi-Assouad dimension is no greater than $m$ shows that the lower Hewitt-Stromberg dimension is preserved under orthogonal projections for almost all $m$-dimensional subspaces.

The purpose of this work is to examine the structure of optimal (alternate) dual fusion frames and get more flexibility in the use of (alternate) dual fusion frames for erasures of subspaces. We deal with optimal alternate dual fusion frames with respect to different definitions of duality. We compare the advantages of alternate and component preserving dual fusion frames. Then, we further study optimal alternate dual fusion frames. We also introduce a new concept of the so-called partial optimal dual which involves less computations for detecting optimal dual for erasures in known locations. Then we study the relationship between local and global optimal duals by partial optimal duals which leads to some applicable results. In the sense that, we obtain an overcomplete frame and a family of associated optimal duals by a given Riesz fusion basis. We present some examples to exhibit the effect of error rate when alternate dual fusion frames are applied in reconstruction.

If local forms of Littlewood's three principles are stated as axioms for an ordered field, then each principle is equivalent to the completeness axiom.

The question posed in the title is positively answered for any continuous, real function defined on any crowded subset of real numbers.

We prove a transfer of differential properties within the framework of non-negative functions from a translation invariant convex density basis $\textbf{B}$ to a translation invariant convex basis $\textbf{H}$ provided each set $H$ forming $\textbf{H}$ can be approximated by some set $B$ forming $\textbf{B}$ in the sense that the estimate $|H\triangle B|\leqslant c\mkern1mu |H\,{\cap}\, B|$ is valid, where $c$ is a positive constant not depending on $H$. An application of this transference result shows that the study of differential properties in the class of non-negative functions for translation invariant convex density bases can be reduced to that for translation invariant Busemann-Feller density bases formed of multi-dimensional rectangles.

Schwarz gradients and differentiability are introduced for functions of two variables. It is proved that for the Schwarz differentiability the existence of a generalized angular Schwarz gradient is necessary and sufficient. An alternative necessary and sufficient condition and a sufficient condition are established.

It is known that the controlled convergence theorem is one of the convergence theorems for $\mathcal{HK}$-integrable functions, which deals with nonabsolute $\mathcal{HK}$-integrable functions. This convergence theorem uses the concept of $U AC_\delta^{\star\star}(X)$, and the proof of the convergence theorem in the $n$-dimensional space is rather involved. In this paper, we shall give a necessary and sufficient condition for $UAC_\delta^{\star\star}(X)$, which is in terms of Lebesgue integrable functions on $X$ and primitive functions, see Theorem 9. Furthermore, we shall give three convergence theorems, Theorems 5, 7 and 10, for $\mathcal{HK}$-integrable functions in the $n$-dimensional space, which may be easier to apply.

We consider the Volterra integration operator $V$, $Vf( x)=\int_{0}^{x} f( t)\mkern1mu dt$, on the following subspace of the Wiener algebra $W[ 0,1]$:\[W^{( 1) }[ 0,1] :=\{ f\in W[ 0,1] : f^{\prime}\!\in W[ 0,1] \} .\]We investigate solvability of the operator equations $VA=\lambda AV$ and $VA=\lambda A^{2}V$, where $\lambda\in\mathbb{C}$ is a complex number. Our proof is based onthe Duhamel product of functions defined by\[( f \circledast g) ( x) :=\frac{d}{dx}\int_{0}^{x}f( x-t) g( t) \, dt.\]

In this paper, we prove a uniform local comparison principle for differential inequality of higher order. As a result, we derive a Hopf lemma for higher order differential operators with $L^1$ coefficients. In particular, an application of the obtained results to powers of the Laplacian is provided.

We define a constructive method of 'approximately sorting' a non-negative continuous function $f$ on $\mathbb{R}^n$. Specifically the symmetric decreasing rearrangement $f^*$ can be considered the sorted version of $f$ and certain equimeasurable rearrangements of $f$ are then sorted and shown to converge uniformly to $f^*$.

Let $f\colon \mathbb R\to\mathbb R$ and let $D_f$ denote the set of points of discontinuity of $f$. First it is proved that if $f|_{D_f}$ is continuous, then $D$ is a nowhere dense, $F_\sigma$ set. The major result is that if $D$ is a nowhere dense, $F_\sigma$ set then there is a function $f$ such that $D_f=D$ and $f|_D$ is continuous. Finally it is shown that such functions are of Baire class one.

We provide an elementary proof of the Schauder-Tychonoff Fixed Point Theorem, using basic notions of General Topology and Convex Analysis.

We simplify in this note Galbis' proof of certain norm estimates for self-adjoint Toeplitz operators on the Fock space.This relies on an extension (and a unification) of his integration lemmas, yet with a simpler proof in the same spirit.