Banach Journal of Mathematical Analysis is named after one of the most outstanding mathematicians in the XXth century, Stefan Banach. Thus it is natural to recall in the first issue of the journal some information about Banach. A very short biography, some of his most eminent results and some stories will be presented in this article.

Let $X$ and $Y$ be normed linear spaces. A mapping $T:X \rightarrow Y$ is called preserving the distance $r$ if for all $x,y$ of $X$ with $\|x-y\|_X=r$ then $\|T(x)-T(y)\|=r$. In this paper, we provide an overall account of the development of the Aleksandrov problem, the Aleksandrov-Rassias problem for mappings which preserve distances and details for the Hyers-Ulam-Rassias stability problem.

Let $q$ be a positive rational number and $n$ be a nonnegative integer. We prove the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras and of generalized derivations on quasi-Banach algebras for the following functional equation: \begin{eqnarray*} \sum_{i=1}^{n} f \left( \sum_{j=1}^{n}q (x_i-x_j) \right) + n f \left(\sum_{i=1}^{n} q x_i \right) = nq \sum_{i=1}^{n} f(x_i) . \end{eqnarray*} This is applied to investigate isomorphisms between quasi-Banach algebras.~The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

In [K.-I. Mitani and K.-S. Saito, J. Math. Anal. Appl. 327 (2007), 898-907] we characterized the strict convexity, uniform convexity and uniform non-squareness of Banach spaces using $\psi$-direct sums of two Banach spaces, where $\psi$ is a continuous convex function with some appropriate conditions on $[0,1]$. In this paper, we characterize the $B_n$-convexity and $J_n$-convexity of Banach spaces using $\psi$-direct sums of $n$ Banach spaces, where $\psi$ is a continuous convex function with some appropriate conditions on a certain convex subset of $\mathbb R^n$.

In this paper, we study the stability of the system of functional equations $f(xy)+f(xy^{-1})=2f(x)+f(y)+f(y^{-1})$ and $f(yx)+f(y^{-1}x)=2f(x)+f(y)+f(y^{-1})$ on groups. Here $f$ is a real-valued function that takes values on a group. Among others we proved the following results: 1) the system, in general, is not stable on an arbitrary group; 2) the system is stable on Heisenberg group $UT(3, K)$, where $K$ is a commutative field with characteristic different from two; 3) the system is stable on certain class of $n$-Abelian groups; 4) any group can be embedded into a group where this system is stable.

In the present work we continue studying the solvability of the linear functional equations ${\sum_{j=1}^N }c_j F\circ a_j=H$ and also the strong and weak stability of the corresponding operator $\mathcal{P}$ (see the definition below). By analogy with the Cauchy and Jensen operators once more model operator $\widehat{\mathcal{P}}$ is considered, and the stability problems as well as some solvability problems for $\widehat{\mathcal{P}}$ are studied. Several unsolved problem of a general character are formulated.

New sharp Kolmogorov type inequalities for hypersingular integrals with homogeneous characteristic of multivariate functions from Holder spaces are obtained. Proved inequalities are used to solve the Stechkin's problem on the best approximation of unbounded hypersingular integral operator by bounded ones on functional classes which are defined by a majorant of the modulus of continuity.

There are many known Cauchy-Schwarz-type inequalities which are valid in different frameworks. In this paper we consider the $A$-valued Cauchy-Schwarz inequality and its reverse in a semi-inner product $A$-module over the $C^*$-algebra $A$. Some remarks on the $A$-valued Cauchy-Schwarz inequality in a semi-inner product $A$-module over the $H^*$-algebra $A$ are also given.

Abram Haimovich Turetzkii [Uchenye Zapiski, 1 (149) (1959), 31-55 (translation in English in East J. Approx. 11 (2005), 337-359)] considered interpolatory quadrature rules which have the following form $\int_0^{2\pi}f(x)w(x)d x\approx \sum_{\nu=0}^{2n}w_\nu f(x_\nu)$, and which are exact for all trigonometric polynomials of degree less than or equal to $n$. Maximal trigonometric degree of exactness of such quadratures is $2n$, and such kind of quadratures are known as quadratures of Gaussian type or Gaussian quadratures for trigonometric polynomials. In this paper we prove some interesting properties of a special Gaussian quadrature with respect to the weight function $w_m(x)=1+\sin mx$, where $m$ is a positive integer.

In this paper we consider the general solution of a Jensen type functional equation. Moreover we prove the stability theorem of this equation in the spirit of Hyers, Ulam, Rassias and Gavruta.

Professor Themistocles M. Rassias' special predilection and contribution to the study of Mobius transformations is well known. Mobius transformations of the open unit disc of the complex plane and, more generally, of the open unit ball of any real inner product space, give rise to Mobius addition in the ball. The latter, in turn, gives rise to Mobius gyrovector spaces that enable the Poincare ball model of hyperbolic geometry to be approached by gyrovector spaces, in full analogy with the common vector space approach to the standard model of Euclidean geometry. The purpose of this paper, dedicated to Professor Themistocles M. Rassias, is to employ the Mobius gyrovector spaces for the introduction of the hyperbolic square in the Poincare ball model of hyperbolic geometry. We will find that the hyperbolic square is richer in structure than its Euclidean counterpart.

We consider the Birkhoff-James orthogonality in normed spaces and classes of linear mappings exactly and approximately preserving this relation. Some related stability problems are posed.

In this paper, we will consider Hyers-Ulam-Rassias stability of multipliers and ring derivations between Banach algebras. As a corollary, we will prove superstability of ring derivations and multipliers. That is, approximate multipliers and approximate ring derivations are exact multipliers and ring derivations.

The main object of this paper is to present several bounding inequalities for the classical Jacobi function of the first kind. A number of closely-related inequalities for such other special functions as the classical Laguerre function are also considered.

In this paper, by using the theory of integral bordism groups in PDE's, previously introduced by Prastaro, we give a new interpretation of the concept of (un)stability in the framework of the geometric theory of PDE's. A geometric criterium to identify stable PDE's and stable solutions of PDE's is given.