Related papers: Analysis in $R^{1,1}$ or the Principal Function Th…
A fundamental problem from invariant theory is to describe the endomorphism algebra of multilinear functions on a representation V invariant under the action of a group G. According to Weyl's classic, a first main (later: fundamental)…
We analyze spectral properties of the ultrarelativistic (Cauchy) operator $|\Delta |^{1/2}$, provided its action is constrained exclusively to the interior of the interval $[-1,1] \subset R$. To this end both analytic and numerical methods…
The character problems of SU(2) and SU(1,1) are reexamined from the standpoint of a physicist by employing the Hilbert space method which is shown to yield a completely unified treatment for SU(2) and the discrete series of representations…
Conformal field theory and its axiomatisation in terms of vertex operator algebras or chiral algebras are most commonly considered on the Riemann sphere. However, an important constraint in physics and an interesting source of mathematics…
Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several…
A group theoretical description of basic discrete symmetries (space inversion P, time reversal T and charge conjugation C) is given. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex…
In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like…
In this note we study $SL(2,\mathbb{Z})$-invariant functions such as modular graph functions or coefficient functions of higher derivative corrections in type IIB string theory. The functions solve inhomogeneous Laplace equations and we…
In these lectures we provide a basic introduction into the topic of dispersion relation and analyticity. The properties of 2-point functions are discussed in some detail from the viewpoint of the K\"all\'en-Lehmann and general dispersion…
We develop a Fourier analysis for a generalization of the class of periodic functions, often referred to as $(\theta, T)$-periodic functions, and prove several properties and inequalities related to the Fourier transform, including a type…
This is the second of three papers math.DG/0111324, math.DG/0204343 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using…
Here we will consider examples of conformally flat manifolds that are conformally equivalent to open subsets of the n-dimensional sphere. For such manifolds we shall introduce a Cauchy kernel and Cauchy integral formula for sections tasking…
In this article we investigate the Fourier series and transforms for the functions defined on the $ [0, 2 \pi]^ d $ or $ R^d $ and belonging to the exponential Orlicz and some other rearrangement invariant (r.i.) spaces.
A variation of Dirac equation based on SO(2,1) group is suggested for treating low dimensional systems in the three dimensional x,y,t space. Non-unitary representations are developed in an analogous way to those used in the ordinary Dirac…
Hille in 1959 gave a stronger form for the classical implicit function theorem of Cauchy for equations of the type $y = \psi(x,y)$, $x,y$ real or complex, $\psi$ analytic with a condition which determines a radius of convergence of the…
These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy,…
In this paper, strongly $(\alpha,T)$-convex functions, i.e., functions $f:D\to \R$ satisfying the functional inequality $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-t\alpha\big((1-t)(x-y)\big)-(1-t)\alpha\big(t(y-x)\big)$$ for $x,y\in D$ and $t\in…
We consider the Cauchy problem for homogeneous linear $q$-difference-differential equations with constant coefficients. We characterise convergent, $k$-summable and multisummable formal power series solutions in terms of analytic…
Modular double of quantum group U_q (sl(2)) with deformation parameter q=e^{i\pi\tau} is a natural object explicitly taking into account the duality \tau -> 1/\tau. The use of the modular double in CFT allows to consider the region 1<c<25…
This is the first volume of a textbook for a two-semester course in mathematical analysis. This first volume is about analysis of functions of a single variable. The topics covered include completeness axiom, Archimedean property,…