Related papers: On the Bloch Theorem and Orthogonality Relations

The almost periodic functions form a natural example of a non-separable normed space. As such, it has been a challenge for constructive mathematicians to find a natural treatment of them. Here we present a simple proof of Bohr's fundamental…

We construct new families of discrete vector orthogonal polynomials that have the property to be eigenfunctions of some difference operator. They are extensions of Charlier, Meixner and Kravchuk polynomial systems. The ideas behind our…

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras. In particular, we give several applications of operator space theory, based on the surprising fact that certain maps are…

The aim of this work is to study the existence of a periodic solutions of integro-differential equations d dt [x(t)-- L(x t)] = A[x(t)-- L(x t)]+ G(x t)+ t --$\infty$ a(t-- s)x(s)ds+ f (t), (0 $\le$ t $\le$ 2$\pi$) with the periodic…

We answer, by counterexample, several open questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator…

We prove some formulas relating Cauchy-Riemann operators defined on hypercomplex subspaces of an alternative *-algebra to a differential operator associated with the concept of slice-regularity and to the spherical Dirac operator. These…

We describe a new operator space structure on $L_p$ when $p$ is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder's…

This paper gives an analysis of the periodic solutions of a ring of $n$ oscillators coupled to their neighbors. We prove the bifurcation of branches of such solutions from a relative equilibrium, and we study their symmetries. We give…

Consider an elliptic operator in divergence form with symmetric coefficients.If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the…

We consider eigenvalue problems in quantum mechanics in one dimension. Hamiltonians contain a class of double well potential terms, x^6 + \alpha x^2, for example . The space coordinate is continued to a complex plane and the connection…

In this paper we study an algebraic and topological structure inside the following sets of special functions: Bloch functions defined on the open unit disk that are unbounded and analytic functions of bounded type defined a Banach algebra E…

Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they…

We study one-dimensional linear hyperbolic systems with $L^{\infty}$-coefficients subjected to periodic conditions in time and reflection boundary conditions in space. We derive a priori estimates and give an operator representation of…

In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are…

We study the nonlinear Schr\"odinger equation with a periodic delta-function potential. This realizes a nonlinear Kr\"onig-Penney model, with physical applications in the context of trapped Bose-Einstein condensate alkaly gases and in the…

In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the…

The main aim of this note is to introduce the notion of an almost anti-periodic function in Banach space. We prove some characterizations for this class of functions, investigating also its relationship with the classes of anti-periodic…

In this paper, we investigate some properties on harmonic functions and solutions to Poisson equations. First, we will discuss the Lipschitz type spaces on harmonic functions. Secondly, we establish the Schwarz-Pick lemma for harmonic…

The concept of monogenic functions over real alternative $\ast$-algebras has recently been introduced to unify several classical monogenic (or regular) functions theories in hypercomplex analysis, including quaternionic, octonionic, and…

In this paper we investigate when linearly independent eigenfunctions of the Schr\''odinger operator may have the same modulus. General properties are established and the one-dimensional case is treated in full generality. The study is…