Elements of q-harmonic analysis

Elements of harmonic analysis

Elements of harmonic analysis , 5 Stephen William Semmes Rice University

are also bounded functions on R for all multi-indices α, β, where d(β) denotes the degree of β, which is to say the sum of its components. Thus S(R) contains the smooth functions on R with compact support as a linear subspace. Also, S(R) contains the Gaussian functions exp(−A(x)·x), where A is a positive-definite symmetric linear transformation on R, and x · y is the usual inner product defined by

Elements of Harmonic Analysis, 3 Locally Compact Abelian Topological Groups

Let A be an abelian group. Thus A is a set equipped with a binary operation + which is commutative and associative, there is an identity element 0 ∈ A such that 0 + a = a for all a ∈ A, and each a ∈ A has an inverse −a characterized by a + (−a) = 0. As basic examples, the integers, real numbers, and complex numbers are abelian groups under addition, and for each positive integer n we have the i...

S ep 1 99 9 Harmonic Analysis on the quotient Q × ⋉ Q \ A 1 ⋉

Abstract. Let G be the semidirect product A1 ⋉ A of the adeles and the norm 1 ideles. Let Γ be the discrete subgroup Q× ⋉Q. In this paper the trace formula for this setting is established and used to give the complete decomposition of the G-representation on L2(Γ\G). It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those con...

A q-Deformation of the Harmonic Oscillator

The q-deformed harmonic oscillator is studied in the light of q-deformed phase space variables. This allows a formulation of the corresponding Hamiltonian in terms of the ordinary canonical variables x and p. The spectrum shows unexpected features such as degeneracy and an additional part that cannot be reached from the ground state by creation operators. The eigenfunctions show lattice structu...