The internal and external Robin problems for the Laplace equation in bounded starlike domains are addressed. We show how to derive the relevant solutions by using a suitable Fourier series-like method. Numerical results are specifically obtained considering threedimensional domains whose boundary is defined by a generalization of the so-called ‘‘superformula’’ introduced by Gielis. By using the...

The classical Hausdorff-Young inequality for locally compact abelian groups states that, for 1 ≤ p ≤ 2, the L-norm of a function dominates the L-norm of its Fourier transform, where 1/p + 1/q = 1. By using the theory of non-commutative L-spaces and by reinterpreting the Fourier transform, R. Kunze (1958) [resp. M. Terp (1980)] extended this inequality to unimodular [resp. non-unimodular] groups...

In this paper we extend analytic signal method to the functions in many dimensions. First it is shown how to obtain separate phase-shifted components and how combine them to obtain signal's envelope, instantaneous frequencies and phases in many dimensions. Second, we show that phase-shifted components may be obtained by positive frequency restriction of the Fourier transform defined in the alge...

Singular Value Decomposition (SVD) is one of the most useful techniques for analyzing data in linear algebra. SVD decomposes a rectangular real or complex matrix into two orthogonal matrices and one diagonal matrix. In this work we introduce a new approach to improve the preciseness of the standard Quantum Fourier Transform. The presented Quantum-SVD algorithm is based on the singular value dec...

Fractional Fourier transforms (FrFT) are a natural one-parameter family of unitary transforms that have the ordinary Fourier transform embedded as a special case. In this paper, following the efforts of several authors, we explore the theory and applications of FrFT, from the standpoints of both quantum mechanics and analysis. These include the phase plane interpretation of FrFT, FrFT’s role in...

We define a Fourier transform S for the quantum double D(G) of a finite group G. Acting on characters of D(G), S and the central ribbon element of D(G) generate a unitary matrix representation of the group SL(2, Z). The characters form a ring over the integers under both the algebra multiplication and its dual, with the latter encoding the fusion rules of D(G). The Fourier transform relates the...

A new approach to the algebra Gτ of temperate nonlinear generalized functions is proposed, in which Gτ is based on the space OM endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of Gτ . This is illustrated by the natural introduction of a re...

A new approach to the algebra Gτ of temperate nonlinear generalized functions is proposed, in which Gτ is based on the space OM endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of Gτ . This is illustrated by the natural introduction of a re...

1. A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Bull. Amer. Math. Soc. 67 (1961), 579-583. 2. , Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485-535. 3. Harish-Chandra, On the characters of a semisimple Lie group, Bull. Amer. Math. Soc. 61 (1955), 389-396. 4. , Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87-12...

This paper concerns a class of orbital integrals in Lie algebras over p-adic fields. The values of these orbital integrals at the unit element in the Hecke algebra count points on varieties over finite fields. The construction, which is based on motivic integration, works both in characteristic zero and in positive characteristic. As an application, the Fundamental Lemma for this class of integ...