POSITIVITY OF THE INTERTWINING OPERATOR AND HARMONIC ANALYSIS ASSOCIATED WITH THE JACOBI-DUNKL OPERATOR ON ${\mathbb R}$

We consider a differential-difference operator Λα,β, [Formula: see text], [Formula: see text] on [Formula: see text]. The eigenfunction of this operator equal to 1 at zero is called the Jacobi–Dunkl kernel. We give a Laplace integral representation for this function and we prove that for [Formula: see text], [Formula: see text], the kernel of this integral representation is positive. This result permits us to prove that the Jacobi–Dunkl intertwining operator and its dual are positive. Next we study the harmonic analysis associated with the operator Λα,β (Jacobi–Dunkl transform, Jacobi–Dunkl translation operators, Jacobi–Dunkl convolution product, Paley–Wiener and Plancherel theorems…).

TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A SINGULAR DIFFERENTIAL-DIFFERENCE OPERATOR ON THE REAL LINE

We consider a singular differential-difference operator Λ on the real line which includes, as particular case, the Dunkl operator associated with the reflection group Z2 on R. We exhibit a Laplace integral representation for the eigenfunctions of the operator Λ. From this representation, we construct a pair of integral transforms which turn out to be transmutation operators of Λ into the first derivative operator d/dx. We exploit these transmutation operators to develop a new commutative harmonic analysis on the real line corresponding to the operator Λ. In particular, we establish a Paley–Wiener theorem and a Plancherel theorem for the Fourier transform associated to Λ.

Modified Laplace Transformation Method and ITS Application to the Anharmonic Oscillator

We apply a recently proposed approximation method to the evaluation of non-Gaussian integral and anharmonic oscillator. The method makes use of the truncated perturbation series by recasting it via the modified Laplace integral representation. The modification of the Laplace transformation is such that the upper limit of integration is cutoff and an extra term is added for the compensation. For the non-Gaussian integral, we find that the perturbation series can give an accurate result and the obtained approximation converges to the exact result in the N → ∞ limit (N denotes the order of perturbation expansion). In the case of the anharmonic oscillator, we show that several order result yields good approximation of the ground state energy over the entire parameter space. The large order aspect is also investigated for the anharmonic oscillator.

The generation of surface waves over a sloping beach by an oscillating line-source. I. The general solution

The problem of wave generation by a line source of sinusoidally varying strength situated in water above a beach of arbitrary angle α(0 < α ≤ π) is solved by the use of a Laplace-integral representation of the solution. It is shown that a solution can be constructed which is regular at the shoreline and gives an outgoing wave-train at infinity.

On resistive instabilities

It has been established by Furth, Killeen & Rosenbluth (1963), and by Johnson, Greene & Coppi (1963), that a hydromagnetic equilibrium which is stable on a theory in which electrical resistance is ignored, may yet be unstable through finite conductivity effects. These authors have isolated and categorized several types of such instabilities which, they show, originate from the critical layer in which the perturbation wavefront is perpendicular to the equilibrium magnetic field. In this paper, the asymptotic properties of the critical layer equations, for large values of the critical layer coordinate, are obtained in a number of cases of interest, using the sheet pinch model with uniform resistivity. The mathematical approach is a novel variant of the Laplace integral representation, which allows results of greater generality to be obtained than those given by previous authors. The technique is applied first to the slow interchange mode, and the restricted (but most significant) class of solutions found by Johnson et al . is recovered. It is also shown that modes entirely localized within the critical layer do not occur. Such modes do exist for the more rapid interchange modes, and a new discussion of these is presented. Finally, the oscillatory resistive modes, which arise when the perturbation wavefront is not perpendicular to the equilibrium magnetic field, are studied by a similar mathematical method, and a class of eigenvalues is obtained.