We solve the Cauchy problem for the n-dimensional wave equation using elementary properties of the Bessel functions.

We solve the Cauchy problem for the n-dimensional wave equation using elementary properties of the Bessel functions.

We discuss a sharpened Hausdorff–Young inequality for n-dimensional Hermite expansions.

We compute the integral of monomials of the form x^2β over the unit sphere and the unit ball in R^n where β = (β1, . . . , βn) is a multi–index with real components βk > −1/2, 1 ≤ k ≤ n, and discuss their asymptotic behavior as some, or all, βk → ∞. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in Rn. We also consider the Fourier transform of monomials xα restricted to the unit sphere in Rn, where the multi–indices α have integer components, and discuss their behaviour at the origin.