and so may have more singular solutions. As a result, the nascent delta functions that arise as fundamental solutions of the associated Cauchy problems are generally oscillatory integrals. An example, which comes from a solution of the Euler–Tricomi equation of transonic gas dynamics, is the rescaled Airy function Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the Dirichlet kernel below), rather than in the sense of measures. Another example is the Cauchy problem for the wave equation in R1+1: The solution u represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin. Other approximations to the identity of this kind include the sinc function (used widely in electronics and telecommunications) and the Bessel function Plane wave decomposition One approach to the study of a linear partial differential equation where L is a differential operator on Rn, is to seek first a fundamental solution, which is a solution of the equation When L is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned). For more complicated operators, it is sometimes easier first to consider an equation of the form where h is a plane wave function, meaning that it has the form for some vector ξ. Such an equation can be resolved (if the coefficients of L are analytic functions) by the Cauchy–Kovalevskaya theorem or (if the coefficients of L are constant) by quadrature. So, if the delta function can be decomposed into plane waves, then one can in principle solve linear partial differential equations. Such a decomposition of the delta function into plane waves was part of a general technique first introduced essentially by Johann Radon, and then developed in this form by Fritz John (1955). Choose k so that is an even integer, and for a real number s, put Then δ is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure dω of for ξ in the unit sphere Sn−1: The Laplacian here is interpreted as a weak derivative, so that this