Plane Wave Decomposition of Even-Dimensional Brownian Local Times

We study the one-dimensional representations of even-dimensional Brownian local time in two ways. The first result is obtained by the application of the odd-dimensional case. The second is led by the plane wave decomposition of the δ-function. The Hilbert transform of derivatives of local time is considered in the second result, in which Hermite functions of the second kind appear in the kernel of the Wiener–Itô decomposition.