B.S.T.J. Brief: A Note on a Special Class of One-Sided Distribution Sums

Occasionally encountered in the calculation of power spectra are limits of the form 1 A - 0 H=0 T. lim 21 U + F(n,N)]einx, N where F(n,N) ^ t am(N)nm m=0 M 0 where lim (D) and $(·) denote respectively a distribution limit2,3 and the Dirac delta function. I I . ANALYSIS Concerning notation, let C°° represent the space of infinitely differentiable scalar functions defined on the real line (-- Cd , the space of "rapidly decaying" test functions, viz., the linear vector space Cd = WW 6 C00, xVkx) 203 0( |x| oo)Vj, k ^ 0}; 204 T H E BELL SYSTEM TECHNICAL J O U R N A L , JANUARY 1965 and G, the space of generalized functions defined relative to the test functions of Cd · Finally, let Fg signify the generalized Fourier transform 2 3 of g 6 0 with F