Spectral Decomposition
Encyclopedia of Statistical Sciences · 2005
Let π 1 , π 2 be cuspidal automorphic representations of PGL 2 (R) of conductor 1 and Hecke eigenvalues λπ 1,2 (n), and let h > 0 be an integer.For any smooth compactly supported weight functions W 1,2 : R × → C and any Y > 0 a spectral decomposition of the shifted convolution sumis obtained.As an application, a spectral decomposition of the Dirichlet seriesis proved for ℜs > 1/2 with polynomial growth on vertical lines in the s aspect and uniformity in the h aspect.τ