Towards Optimal Measurement of Power Spectra II: A Basis of Positive, Compact, Statistically Orthogonal Kernels

This is the second of two papers which address the problem of measuring the unredshifted power spectrum of fluctuations from a galaxy survey in optimal fashion. A key quantity is the Fisher matrix, which is the inverse of the covariance matrix of minimum variance estimators of the power spectrum of the survey. It is shown that bases of kernels which give rise to complete sets of statistically orthogonal windowed power spectra are obtained in general from the eigenfunctions of the Fisher matrix scaled by some arbitrary positive definite scaling matrix. Among the many possible bases of kernels, there is a basis, obtained by applying an infinitely steep scaling function, which leads to kernels which are positive and compact in Fourier space. This basis of kernels, along with the associated minimum variance pair weighting derived in the previous paper, would appear to offer a solution to the problem of how to measure the unredshifted power spectrum optimally. Illustrative kernels are presented for the case of the PSCz survey.