Conjugate Duality and the Exponential Fourier Spectrum For some fields such as econometrics (Shore 1980) oil prospecting (Claerbout 1976) speech recognition (Levinson and Lieberman 1981) satellite monitoring (Lavergnat et al. 1980) epilepsy diagnosis (Gersch and Tharp 1977) and plasma physics (Bloomfield 1976) there is a need to obtain an estimate of the spectral density (when it exists) in order to gain at least a crude understanding of the frequency content of time series data. An outstanding tutorial on the classical problem of spectral density estimation is given by Kay and Marple (1981). For an excellent collection of fundamental papers dealing with modern spec­ tral density estimation as well as an extensive bibliography on other fields of application see Childers (1978). To devise a high performance sample spectral density estimator one must develop a rational basis for its construction provide a feasible algorithm and demonstrate its performance with respect to prescribed criteria. An algorithm is certainly feasible if it can be implemented on a computer possesses computational efficiency (as measured by compu­ tational complexity analysis) and exhibits numerical stability. An estimator shows high performance if it is insensitive to violations of its underlying assumptions (i.e. robust) consistently shows excellent frequency resolutipn under realistic sample sizes and signal to noise power ratios possesses a demonstrable numerical rate of convergence to the true population spectral density and or enjoys demonstrable asymp­ totic statistical properties such as consistency and efficiency. 1. Introduction 2. Imposing the constraints 3. Selecting the objective functional: conjugate duality 4. Choosing a truncation point for (c*c)k 5. Solving for Solving for $$\\vec{\\theta}{\\rm^{(n)}: n=1}$$ 6. Solving for $$\\vec{\\theta}{\\rm^{(n)}: n > 1}$$ 7. Obtaining an initial estimate $$\\hat{\\vec{\\theta}}^{(\\rm n)}$$ of $$\\vec{\\theta}^{(\\rm n)}$$ 8. Numerical asymptotics 9. Assessing the sample efficiency of $$\\hat{\\vec{\\rm b}}^{(\\rm n)}$$ and $$\\hat{\\vec{\\theta}}^{(\\rm n)}$$ 10. The numerical experiment: preliminaries 11. The numerical experiment: results 12. Tables and graphs Fig. 1 The Sunspot Cycle: 1610–1976 Fig. 2 The Autocorrelogram of the untransformed data Fig. 3 The Autocorrelogram of the transformed data Table 1 Summary analysis for fn* Table 2 Summary analysis for fn Table 3 Order determination of fn* on the Akaike (1977) criterion Table 4 Summary analysis of fn* Table 5 Asymptotic analysis of Zn (11  n  20) Figs. 4–23 Simultaneous plot of fn* (k = 1) 13. Conclusion Appendix A: Comparison of Steffensen(1) simplified N R(2) unmodified N R(3) and Laguerre(4) update strategies for solving A(x) = r for x given r = 0(0.01)0.93 Appendix C: ANSI ‘77 FORTRAN source code listing (MAIN) Appendix D: ANSI ‘77 FORTRAN source code listing (FOLLOW UP) Appendix H: ANSI ‘77 FORTRAN source code (Korovkin).