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This paper is concerned with maximizing the square integral over certain classes of periodic positive-definite functions. The question arose at Jet Propulsion Laboratory in connection with maximizing the average power of the received signal in radar exploration of the planets. We present computational evidence that the maximizing function exists and, in most but not all cases, is unique. Upper and lower bounds for the maximum square integral are computed, and formulas are conjectured for thedoi:10.1090/s0025-5718-1973-0411168-7 fatcat:hbhbhuwhivehrfhsfwjifchgci