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A formula is given for the maximum gain of a class of antennas having fields expressible as a finite number of spherical wave functions. This maximum gain can be achieved for arbitrarily polarized radiation fields, and it can be related to antenna size by requiring the near fields to be small in magnitude. Also, the radiation field is in the form of a polynomial, so that patterns optimum in the Tchebycheff sense can be defined. Formulas for the relationship of beamwidth to sidelobe level are given.