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Bounds on the optimum transient radiation from an arbitrary antenna enclosed by a spherical surface of specified size are presented. The optimization criterion is either maximization of the radiated electric field amplitude at a specified time and far-zone position, or maximization of the radiated energy density in a specified time interval for a particular far-zone position. The latter optimization results in the prolate spheroidal wave functions. In both cases, the total energy radiated by the antenna is constrained to be 1 J, and the antenna excitation is assumed to be bandlimited. The performance of the optimum arbitrary antenna is compared to that of optimized practical antennas, such as dipoles and arrays of dipoles. The effect of a sidelobe constraint is also studied.