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The well-known limits on the radiation Q of electrically small antennas do not lead to a straightforward, rigorous computation of wide-band impedance matching limits, especially when higher order spherical modes are involved and higher order tuning networks are considered. Fano's formulation of the wide-band matching of arbitrary impedances does provide a rigorous solution and has been previously applied to the lowest order modes. In this paper, we apply Fano's theory to higher order spherical modes. Graphs of numerical limits on high-pass and bandpass tuning versus size are presented. It is shown that in the case where the relative bandwidth is large (multiple octaves), minimum size requirements are determined primarily by the return loss requirement and the frequency of the lower band edge; thus limits on high-pass tuning differ little from those on bandpass tuning. In the case of antennas required to radiate multiple modes, bounds on size versus bandwidth can be obtained from the limits on individual modes, with the highest order mode tending to set the most restrictive limit.