US Pat. 512,340 to Tesla (1894) shows a bifilar inductor comprising two coils in series, interleaved such that between adjacent windings the current always flows in the same direction and across the gap between them is a voltage. The internal capacitor thus configured serves to neutralize self-inductance by providing means for greatly increasing the capacity for energy storage in the electrical field. I suppose that is by shunting the back-emf from self-inductance to the parasitic capacitor.
Self-resonance (XL=XC) was known to Tesla but he does not discuss the advantage of his invention with regard to resonant frequency, so I suppose resonant frequency is immaterial. His purpose was to neutralize the effect of self-inductance, at any frequency.
The Tesla Inductor (not to be confused with the Tesla Coil or the Tesla Antenna) is a pancake coil which when oscillated should cause displacement currents in conductive material in the near field. Please note that this is not the bifilar design where currents flow in opposite directions in adjacent windings.
Does the Tesla Inductor need to be operated only at a resonant frequency if there is no intent to broadcast energy to the far field and only near-field effects are desired?
http://www.freepatentsonline.com/0512340.pdf