What is the precise meaning of "matching impedance" for an ultra-wide band antenna, and how to do that?
... I'm not sure I really understand what is the meaning of matching impedance, even for an ordinary antenna


Let's go here first.  Impedance matching means that the internal resistance/impedance of the source equals the resistance/impedance of the load.  When that is true, the maximum energy is transferred from the source to the load.  

That is what we are looking for in an antenna.  We want maximum energy transferred from the transmitter to the antenna.  So the transmitter impedance should match the line and antenna for maximum transmitted energy.


  I did an ultra-wideband self-grounded bowtie antenna, (400MHz - 3GHZ),


By definition, "wideband" means that the antenna matches the transmitter and feed line, over the range of frequencies that you provide (400 MHz to 3 GHz).  So, if you say that your antenna is "wideband" then you are saying that it matches over the band.  

The ultimate "wideband" antenna is a dummy load.  It matches at any frequency.  Of course, it doesn't radiate.  (Y0u can't have everything!)  But it certainly absorbs the maximum amount of energy from the transmitter and feed line.  


  What is the precise meaning of “matching impedance” for an ultra-wide band antenna, and how to do that?


So, the answer to your FIRST question is as I stated in the beginning.  And as to the second question "how to do that", the answer is "nothing".  The information and conditions of the antenna that you provide already says that you don't need to do anything to match the antenna impedance to the transmitter and line.  As you say, the antenna is "wideband".

If you want, you can test the antenna, measuring the SWR for all of the frequencies in the "wideband".  That test is up to you to verify that your antenna is really "wideband".