My group theory text (and other sources) say that the character of a n operator in a representation is the trace of the matrix corresponding to the operator. ie. Given a group operator in some representation we have .
My newest text says a function (in a space) ca be expanded into a sum/integral of characters
(To put this into context the section is talking about inverse Fourier transforms and that the inverse FT diagonalizes the representation for f.)
Are these two character definitions related to each other in some way? Or are they representing (no pun intended) different concepts? My text, at least, uses the same symbol for both.