This a very good point, which I decided to neglect when I answered hellfire's question. Hellfire asked specifically about quantum fields, and, using print.google.com, I saw that Wald p. 393 (access to p. 394 was restricted ) talks about operators, so I answered in terms Fourier expansions of quantum field operators, which don't even converge in norm.reilly said:Fourier series and integrals converge in the mean. ... The issue of wave functions for a localized particle is not the only manifestation of convergence issues.
I first ran into this when I learned in a high school electronic course that a square wave "equals"For example, there's the Gibbs Phenomena, in which the Fourier representation always oscillates around a point of discontinuity -- just look at a square waveform in an oscilloscope ... . You get a good RMS fit at the corners, and a lousy pointwise fit at the corners