This is a problem I ran into doing some physics research. Unfortunately, it is a bit outside my field of expertise. So, I'd appreciate some help.
From what I could find out, the cardinality of the space of all functions is strictly larger than the cardinality of the continuum. However, I also found out that if the functions are restricted to be only those that satisfy a specific property, such as the analytic functions, then the cardinality of the space of such functions could drop down to be that of the continuum.
To have a well-defined Fourier transform, a function needs to be square integrable. My question is, does this requirement reduce the number of functions to the extent that the cardinality of the space of such functions reduces to be that of the continuum?