The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.Depending on the application a Lebesgue integral, distributional, or other approach may be most appropriate. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint. provides solid justification for these formal procedures without going too deeply into functional analysis or the theory of distributions.
The Fourier transform can be formally defined as an improper Riemann integral making it an integral transform although this definition is not suitable for many applications requiring a more sophisticated integration theoryDepending on the application a Lebesgue integral distributional or other approach may be most appropriate For example many relatively simple applications use the Dirac delta function which can be treated formally as if it were a function but the justification requires a mathematically more sophisticated viewpoint provides solid justification for these formal procedures without going too deeply into functional analysis or the theory of distributions