A crude explanation might be like this - if you look at graphs of sin and cos, you'd instantly recognize their symmetries, but what if you're given the graph of a linear combination of them, and asked to decipher the coefficients?
Naively, you'd evaluate the functions at every point by trial & error until they much the shape of the given graph. Or use the symmetry of sin & cos to combine them constructively and destructively (peaks and valley) and to match the given shape.
FT & QFT are "shortcuts" that help to decipher the correct combination of basis functions.
Naively, you'd evaluate the functions at every point by trial & error until they much the shape of the given graph. Or use the symmetry of sin & cos to combine them constructively and destructively (peaks and valley) and to match the given shape.
FT & QFT are "shortcuts" that help to decipher the correct combination of basis functions.