There is both Fourier series and the Fourier transform. For each, need to establish that they exist. Then will usually want to know that the inverse transform DOES approximate the original function. Details are in 4 books by W. Rudin. One of these, on groups, I ignored. I did too much group theory, e.g., wrote my honors paper on it, and didn't want to do more.
Principles does Fourier series and does not use measure theory. Real and Complex Analysis does the Fourier transform and uses the Lebesgue integral, that is, measure theory. Functional Analysis does Fourier theory with distributions.
I had serious courses from Principles and R & CA. Also Royden and Neveu. All from a star student of Cinlar, long at Princeton. I read FA, quickly. I don't much care to bother with distributions.
I got into applications via the fast Fourier transform and power spectral estimation as in Blackman and Tukey.
There is an intuitive view that sort of works: The given function is a point in a vector space, with an inner product. The sine waves are coordinate axes. They are orthogonal. The Fourier things are projections of the point onto the coordinate axes. The inverse Fourier thing reconstructs the original function. If use only some of the coordinate axes, then get a least squares approximation of the original function. This all works exactly in ordinary linear algebra, e.g., as in Halmos, Finite Dimensional Vector Spaces, sometimes given to physics students studying quantum mechanics.
But these Fourier things have infinitely many coordinate axes, countably infinite for Fourier series and uncountably infinite for the transform, and there the finite dimensional things don't always work. So, Rudin has to be very careful in presenting what does work and proving it -- so with the details it's not easy reading. What fails is a fairly general situation in a fully general Hilbert space. E.g., are not locally compact, but can get some help from a clever use of the parallolgram inequality (somewhat relevant in my startup).
I'm no teacher. I do not now nor have I ever had any desire to be a prof.
For more, get copies of Rudin's books and dig in.
Wonder of wonders, not all physics profs have done that.
Principles does Fourier series and does not use measure theory. Real and Complex Analysis does the Fourier transform and uses the Lebesgue integral, that is, measure theory. Functional Analysis does Fourier theory with distributions.
I had serious courses from Principles and R & CA. Also Royden and Neveu. All from a star student of Cinlar, long at Princeton. I read FA, quickly. I don't much care to bother with distributions.
I got into applications via the fast Fourier transform and power spectral estimation as in Blackman and Tukey.
There is an intuitive view that sort of works: The given function is a point in a vector space, with an inner product. The sine waves are coordinate axes. They are orthogonal. The Fourier things are projections of the point onto the coordinate axes. The inverse Fourier thing reconstructs the original function. If use only some of the coordinate axes, then get a least squares approximation of the original function. This all works exactly in ordinary linear algebra, e.g., as in Halmos, Finite Dimensional Vector Spaces, sometimes given to physics students studying quantum mechanics.
But these Fourier things have infinitely many coordinate axes, countably infinite for Fourier series and uncountably infinite for the transform, and there the finite dimensional things don't always work. So, Rudin has to be very careful in presenting what does work and proving it -- so with the details it's not easy reading. What fails is a fairly general situation in a fully general Hilbert space. E.g., are not locally compact, but can get some help from a clever use of the parallolgram inequality (somewhat relevant in my startup).
I'm no teacher. I do not now nor have I ever had any desire to be a prof.
For more, get copies of Rudin's books and dig in.
Wonder of wonders, not all physics profs have done that.