Answer. You need to tell why you chose the type of convergence, but this only involves stating that the conditions of a particular theorem are met. The ones that get used are Theorems 1.27, 1.29, and 1.32. In each case, look at the periodic extension of the function (scale to change the period from 2*pi to the one you need), and make a rough sketch for it. Then, answer the question the way we did in class Thursday, 2/3/00.
Question (2/5/00). We are having some problems figuring out how to do problem 18, §1.4. We are not sure how to find the Fourier series because we can not do the integrals, since the intervals are open and not closed.
Answer. You do not need to compute the Fourier series involved to answer the question. Also, the endpoints for the piecewise continuous functions are immaterial here, as far as integrals go. Just add in the right endpoint, and think of the functions as p eriodic. This is essentially the same sort of thing you did in #16.
Question (2/6/00). I am still a little bit confused about the different types of convergence. My view of uniform convergence for the Fourier series of a function f(x) is this. Choose epsilon. Draw a tube of size epsilon around the graph of the function. If the partial sums stay in the tube for all N large enough, no matter what epsil on is, the Fourier series is uniformly convergent. Is this correct? What about mean convergence and pointwise convergence for the Fourier series of a function with jumps?
Answer. Your idea of uniform convergence is correct. For pointwise convergence you (1) fix a point x and (2) take the limit as N -> infinity. If the limit exists, then you have a pointwise limit at x. If the pointwise limit exists for each x in some interval, then you have pointwise convergence on the interval. This is the kind of convergence that you get for a function that has jump discontinuities.
The partial sums for the Fourier series for such a function are not uniformly convergent, however. If you want to really see why, try this experiment. Pick epsilon. For each x in an interval near the jump, find the first N for which | SN(x) - f(x) | < epsilon. Call this Nx. Plot it. You should see that as you get close to the jump, Nx gets large. Make epsilon smaller and repeat.
Convergence in the mean is statistical. About all that one needs for it is that the function have finite ``energy'' -- that is, the integral of its square is finite. More on this on Tuesday, 2/8/00.
Question (2/8/00). I am confused about uniform convergence of the Fourier series for two of the the functions in §1.4, Problem 18. The first concerns sin(x)/x. The Fourier series for sin(x)/x should not converge uniformly, since the function is discontinuous at x=0. It seems strange though, since discounting that point it would uniformly converge. The other is ex. I thought about it one way, and if we just take ex on [-1,1] and repeat it, then it would be discontinuous and thus not uniformly convergent. Then I thought about it another way. If instead of ex on [-1,1], I consider a piecewise function that is ex on [-1,1] and e2-x on [1,3] and make a periodic repetition of this, it would be piecewise smooth and so the fourier series would uniformly converge. How should I deal with this?
Answer. The series for sin(x)/x does converge uniformly. If we set sin(x)/x = 1 for x=0, the discontinuity goes away. such discontinuities do not affect the Fourier series for a function, nor do they affect convergence properties. Concerning ex, you are given the function only on [-1,1]. When you are asked to compute the Fourier series for such a function, you are really computing it for the periodic extension of it, and no other extension. Thus the periodic extension that one needs to use is the one for which ex is the template. On [1,3], this would have values given by ex-2, not e2-x.
Question (4/26/00). I just wanted to clarify what needs to be understood for the upcoming test, not because you were unclear, but just so I feel confident in what I am studying and what I know. It is my understanding that to be properly prepared for the test we should memorize the theorems, know what they are saying, and know how to apply basic concepts such as decomposition, recomposition, etc. Is this correct? Again, I just want to be clear since we really didn't review anything and that tells me that you either have tremendous confidence in us or you are going to use this test as an excuse for a grade, as you hinted at in class. Any insight that could possibly steer me in the right direction or relieve my nerves would be greatly appreciated.
Answer. Here is a brief review of the sections that I said would be on the exam; I hope it helps out.