MATH 321:201: Real Variables II (Term 2, 2010) Home work assignment # 8 Due date: Friday March 26, 2010 (hand-in in class) This HW assignment has two pages. Problem 1 (Fejér’s theorem): Do [Rudin, Ch 8. Exercise #15]. Problem 2 (Sobolev imbedding theorem) : Suppose f : R → R is a 2π-periodic Riemann integrable function. (a) Assume ∞ X |n|2 |fˆ(n)|2 < ∞. −∞ Show that there exists a 2π-periodic continuous function g : R → R such that Z π |f (x) − g(x)|2 dx = 0. −π (b) Assume for k ∈ N, ∞ X |n|2(k+1) |fˆ(n)|2 < ∞. −∞ Show that there exists a 2π-periodic function g : R → R, which is differentiable k times, such that Z π |f (x) − g(x)|2 dx = 0. −π Problem 3: Use the same notation and assumptions as in Problem 4 in HW # 7, except now we assume ONLY that g is Riemann integrable: in particular, g may NOT be differentiable. Show that the following holds: Z π |f (x, t) − g(x)|2 dx → 0 as t → 0+. −π Problem 4: (Poincaré inequality) (a) Show that there exists a constant C > 0 with the following property: Let fR : R → R be a π 2π-periodic function such that its derivative f 0 exists and integrable. Assume −π f (x)dx = 0. Then Z π Z π |f 0 (x)|2 dx. |f (x)|2 dx ≤ C −π −π Find the smallest constant C for which this inequality holds (independently of f ). (Hint: Use Parseval’s theorem.) (b) Show that there exists a constant C > 0 with the following property: Let f : R → R be a differentiable function such that its derivative f 0 is integrable. Assume f (a) = f (b) = 0 for some a < b ∈ R. Then, Z Z b b |f 0 (x)|2 dx. |f (x)|2 dx ≤ C a a Find the smallest constant C for which this inequality holds (independently of f ). (Hint: Use part a).) Essay Problem (Optional, for extra mark (≤ 20)): Hand-in in a separate paper (maximum three pages). To earn marks your answer should be INTERESTING. Give an intuitive or physical explanation of the following facts that for 2π-periodic Riemann integrable functions f, g : R → R, f[ ∗ g(n) = fˆ(n)ĝ(n) for all n ∈ Z, and fcg(n) = X fˆ(n − k)ĝ(k) for all n ∈ Z. k∈Z The following are suggested exercises. Please DO NOT hand-in, but, it is important for you to do these suggested exercises! Problem: Do Rudin, Ch. 8, Exercises # 16, # 17, #19. Problem: Do Rudin, Ch. 8, Exercises # 1. This important exercise is not about Fourier series but about power series. Use the definition of ex : ∞ X xk x e = . k! k=0 Here 0! = 1. 2