AMATH/PMATH 331 Real Analysis, Assignment 5 Due Mon Dec 7
Read Chapters 7 and 8 in the Lecture Notes, work on Problem Set 5 and read the solutions, then solve each of the following problems.
1: (a) The polynomial p(x) = x3 − 3x + 1 has a unique root in 0, 1 . Approximate this 2
root using the Banach Fixed Point Theorem as follows: Let f (x) = 1 (x3 + 1). Show that 11 3
f : 0, 2 → 0, 2 is a contraction map whose unique fixed point is the desired root of p. Approximate the root by using a calculator to find x5 where x0 = 0 and xn+1 = f(xn).
(b) Solve the differential equation y′ = 1 + x2y with y(0) = 0 in the interval [−1, 1] as x
t2f(t)dt. Show that F is
a contraction map (using the supremum norm) whose unique fixed point is the desired solution. Express the solution as a power series by finding a formula for fn(x) where f0(x) = 0 and fn+1(x) = F(fn)(x).
1 1
(c) Let f ∈ C[0,1]. Suppose that f(x)dx = 0 and x12+3nf(x)dx = 0 for all n ∈ Z+. 00
Use the Stone-Weierstrass Theorem to show that f (x) = 0 for all x ∈ [0, 1].
2: (a) Find the (real) Fourier series for f (x) = sin x, prove that the Fourier series converges uniformly to f on R, then evaluate at π to find the sum ∞ (−1)k+1 .
follows: Define F : C[−1,1] → C[−1,1] by F(f)(x) = x +
0
4k2−1
(b) Find the (real) Fourier series for the 2π-periodic function f : R → R given by f(x) = x
2
when −π < x ≤ π, then use Parseval’s Identity to find the sum 1 .
(c) Prove that for every f ∈ R(T) we have sn(f) → f in R(T),∥ ∥1. Hint: use Theorem 8.23 and the Cauchy-Schwarz Inequality).
mm
3: Let (an)n≥0 and (bn)n≥1 be sequences in R, let sm(x) = a0 + an cosnx+ bn sinnx
n=1 n=1
(a)Showthatan(σl)=l+1−nan for0≤n≤landbn(σl)=l+1−nbn for1≤n≤l. l+1 l+1
(b) Show that the sequence of functions (σl)l≥0 converges uniformly on R if and only if thereexistsf∈C(T)suchthatan =an(f)foralln≥0andbn =bn(f)foralln≥1.
(c) Show that there is no f ∈ C(T ) whose Fourier series is ∞ (−1)n cos nx + ∞ 1 sin nx. n n2
n=1 n=1
l form≥0,andletσl(x)= 1 sm(x)forl≥0.
k=1
∞
n2 n=1
l+1
m=0