**Assignment 4** - Due Wednesday, September 26, 2018.

- Read the notes on Lebesgue integration and on Orthonormal sets and expansions.
- Do the following problems.
- Section 2.1: 10
- Section 2.2: 1 (Use $w=1$.), 8(a,b,c) (FYI: the formula for
$T_n(x)$ has an $n!$ missing in the numerator.), 9
- This problem is aimed at showing that the Chebyshev polynomials
form a complete set in $L^2_w$, which has the weighted inner product
\[ \langle f,g\rangle_w := \int_{-1}^1
\frac{f(x)\overline{g(x)}dx}{\sqrt{1 - x^2}}. \]
- Show that the continuous functions are dense in $L^2_w$. Hint: if $f\in L^2_w$, then $ \frac{f(x)}{(1 - x^2)^{1/4}}$ is in $L^2[-1,1]$.
- Show that if $f\in L^\infty[-1,1]$, then $\|f\|_w \le \sqrt{\pi}\|f\|_\infty$.
- Follow the proof given in the notes on Orthonormal Sets and Expansions showing that the Legendre polynomials form a complete set in $L^2[-1,1]$ to show that the Chebyshev polynomials form a complete orthogonal set in $L^2_w$.

- Let F(s) = ∫
_{ 0}^{∞}e^{ − s t}f(t)dt be the Laplace transform of f ∈ L^{1}([0,∞)). Use the Lebesgue dominated convergence theorem to show that F is continuous from the right at s = 0. That is, show that

lim_{ s↓0}F(s) = F(0) = ∫_{ 0}^{∞}f(t)dt - Let f
_{n}(x) = n^{3/2}x e^{-n x}, where x ∈ [0,1] and n = 1, 2, 3, ....- Verify that the pointwise limit of f
_{n}(x) is f(x) = 0. - Show that ||f
_{n}||_{C[0,1]}→ ∞ as n → ∞, so that f_{n}does*not*converge uniformly to 0. - Find a constant C such that for all n and x fixed
f
_{n}(x) ≤ C x^{−1/2}, x ∈ (0,1]. - Use the Lebesgue dominated convergence theorem to show that

lim_{ n→∞}∫_{ 0}^{1}f_{n}(x)dx = 0.

- Verify that the pointwise limit of f
- Let $U:=\{u_j\}_{j=1}^\infty$ be an orthonormal set in a Hilbert
space $\mathcal H$. Show that the two statements are
equivalent. (You may use what we have proved for o.n. sets in
general; for example, Bessel's inequality, minimization properties,
etc.)
- $U$ is maximal in the sense that there is no non-zero vector in $\mathcal H$ that is orthogonal to $U$. (Equivalently, $U$ is not a proper subset of any other o.n. set in $\mathcal H$.)
- Every vector in $\mathcal H$ may be uniquely represented as the series $f=\sum_{j=1}^\infty \langle f, u_j\rangle u_j$.

- Section 2.1: 10

Updated 9/19/2018.