User:MCPOliseno /Math710 FINAL

MICHELLE POLISENO

FINAL

1) Let $$ f \ $$ have bounded variation on [0, 1] and define $$ g(x) = f(a) + \int_{a}^{x} f'(t)dt \ $$. Support the following assertions: (a) $$ f'(x) = g'(x) \ $$ a. e.; (b) A function of bounded variation can be expressed uniquely (up to additive constant) as the sum of an absolutely continuous function and a singular function (A singular function is a function $$ s \ $$ for which $$ s'(x) \ $$ = 0 a.e.)

(a) Let $$ f \ $$ have bounded variation on [0, 1] and define $$ g(x) = f(a) + \int_{a}^{x} f'(t)dt \ $$. Then $$ g(x) \le f(a) + f(x) - f(a) \ $$ = $$ f(x) \ $$. Note that since $$ f \ $$ is of bounded variation, $$ f'(x) \ $$ exists for almost all $$ x \in \ $$ [0, 1]. Thus $$ g'(x) = f'(x) \ $$ almost everywhere.

(b) A function $$ f \ $$ has bounded variation if and only if $$ f = g + h \ $$ where $$ g, h \ $$ are bounded monotone increasing functions, and $$ f' \ $$ exists almost everywhere on [0, 1]. Set $$ h= f-g \ $$, then since $$ g(x) = f(a) + \int_{a}^{x} f'(t)dt \ $$, we have that $$ g \in AC[0, 1] \ $$. (i.e. $$ g \ $$ is absolutely continuous on [0 ,1]). And by (a), $$ g' = f' \ $$ almost everywhere. Thus $$ h' = 0 \ $$ almost everywhere. And therefore $$ h \ $$ is singular.

To show uniqueness, suppose we also had $$ f = g_1 + h_1 \ $$ with $$ g_1 \ $$ absolutely continuous and $$ h_1 \ $$ singular. Then $$ g - g_1 = 0 = h - h_1 \ $$ and so $$ g - g_1 \ $$ and $$ h - h_1 \ $$ are both absolutely continuous and singular and therefore are constant.

2) Let $$ f \in L^1 (\R) \ $$ be non-negative. For each measurable set E, define $$ \mu (E) = \int_E f \ $$. Show that $$ \mu \ $$ is countable additive on the sigma algebra of measurable sets.

Let $$ E \ $$ be a measurable set such that $$ \mu (E) = \int_E f \ $$.

3) Compute $$ \int_{0}^{1} f \ $$ for the Cantor function $$ f \ $$.

Let the Cantor function be $$ f: [0, 1] \to \R \ $$ be Lebesgue measurable. Note that $$ 1-f(x) = f(1-x) \ $$. Then 1 = $$ \int_{0}^{1} (1) = \int_{0}^{1} (f + (1-f)) \ $$ = $$ \int_{0}^{1} f - \int_{0}^{1} (1-f) \ $$ = $$ \int_{0}^{1}f + \int_{0}^{1}(f(1-x))dx \ $$ = $$ \int_{0}^{1}f + \int_{0}^{1} f = 2 \int_{0}^{1} f \ $$, thus $$ \int_{0}^{1} f \ $$ = 1/2.

4) Suppose $$ E \ $$ has a finite measure. Show that $$ L^2 (E) \subseteq L^1 (E) \ $$ and that the map $$ \psi: L^2(E) \to L^1(E) \ $$ defined by $$ \phi(f) = f \ $$ is continuous.

Suppose $$ E \ $$ has a finite measure. Let $$ \psi: L^2(E) \to L^1(E) \ $$ be defined by $$ \phi(f) = f \ $$.

5) A trigonometric polynomial is a function of the form

$$ p(x) = a_0 + \sum_{k=1}^{n} (a_k cos kx + b_k sin kx) \ $$.

Let $$ P \ $$ denote the space of trig polynomials.

Lemma 1 ''Let $$ f \ $$ be a continuous 2$$ \pi \ $$ periodic real-valued function on $$ \R \ $$. For each positive number $$ \epsilon \ $$ there exists a trigonometric polynomial $$ p \ $$ such that |$$ f(x) - p(x) \ $$| < $$ \epsilon \ $$ for all $$ x \ $$.

(a) Let $$ -\pi \le a < b \le \pi \ $$. Use the above lemma to prove there exists $$ p \in P \ $$ such that $$ \int_{-\pi}^{\pi} \ $$ $$|X_{[a,b]} (x) - p(x)|^2 < \epsilon \ $$.

(b) Let $$ f \in L^2 [-\pi, \pi] \ $$ and $$ [a, b] \ $$ be as in part (a). Use part (a) and the Cauchy Schwarz inequality to prove there exists a trigonometric polynomial $$ p \ $$ such that $$ | \int_{a}^{b} f - \int_{-\pi}^{\pi} fp | \ $$ < $$ \epsilon \ $$.

(c) Suppose $$ f \in L^2 [-\pi, \pi] \ $$ has the property that $$ \int_{-\pi}^{\pi} f(x) cos mx dx = \int_{-\pi}^{\pi} f(x) sin mx dx = 0 \ $$ for $$ m = 0, 1, 2, 3, \dots \ $$. Prove that $$ \int_{a}^{b} f(x) dx = 0 \ $$ on every interval $$ [a, b] \subseteq [-\pi, \pi ] \ $$.

(d) Let $$ f \ $$ be as in part (c). Prove that $$ f = 0 \ $$ almost everywhere.

(e) Show that the functions $$ {1/ \sqrt{2\pi}, 1/\sqrt{\pi} cos nx,1/\sqrt{\pi}  sin nx: n = 1, 2, 3, \dots} \ $$ is an orthonormal basis of $$ L^2 [-\pi, \pi] \ $$.

6) Let $$ f \ $$ be an integrable function on a measurable set $$ E \ $$. Define its distribution function $$ F \ $$ as follows: $$ F (x_ = m{t:f(t) \le x} \ $$. Show (a) $$ F \ $$ is a non-negative, non-decreasing, and continuous from the right. (b) $$ lim_{x \to -\infty} F(x) = 0 \ $$.

(7) Let $$ f: [0, 2] \to \R \ $$ be the characteristic function of the interval (1/2, 1]. Find the distribution function for $$ f\ $$.

Let $$ A \ $$ = (1/2, 1]. Then $$ f_A (x) = \ $$$$ \begin{cases} 1,          & x \in A             \\ 0,     & x \notin A   \end{cases} \ $$

(8) Let $$ f \ $$ be a bounded measurable function fro [0, 1] into [0, $$ M \ $$] and let $$ F \ $$ be its distribution function. Show that $$ \int_{0}^{1} f = \int_{0}^{M} x dF(x) \ $$, where the second integral is the Riemann-Stieltjes integral of $$ x \ $$. (An approximating sum for $$ \int_{0}^{M} g(x)dF(x) \ $$ is the Riemann-Stieltjes sum given by $$ \sum_{i=1}^{n} g(x_{i}^{*})(F(x_i)-F(x_{i-1})) \ $$, where $$ {x_0, x_1, \dots, x_n} \ $$ is a partition of [0, $$ M \ $$] and $$ x_{i}^{*} \ $$ denotes a sample point in $$[x_{i-1}, x_i] \ $$.)