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)

(b) Since $$ f \in \ $$ BV[0, 1], then fix a point c $$ \in \ $$ [0, 1] such that $$ f = f_{0c} + f_{s} \ $$ where $$ f_{0c} \in \ $$ AC[0, 1] and $$ f_s \ $$ is singular. Note that $$ f_{0c}(x) = \int_{0}^{x} f'(x) dx \ $$ and the remainder $$ f_s = f - f_{0c} \ $$ is the singular part. Then let $$ D \subset \ $$ [0, 1] denote the set of points of discontinuity of $$ f \ $$. Since $$ f \ $$ is the difference of monotone functions, then it can only contain jump discontinuities at which its left and right limits exist, excluding the left limit at 0 and the right limit at 1. Then $$ D \ $$ is necessarily countable.

So, if $$ c \in D \ $$, let $$ [f](c) = f(c^+) - f(c^-) \ $$ denote the jump of $$ f \ $$ at $$ c \ $$, where $$ f(0^-) = f(0), f(1^+) = f(1) \ $$ if 0, 1 $$ \in D \ $$.

Define $$ f_p(x) = \sum_{c \in D \cup[0,x] [f](c) \ $$ if $$ x \notin D \ $$. Then $$ f_p \ $$ has the same jump discontinuities as $$ f \ $$ and, with an appropriate choice of $$ f_p(c) \ $$ for $$ c \in D \ $$ the function $$ f - f_p \ $$ is continuous on [0, 1]. Decomposing the continuous part into an absolutely continuous and singular continuous part,

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.

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

let $$ f \ $$= Cantor function on [0, 1]. Then $$ f'(x) \ $$ = 0 $$ \forall x \in \ $$ [0, 1]\$$C \ $$. Thus $$ f'(x) \ $$ = 0 almost everywhere. Then $$ \int_{0}^{1} f' = \int_{0}^{1} 0 = 0 \ne f(1) - f(0) = 1- 0 = 1 \ $$

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.

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 $$ f\ $$.

(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 $$[x_{i-1}, x_i] \ $$.)