Space of Integrable Functions

Let $X$ be a positive measure space of total measure $1$. Show that for any $[0,\infty )$-valued measurable function $f$ on $X$,

is a non-decreasing function of $p \in (1,\infty )$. Under what circumstances is $I(p)$ strictly increasing as a function of $p$?

Let $(X, \mathcal{M}, \mu )$ be a fixed measure space, let $\{ p_i\}_{i=1}^{n}$ be positive numbers such that $p_i > 1$, $i = 1, \ldots , n$, $p > 1$ and

If, for $i \in \{ 1, \ldots , n\}$, $f_i \in L^{p_i}(\mu )$, must it be the case that

Justify.

Let $X = \{ a,b,c\}$, let $\mathcal{M}$ be the $\sigma$-algebra of all subsets of $X$, and let $\mu$ be the measure determined by $\mu (\{ a\} ) = 0$, $\mu (\{ b\} ) = 1$ and $\mu (\{ c\} ) = \infty$. What are the dimensions of the vector spaces $L^1(\mu )$, $L^2(\mu )$, and $L^{\infty }(\mu )$? Justify your answers.

Let $f, g \in L^2([0,1])$,

Show that

Let $f$ be a non-negative function in $L^p(\mathbb {R})$ for some $1 \leq p < \infty$ and let $r+s = p$, $r > 0$, $s > 0$. Also let $f_h(x) = f(x+h)$.

Let $f : \mathbb {R} \to \mathbb {R}_+$ be measurable, and let $0 < r < \infty$. Show that for any interval $I$,

Let $f$ be a bounded measurable function on $(0,1)$. Prove that

where

Let $f$ be a non-negative measurable function on $[0,1]$ satisfying

Determine those values of $p$, $1 \leq p < \infty$, for which $f \in L^p$ and find the minimum value of $p$ for which $f$ may fail to be in $L^p$.

With Lebesgue measure on $[0,1]$, prove that

is not compact in $L^1[0,1]$.

Let $\mathcal{H}$ be Hilbert space $L^2(0,2\pi )$, with inner product defined by

Consider the elements $u_n \in \mathcal{H}$, $n = 1, 2, \ldots$, defined by $u_n(x) = \sin (nx)$ for $x \in (0,2\pi )$. Show that

Let $H^1$ be the Sobolev space on the unit interval $[0,1]$, i.e., the Hilbert space consisting of functions $f \in L^2[0,1]$ such that

where

are Fourier coefficients of $f$. Show that there exists constant $C > 0$ such that $\left\| f\right\|_{\infty } \leq C\left\| f\right\|_{H^1}$.

Let $f$ be a periodic function on $\mathbb {R}$ with period $2\pi$ such that $f|_{[0,2\pi ]}$ belongs to $L^2(0,2\pi )$. Suppose

For each $h \in \mathbb {R}$ define the function $f_h$ by $f_h(x) = f(x-h)$.

Let $f_n(x)$ be an orthonormal family of functions in the Hilbert space $L^2(0,1)$. Prove that

for all $x \in [0,1]$, and that this inequality is sharp (equality) if and only if $\{ f_n \mid n = 1, \ldots \}$ span a dense subspace of $L^2(0,1)$.

If $f$ is a function on $\mathbb {R}$, let $f_t$ be the translate $f_t(x) = f(t+x)$. Prove that if $f$ is square integrable with respect to Lebesgue measure, then

A trigonometric polynomial is a function $p$ on $\mathbb {R}$ of the form

for some $c_n \in \mathbb {C}$, $N \geq 0$. Suppose $f$ is continuous and $2\pi$-periodic on $\mathbb {R}$, and let

Show that for every $\varepsilon > 0$, there exists a trigonometric polynomial

such that

and

Prove or disprove the following statements:

Let $\{ f_n\}_{n=1}^{\infty }$ be a sequence in $L^p([0,1])$ (with Lebesgue measure) and $\{ g_n\}_{n=1}^{\infty }$ a sequence in $L^q([0,1])$, where $\frac{1}{p}+\frac{1}{q}=1$. If $\lim_{n \to \infty } f_n = f$ in $L^p$ and $\lim_{n \to \infty } g_n = g$ in $L^q$, is it true that $f_n g_n \to fg$ in measure? Justify.

Let $X$ be a measure space with measure $\mu$ and suppose that $\mu (X) < \infty$. Let

(Here, as usual, two measurable complex functions are equivalent if they agree a.e.) For $f \in S$, define

Show that $d(f,g) = \rho (f-g)$ is a metric on $S$, and that $f_n \to f$ in this metric if and only if $f_n \to f$ in measure.

Let $g$ be a measurable function such that $\int_{\mathbb {R}} \left|fg\right| < \infty$ for every $f \in L^p(\mathbb {R})$ (fixed $p \geq 1$). Prove that there is a constant $M$ such that

for all $f \in L^p(\mathbb {R})$.

Let $S^1$ denote the unit circle (the set of complex numbers with modulus one, or the real numbers modulo $2\pi$). The convolution of two functions on $S^1$ is

Suppose that $f$ is an element of $L^2(S^1)$ with the property that its Fourier coefficient

is non-zero for all $n \in \mathbb {Z}$. Show that the linear space $\{ f * k \mid k \in L^2(S^1)\}$ is dense in $L^2(S^1)$.

For functions $f, g \in L^2(\mathbb {R})$, define the convolution $f*g$ by

where the integral is with respect to Lebesgue measure.

If $f \in L^1(\mathbb {R})$, define

Prove that, for any $f \in L^1(\mathbb {R})$, $\hat{f}(\xi ) \to 0$ as $\left|\xi \right| \to \infty$.

Let $f : [0,1] \to [0,\infty )$ be an essentially bounded function, $\left\| f\right\|_{\infty } > 0$. Show that

Let $(X, \mathcal{M}, \mu )$ be a measure space for which $\mu (X) < \infty$. Let $1 < p < \infty$. Suppose that $\{ f_k\}$ is a sequence in $L^p(X)$ such that $\sup_k \left\| f_k\right\|_{p} < \infty$ and $\lim_{n \to \infty } f_k(x)$ exists for $\mu$-a.e.$x$. Prove that

Give an example of a derivative on $[0,1]$ which is not Lebesgue integrable.

Let $(X, \mathcal{F}, \mu )$ be a finite measure space. Let $f_n$ be a sequence of measurable functions with $f_1 \in L^1(\mu )$ and with the property that

for all $n$ and all $\lambda > 0$. Prove that

Let $g : \mathbb {R}^n \to \mathbb {R}$ be non-negative with $\int_{\mathbb {R}^n} g(y) \, dy = 1$, let $g_{\varepsilon }(x) = \varepsilon^{-n} g(x/\varepsilon )$ for $\varepsilon > 0$, and define

If $f : \mathbb {R}^n \to \mathbb {R}$ is continuous and bounded, then show that $f_{\varepsilon } \to f$ as $\varepsilon \to 0$ uniformly on compact subsets of $\mathbb {R}^n$.

Fix $p \geq 1$. Assume without proof that the set of continuous functions with compact support is dense in $L^p(\mathbb {R}^n)$. Let $f \in L^p(\mathbb {R}^n)$ and, for $t \in \mathbb {R}^n$, let $f_t(x) = f(x+t)$ be the translate of $f$ by $t$. Show

This problem concerns Lebesgue integration of functions from $\mathbb {R}$ to $\mathbb {R}$. For such functions, a trivial form of Holder's inequality says that if $f \in L^{\infty }(\mathbb {R})$ and $g \in L^1(\mathbb {R})$ then