0
votes
1answer
23 views

How to interchange sum and integral when measure is in terms of Dirac measure?

Let $\{c_{k}\}_{k\in \mathbb Z} \subset \mathbb C$ such that, $\sum_{k\in \mathbb Z} |c_{k}| < \infty.$ Let $\delta_{k}$ is the unit Dirac mass at $k $, we note that $\mu = \sum_{k\in \mathbb Z} ...
1
vote
1answer
42 views

The behavior of Fourier transform near the origin

I'm attacking a homework problem, which I have reduced to the following: Let Schwartz function $f \in \mathcal{S}^1(\mathbb{R})$ be nonnegative and $\|f\|_{L^1} = 1$. Assume further that ...
6
votes
1answer
47 views

How to determine measure from the integral equation?

Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ ...
1
vote
0answers
30 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
5
votes
1answer
58 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
1
vote
0answers
37 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
3
votes
0answers
45 views

Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has ...
0
votes
1answer
30 views

Continuity/Differentiability of Fourier Series

Possibly stupid question: I'm wondering if there is some trick for evaluating the continuity/differentiability of a Fourier series. In particular, I'm looking at the function $f(x)=\sum_{n=0}^\infty ...
2
votes
1answer
53 views

Are these operators and the fourier transform compact?

I do not want a proof but rather an explanation. I just read that $T_k:L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ such that $(T_kf)(s) = \int_{\mathbb{R}} k(s,t)f(t) dt $ is compact. (in this ...
4
votes
1answer
164 views

Weak convergence and Fourier transform

For positive integer $k$, let $\mu_k=\dfrac{1}{2}\left(\delta(x)+\delta\left(x-\dfrac{2}{3^k}\right)\right)$. Let $dC_k=\mu_1\ast\cdots\ast\mu_k$. We have that $dC_k$ converges weakly to $\mu_C$, ...
3
votes
2answers
116 views

Show that there exists a constant $c$ such that $\left|\int_0^b\frac{\sin ax}{x}dx\right|\le c$

Show that there exists a constant $c$ such that $$\left|\int_0^b\frac{\sin ax}{x}dx\right|\le c$$ In fact, show that the smallest such number is $c=\int_0^\pi\frac{\sin x}xdx$. Well, I'm thinking ...
3
votes
0answers
52 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) ...
2
votes
1answer
41 views

Finding $E(X^r\mid Y)$ of an exponential function

Let $(X,Y)$ denote a two-dimensional random vector with an absolutely continuous distribution with density function $$p(x,y) = \frac{1}{y}\exp(-y), \qquad 0 < x < y < \infty.$$ Find ...
0
votes
0answers
36 views

What is the error when approximating $L^2([0,1])$ by a finite dimensional space?

Let $X \subset L^2([0,1])$ such that $f([0,1]) \subset [-M,M]$, for some constant $M$, and any $f \in X$. By choosing a finite dimensional basis $V=\left(v_i\right)_{i=0}^n$, where each $v_i \in X$, ...
1
vote
0answers
70 views

Definition of $L^p(\mathbb T)$ with $\mathbb T$ the unit circle

I'm trying to define the $L^p$ spaces in the unit circle(denoted as $\mathbb T$), as Rudin's Real and Complex analysis does in page 88. I've defined a measure in $\mathbb T$ via Riesz's representation ...
2
votes
1answer
65 views

Tails of Fourier Transformed family of functions

I am reading a thesis where on page 39, Definition 4, $\epsilon$-oscillatory is defined as a property for a family of functions $\{f_{\epsilon}\}_{0<\epsilon<1}$ in $L^2(\mathbb{R}^d)$ to have ...
0
votes
1answer
85 views

A basic question about $\operatorname{supp}f$ (support of f).

Is it true that $\operatorname{supp}f$ is the complement of the biggest open set where $f=0 $? Here $\operatorname{supp}f=$ {$x\in \Bbb R^n ; f(x)\not=0$} and $f\in C$ (collection of continuous maps ...
0
votes
0answers
144 views

Problem 25 pg 95, Stein and Shakarchi: $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$.

Show that for any $\epsilon>0$, the function $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$. [Hint: $K_{\delta}(X) = e^{-\pi|x|^{2/\delta}} ...
1
vote
2answers
130 views

Problem #23 pg-94, Stein and Shakarchi

As an application of the Fourier transform, show that there does not exist a function $I\in L^1(R^d,m)$ such that $f*I = f$ for all $f\in L^1(R^d,m)$.
1
vote
1answer
58 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
2
votes
1answer
93 views

How to recover a measure from its Fourier transform?

Let $f$ be the complex function defined on $\mathbb{R}$ by $$ f(t)=\frac{1-it}{1+it}. $$ 1) Does there exist a complex bounded measure $\mu \in M(\mathbb{R})$ such that $\hat{\mu}=f$ (where $\hat{}$ ...
3
votes
1answer
316 views

Real Analysis Qualifying Exam Problem

I think this should be an easy question, and I believe the answer should be in the positive, but I am not sure how to start. I would appreciate some help. Thank you. Suppose that $f_j$ is a ...
1
vote
0answers
170 views

Fourier Coefficients of Complex Measure

For my homework I am trying to prove the following: Suppose $\mu$ is a complex Borel measure on $[0,2\pi)$, and define the Fourier coefficients of $\mu$ by $\hat{\mu}(n)=\displaystyle\int ...
1
vote
0answers
64 views

Probability measures on $\mathbb{T}$ whose Fourier coefficients tend to 1

Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent? $\limsup_{n\to\infty}|\hat{\mu}(n)|=1$. There exists an increasing sequence ...
1
vote
1answer
35 views

An equivalent condition for a measure on $\mathbb{T}$ to be symmetric

Is it correct that a Borel probability measure $\sigma$ on the complex unit circle $\mathbb{T}$ is symmetric (i.e. $\sigma(A)=\sigma(\overline{A})$ for every Borel set) iff ...
4
votes
0answers
166 views

Uniqueness of Haar Measures

Haar measure on a LCA (locally compact abelian) group $G$ is said to be unique ... up to a scaling factor. This is not as elegant as it might be expected because this requires a choice of a unit ...
3
votes
1answer
125 views

Why is the Fourier Transform of a Lévy Process a continuous function? What about the inverse? (Bochners Theorem)

I was confronted with this question when reading "Stochastic Integration and Differential Equations" by Protter. Just after the definition of a Lévy process he says the following: If $X_t$ is a ...
1
vote
1answer
67 views

How similar are the measures if their Fourier transform coincide?

Let $\mu$ and $\nu$ be finite Borel nonnegative measures exponentially decreasing at infinity, i.e. there exists $A > 0$ such that $$ \int\limits_{0}^{\infty} e^{Ax} \mu(dx) < \infty, \;\;\; ...
3
votes
1answer
112 views

Find the Hardy-Littlewood maximal function of $\chi_{[-1,1]}$ on $\Bbb R$

Find the Hardy-Littlewood maximal function $Mf$ of the $\Bbb R^\Bbb R$ function $f=\chi_{[-1,1]}$. How do we find $Mf(x)$ for $|x| > 1$? I see that it should decrease like $1/x$, but I can't find ...
1
vote
1answer
97 views

Show the convolution of a $C_c^\infty (\Bbb R^n)$ function with a $L^p(\Bbb R^n)$ function is in $C^\infty(\Bbb R^n)$, $1\le p\le\infty$

Let $f \in L^p\left(\Bbb R^n\right)$ and $g \in C_c^\infty \left(\Bbb R^n\right)$. Show $f \ast g \in C^\infty\left(\Bbb R^n\right)$ for $1 \le p \le \infty$. Let $x=(x_1,x_2,\ldots,x_n)$ and ...
0
votes
2answers
110 views

Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$

Let $C_c^\infty$ denotes the set of real valued function with compact support. Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$. If ...
0
votes
1answer
80 views

Expressing the cantor function on $[0,1]$ as a function on $\text{Ternary}([0,1])$

I would to link the simple function and probabilistic approach for the calculation of the Fourier transform of the Cantor function. Let $f:[0,1] \to [0,1]$ be the Cantor function. In the simple ...
2
votes
1answer
65 views

Probabilistic calulation of the Fourier transform of the Cantor function

This is on the same theme as in this post, where the Fourier transform was derived using simple function. Let $f:[0,1] \to [0,1]$ be the Cantor function. Then $f$ is the cumulative distribution of ...
2
votes
1answer
313 views

Fourier transform of the Cantor function

Let $f:[0,1] \to [0,1]$ be the Cantor function. Extend $f$ to all of $\mathbb R$ by setting $f(x)=0$ on $\mathbb R \setminus [0,1]$. Calculate the Fourier transform of $f$ $$ \hat f(x)= \int f(t) ...
1
vote
1answer
44 views

The Fourier transform of $x \mapsto |x|^{-1/3}e^{-x^{2}}$ is not in $L^{1}$.

Okay previously my lecturer showed that this is so by proving in the following way: Proof by contradiction. Suppose the transform is in $L^{1}$. Then as $f \in L^1$, we may use Fourier Inversion ...
5
votes
3answers
238 views

if convolution of $f$ with itself remains same, then $f=0$ a.e?

I'm trying to answer the question above.. But I'm not certain in either way. I tried to prove it by giving counter examples.. But it always failed.. Then i also tried to draw contradictions But ...
2
votes
1answer
103 views

Show translation is not continuous in $\text{Lip}_\alpha(T)$

Let $f=\sqrt{|x|} \in \text{Lip}_\alpha(T)$, where $\text{Lip}_\alpha(T)$ is the set of Lipschitz function with Lipschitz constant $\alpha=1/2$ on the unit circle $T$. What is $$ \|f\|=\sup_{t\in T,h ...
4
votes
0answers
246 views

A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
3
votes
2answers
124 views

Does convergence of Fourier transforms imply convergence of measures?

Let $\{\sigma_n\}$ be a sequence of measures on the complex unit circle $\mathbb{T}$ and let $\sigma$ also be such a measure. Suppose that $\hat{\sigma_n}(k) \rightarrow \hat{\sigma}(k)$ as ...
5
votes
1answer
85 views

Inequality for Fourier transform of measure

I am having trouble with the following question. Let $\mu$ be finite measure on $\mathbb{R}$ and let $\hat{\mu}(\xi) = \int_{-\infty}^\infty e^{-ix \xi} d\mu(x)$ be its Fourier transform. Prove that ...
2
votes
3answers
152 views

Continuous probability measures on the unit circle

Is there a continuous probability measure on the unit circle in the complex plane - $\sigma$ with full support, such that $\hat{\sigma}(n_k)\rightarrow1$ as $k\rightarrow\infty$ for some increasing ...
1
vote
1answer
76 views

Absolute continuity with respect to a Rajchman measure

A measure $\sigma$ on $\mathbb{T}$ (the unit circle in the complex plain) is called a Rajchman measure if $ \hat{\sigma}(n)\rightarrow0$ as $|n| \rightarrow \infty$. I want to prove that if ...
1
vote
1answer
139 views

Hilbert spaces other than $L^2$

From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ ...
6
votes
1answer
343 views

What exactly is a Haar measure

I've come across at least 3 definitions, for example: Taken from here where $\Gamma$ is a topological group. Apparently, this definition doesn't require the Haar measure to be finite on compact ...
3
votes
2answers
128 views

Continuous Random Variable with constant moments?

I would like to know if there exists a measure $\rho$ on the positive real line such that its moments $\int_0^{\infty} x^j d\rho(x)$ are equal to a constant (for example equal to one) for all ...
4
votes
2answers
280 views

Measurable homomorphism of $\mathbb{T}$ into $\mathbb{C}^\times$

Working through Katznelson's An Introduction to Harmonic Analysis and have been stumped by the following problem for the past few days: Show that a measurable homomorphism of ...
3
votes
1answer
107 views

Radon measures and sequences

I have a question about Radon meaures: Given a Radon measures $ \mu_{1}, \mu_{2}$, both have compact support: How to show that $\int \hat{\mu_{1}}(x)\,d\mu_{2}(x)=\int \hat{\mu_{2}}(x)\,d\mu_{1}(x)$ ...
3
votes
1answer
174 views

a question on distributions.

Suppose $L\in \mathcal{S}$ is a tempered distribution, and for each $\phi \in \mathcal{S} :\phi \geq 0 \implies L(\phi) \geq 0$. Prove that there exists a borel measure $\mu$ with polynomial growth ...
5
votes
0answers
233 views

Fourier dimension of sets of positive Lebesgue measure

Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
1
vote
0answers
201 views

A function in BMO space

Let $\psi:[0;1]\to\mathbb R$ is a nonnegative measurable function. Let $b_d(x)=1_{B(0,1)}\cdot{\rm sgn}(\sin (\pi d|x|))$, where $d\in\mathbb N$. Here $1_{B(0,1)}$ is the charateristic function of the ...