-1
votes
0answers
31 views

Possion integral for measure dominated by the maximal function of the measure [closed]

Recently, I was thinking a problem about Possion integral for measure dominated by the maximal function of the measure, that is to say, let $\mu$ be a regular Borel measure in $\mathbb{R}$, define its ...
3
votes
0answers
32 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
2
votes
2answers
47 views

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family ...
1
vote
0answers
14 views

Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

It is well-known that, $L^{1}(\mathbb R)$ can be embed into $M(\mathbb R)$ (= The space of complex Borel measure on $\mathbb R$); by identifying $f\in L^{1}(\mathbb R)$ with the measure $d\mu= f dm.$ ...
0
votes
0answers
78 views

Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
2
votes
1answer
29 views

Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?

Let $C$ is a compact subset of $\mathbb R,$ $V\subset \mathbb R,$ and $0<m(V)<\infty,$ where $m$ is a Lebsgue measure on $\mathbb R.$ My Question is: Can we expect to find $k\in ...
0
votes
0answers
19 views

Fourier Transform for option pricing

Can Fourier transforms be used to derive the joint probability density function of stochastic interest rates and sotck price Brownian motions of call options under stochastic interest rates? So lets ...
0
votes
0answers
18 views

Existence of Density in Bochner's Thoerem

Bochner theorem for locally compact abelian group, $G$ and a positive definite function $f$ there exist a unique measure $\mu_f$ such that: $$f(x)=\int\limits_{\hat G}(x,\gamma)d\mu_f(\gamma)$$ Where ...
3
votes
1answer
36 views

counterexample of Riemann-Lebesgue lemma for non-Borel functions

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a Borel measurable function. Then $$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$ I obtain this result by showing that it is ...
2
votes
1answer
42 views

If the Fourier transform of a probability measure goes to zero at infinity, can the measure have a point mass?

Let $\mu$ be a probability measure on $\mathbb{R}$. Is the following implication true? $$ \widehat{\mu}(y) \rightarrow 0 \text{ as } |y| \rightarrow \infty \quad \Rightarrow \quad \mu(\{x\})=0 \quad ...
0
votes
1answer
45 views

Convolution of L1 & L2 function: definition

A book that I'm reading makes the following statement that I'm not sure how to understand: On $\mathbb R^n$, if $f\in L1$ and $g\in L2$, we have: $$\widehat{f*g}=\hat f \hat g$$ How do I read it? I ...
0
votes
1answer
47 views

Computing Fourier transform of a surface measure.

I am almost beginner in the topics of Fourier transform. So, I am asking this question here. Let $n=3$ and let $\mu_t$ denote surface measure on the sphere $|x|=t$. Then how do we show that $$ ...
0
votes
1answer
40 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
46 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
57 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
45 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 ...
6
votes
1answer
80 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
44 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
49 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
33 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
65 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
178 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
129 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
56 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
42 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
37 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
77 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
72 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
93 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
148 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
135 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
61 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
109 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
337 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
196 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
66 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
38 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
189 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
147 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
72 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
121 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
106 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
118 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
83 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
66 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
352 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
287 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
106 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
1answer
262 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 ( $\sigma\ll\mu$)?