1
vote
0answers
32 views

Use the uncertainty principle to prove that $\int_{\mathbb R^n} (f(x))^2|\hat{f}(x)|^2 \,dx=0$ if and only if $f\equiv 0$.

Let $f \in L^2(\mathbb R^n)$. Use the uncertainty principle to prove that $\int_{R^n} (f(x))^2|\hat{f}(x)|^2 \,dx=0$ if and only if $f\equiv 0$. $\leftarrow$: If $f \equiv 0$ then $\int_{\mathbb R^n} ...
1
vote
0answers
38 views

$f_x(y) = f(y-x)$, $L^p(\mathbb{R}^n)$ [on hold]

Let $x \in \mathbb{R}^n$ and $f \in L^p(\mathbb{R}^n)$, $f_x$ function on $\mathbb{R}^n$ defines $f_x(y) = f(y-x)$.Let fix $f$ and $1 \leq p < \infty$. Prove that is mapping $x \mapsto f_x$ ...
1
vote
0answers
19 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
1
vote
1answer
17 views

can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
3
votes
0answers
50 views

$\|\phi_{\lambda}- \phi_{\lambda} \ast f \|_{L^{2}(\mathbb R)}\to 0$ as $\lambda \to \infty$? ($\phi_{\lambda}(x)=\lambda^{-1} \phi(x/\lambda).$)

For $f\in L^{1}(\mathbb R),$ we define its Fourier transform as follows: $\hat{f}(t)=\int_{\mathbb R} f(x) e^{-ix\cdot t} dx ,(t\in \mathbb R).$ Suppose that $f\in L^{1}(\mathbb R)$ with ...
4
votes
2answers
45 views

Can we expect, $S(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1<p<\infty) $?

It is well-known that $L^{1}(\mathbb R)$ is a closed with respect to convolution(product), that is, $L^{1}(\mathbb R)\ast L^{1}(\mathbb R)\subset L^{1}(\mathbb R),$ more specifically, if $f, g\in ...
5
votes
1answer
72 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
0
votes
0answers
25 views

Show that the Fourier transform of f is in $L^p(R)$ for every $2 \leq p \leq \infty$.

Let $$f(x)=\sum_{n=1}^\infty \sqrt{n} \chi_{(\frac{1}{n+1},\frac{1}{n})}(x)$$. The Fourier transformation of f is $$\hat{f}(y)=\sum_{n=1}^\infty ...
1
vote
1answer
32 views

The derivative of a function is square integrable assuming Fourier transform dominated

I am struggling in solving the second part of this problem. Let $g$ be a continuous function in $L^1(\mathbb{R})$ whose Fourier transform is the function $F$. Suppose $|F(x)|\leq (1+x^2)^{-2}$. Prove ...
3
votes
1answer
44 views

How to show the completeness of the space of Fourier transforms $\mathcal{F}L^{1}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
1
vote
2answers
41 views

$f_{n}\to f$ in $L^{1}\implies \hat{f_{n}}\to \hat{f}$ in $L^{1}$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R .$ Suppose there exists $f_{n}, \in L^{1}(\mathbb R)$ such ...
2
votes
1answer
144 views

$g, f, \hat {f} \in L^{1}(\mathbb R)\cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R) \implies \widehat{(fg)}= \hat{f} \ast \hat{g} ? $

Let $f, g\in L^{1}(\mathbb R)$ and it Fourier transform of $f$, $\hat{f} (y) = \int _ {\mathbb R} f(x) e^{-2\pi i x \cdot y} dx, \ (y\in \mathbb R)$ and the convolution of $f $ and $g$; $f\ast g ...
3
votes
0answers
70 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
6
votes
1answer
139 views

Extend a function by convolution

Let $f \in \mathcal{C}^{\infty}(\mathbb{R})$ be a compactly supported function ($supp(f)\Subset\mathbb{R})$. I am wondering about the existence of a $g \in L^p(\mathbb{R})$, for some $p$, such that ...
2
votes
2answers
33 views

Let $F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha$. Is it true $F, F^{-1}\in L^{1}(\mathbb R)$?

Define $$F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha, \ (x\in \mathbb R).$$ It is clear to me that, the integral converges for every real $x$ (as near origin integrand is ...
2
votes
1answer
61 views

Subspace of Tempered Distributions

Let ${S_{h}}'(\mathbb{R}^{n})$ be the space of tempered distributions such that if $u\in {S_{h}}'(\mathbb{R}^{n})$, then $\lim_{\lambda\rightarrow \infty}{||\phi(\lambda D)u||_{\infty}} = 0$ for all ...
0
votes
1answer
44 views

Upper bound for the norm of inverse Fourier tansform

Recall Hausdorff-Young inequality: For any $f\in L^p(\mathbb{R}^n)$, we have $||\hat{f}||_q\le ||f||_p$, where $p$ and $q$ are conjugate exponents and $p\in[1,2]$. It seems to me that it follows ...
2
votes
1answer
116 views

Fourier transform of convolution for $L^2$ functions

If $f,g\in L^1(\mathbb{R})$, it is not hard to show by definition that $$(\hat{f\ast g)}(t)=\hat{f}(t)\hat{g}(t).$$ But what about if $f,g\in L^2(\mathbb{R})$? The Fourier transform on ...
0
votes
1answer
48 views

Boundedness of continuous summable function

Let $f\colon\mathbb{R}\to\mathbb{C}$ be a continuous function. If we suppose that $f$ is a $L^1(\mathbb{R;C})$ function too, then can we conclude that $f$ is bounded? ADD: I asked the preceding ...
1
vote
0answers
76 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
450 views

What will be the support of the convolution of two test functions.

If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$. Regularization of $g$ is ...
0
votes
1answer
91 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 ...
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 $ ...
1
vote
1answer
46 views

there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each ...
0
votes
1answer
85 views

Dual space of the function $f$ in Fourier Transform

Let $f\in L^1{(\mathbb{R})}$. Why the Fourier Transform $\hat{f}\in L^{\infty}{(\mathbb{R})}$. Is it because $(L^1{(\mathbb{R})})^*=L^{\infty}{(\mathbb{R})}$?