3
votes
1answer
45 views

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

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
2
votes
1answer
67 views

Identifying the dual group of a locally compact abelian group with the spectrum of $ {L^{1}}(G) $.

Folland stated the following theorem in his book A Course in Abstract Harmonic Analysis on Page 88. The dual group of a locally compact abelian group can be identified with the spectrum of $ ...
0
votes
1answer
33 views

Locally compact group and continuous function with compact support

Suppose that $G$ is a locally comact group, let $H$ be a open subgroup of $G$ and let $\phi:G\to\Bbb C$ be a continuous function such that $Supp\phi=:\overline{\{x\in G: \phi(x)\neq 0\}}$ is compact. ...
0
votes
1answer
112 views

On the mean value theorem in $\mathbb R^2$

Consider the following claim: If $A$ is a (complex) unital Banach algebra and $f: \mathbb R \to A$ is differentiable with $f' = 0$ then $f$ is constant. The proof uses that for $\tau \in A^\ast$: ...
2
votes
0answers
54 views

How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
0
votes
1answer
26 views

Continuity of additive maps

I studied all of previous posts about "when an additive map is continuous?" but I did not get my answer! My question is the following: Let $f:A\longrightarrow B$, be an bijective map from a Banach ...
2
votes
0answers
67 views

Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
0
votes
1answer
51 views

Matrix-valued function

I have a problem about matrix-valued function. Given a function $f:\mathbb{R}^k \rightarrow {\cal M}_{k \times k}$ of class $C^1$, where ${\cal M}_{k \times k}$ is the set of all $k \times k$ ...
2
votes
1answer
184 views

Application of Stone-Weierstrass with a non-unital algebra

Let $X$ be a locally compact Hausdorff space. We say that a function $f\colon X \to \mathbb{R}$ vanishes at $\infty$ if for each $\epsilon >0$ there exists a compact $K_\epsilon \subset X$ such ...
1
vote
0answers
80 views

Approximation of certain continuous functions by analytic functions

Let $f\in C(S^{1},M_{n}(\mathbb{C}))$ be a unitary. Does there exist an analytic unitary function $g$ from $S^{1}$ to $M_{n}(\mathbb{C})$ that approximates $f$?
0
votes
1answer
357 views

Is the space of bounded functions with the Supremum norm a Banach Algebra

X is an arbitrary , non empty set, B(X) the set of bounded functions $f:X\rightarrow \mathbb{R}$ and $||f||_\infty = \sup_{x\in X }|f(x)|$. Is $(B(X),||.||_\infty )$ a Banach Algebra? My attempt ...
8
votes
2answers
240 views

Does the Banach algebra $L^1(\mathbb{R})$ have zero divisors?

Assume that the functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are integrable and equal to zero on $(-\infty,0)$, (i.e $f,g \in L^+$). Then by Titchmarsh's theorem: $f*g$ is zero almost everywhere ...