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can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
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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 ... 1answer 47 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)$, ... 3answers 82 views For which$s\in\mathbb R$, is$H^s(\mathbb T)$a Banach algebra? According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If$mp>n$,$m\in\mathbb N$, then$W^{m,p}(\Omega)$is a Banach algebra, provided that$\Omega\subset\mathbb R^n$satisfies the ... 0answers 14 views How to prove the set of fourier multipliers is a banach algebra? Hi I am new here at math stack Exchange, this is my first question, hope you guys can help me out:) Suppose$F\colon L^2(\mathbb{R} ) \to L^2(\mathbb{R})$is the Fourier transform given by ... 0answers 22 views Can we expect,$\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$in the Banach algebra$A(\mathbb R)$? Let$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 ;$and consider Fourier algebra$$A(\mathbb R):= \{f\in ... 0answers 14 views How to use$(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in Banach sapace$C([0, T];M^{p,1})$? (For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ... 1answer 66 views Continuity of double centralizers in Banach algebras I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let$A$be a ... 0answers 88 views Open map in Banach algebra I'm having trouble showing a certian function is open and can be extended. Let$\Omega$be a completely regular topological space and$A=C_b(\Omega)$the space of all complex-values bounded ... 2answers 134 views Banach-algebra homeomorphism. Let$ A $be a commutative unital Banach algebra that is generated by a set$ Y \subseteq A $. I want to show that$ \Phi(A) $is homeomorphic to a closed subset of the Cartesian product$ ...
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 ...
Prove that if $A$ is algebra generated by $\sin(x)$ and $\cos(x)$ then $A = \{ f\in C_b(\mathbb R ) : f(t) = f(t + 2\pi )$ for all $t \in \mathbb R\}$ [duplicate]
Possible Duplicate: Finding a closed subalgebra generated by functions. Let $A$ be the uniformly closed subalgebra of $C_b(\mathbb{R} )$ generated by $\sin(x)$ and $\cos(x)$. Prove: \$A = ...