All Questions
Tagged with convolution lp-spaces
115
questions
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19
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Weighted inequality on torus
In the Torus (circle). Let $[0,2\pi]\to\mathbb ]0,\infty[\colon \theta\mapsto w(\theta)$ a weight function, i.e. nonnegative and integrable on $[0,2\pi]$. If $\mathbb{Z}\to\mathbb{R}\colon k\mapsto m(...
- 3,650
3
votes
1
answer
75
views
What is wrong with this proof that a linear, bounded, time invariant operator on $L_p$ must be a convolution?
I'm trying to understand if this is true and how to prove it, "If $T$ is a bounded, time invariant operator on $L_p(\mathbb{R})$, then $T$ is a convolution operator.''
Here's an attempt at a ...
- 286
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17
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Prove that the derivative of the mollification approaches the strong $L^p$ derivative
Let $g \in C_c(\mathbb{R})$ be such that $\int_\mathbb{R} g dx = 1$. Suppose that $f \in L^p(\mathbb{R}$ has a strong $L^p$ derivative, i.e. there is some $h$ such that $\displaystyle\lim_{y \to 0} \...
- 3,800
2
votes
2
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88
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Does convolution preserve local integrability in the sense that if $f \in L^p_{\text{loc}}$ and $g \in L^1$, then $f \ast g \in L^p_{\text{loc}}$?
Consider the Euclidian space $\mathbb R^n$, where $n \in \mathbb N$ is a fixed integer, equipped with the Lebesgue measure. Let $L^p_{\text{loc}}(\mathbb R^n)$($1 \leqslant p < \infty$) denote the ...
- 1,141
3
votes
1
answer
98
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Summability of the Fourier Transform.
I was reading the notes "Introduction to Complex Analysis" by Michael E. Taylor, and I am stuck on the following exercise about the Fourier Transform and the space $\mathcal{A}(\mathbb{R})$ ...
1
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119
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Define multiplication on $L^1(\mathbb{R})$
I want to define multiplication on $L^1(\mathbb{R})$, of course, convolution is an allowed multiplication which makes $(L^1(\mathbb{R}),*)$ is a Banach algebra, I want to know if there are other ...
- 11
2
votes
1
answer
75
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Differentiability of convolution without compact support
Let $g \in C^1(\mathbb R), f, g, g' \in L^1(\mathbb R)$.
How do I then show $f * g$ is differentiable? Variations of this question have been asked where $g$ or $g'$ have compact support, but these ...
- 878
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64
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Uniform and $L^p$ convergence of convolution with an approximate identity
A family of functions $(K_\delta)_{\delta>0}$ is called an approximate identity on $\mathbb{R}^d$ if
(i) $\int_{\mathbb{R}^d}K_\delta(x)\ dx=1$.
(ii)$\int_{\mathbb{R}^d}|K_\delta(x)|\ dx\leq A$ for ...
- 53
2
votes
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54
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The adjoint of $T_a: L^2 (\mathbb R^d) \to L^2 (\mathbb R^d), u \mapsto a * u$
Let $a \in L^1 (\mathbb R^d)$ be a fixed function. Consider a linear operator
$$
T_a: L^2 (\mathbb R^d) \to L^2 (\mathbb R^d), u \mapsto a * u.
$$
By Young's inequality, $\|a*u\|_2 \le \|a\|_1 \|u\|_2$...
- 5,817
2
votes
1
answer
69
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The map $f*g$ is uniformly continuous
Let $p, q \in [1, \infty]$ such that $\frac{1}{p}+\frac{1}{q} = 1$. We define the convolution operator
$$
* : L^{p} (\mathbb R^d) \times L^{q} (\mathbb R^d) \to L^\infty (\mathbb R^d)
$$
by
$$
(f*g) (...
- 17.6k
0
votes
1
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117
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Brezis's Theorem 4.26: how to obtain $\|\rho_n \star f\|_{L^\infty (\mathbb{R}^N)} \le C_n\|f\|_{L^p(\mathbb{R}^N)}$?
A sequence of mollifiers $\left(\rho_n\right)_{n \geq 1}$ is any sequence of functions on $\mathbb{R}^N$ such that
$$
\rho_n \in C_c^{\infty}\left(\mathbb{R}^N\right), \quad \operatorname{supp} \rho_n ...
- 17.6k
1
vote
1
answer
75
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Proposition 4.19 in Brezis's Functional Analysis
I'm trying to prove Proposition 4.19. in Brezis's Functional Analysis, i.e.,
Theorem Let $f \in \mathcal C_c (\mathbb R^n)$ and $g \in L_{\text{loc}}^1 (\mathbb R^n)$. Then $(f*g) (x)$ is well-...
- 17.6k
1
vote
1
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141
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Connecting piecewise smooth functions using mollification.
Let's say I am given a function $f: \mathbb{R}\setminus (0,1) \to \mathbb{R}$ defined piecewise as
$$f(x)= \begin{cases}
g(x) \quad x \geq 1 \\
h(x) \quad x \leq 0
\end{cases},$$
where $g$ and $h$ ...
- 1,063
0
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46
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Finding a suitable mollifier $\phi$ so that $\lim_{k\to\infty}||f - h\ast\phi||_\infty=0$ for a continuous function $f$ vanishing at the infinity
I am asking for pointers on how to move forward with the following proof, be that extended details regarding the current work or an alternative approach regarding the chosen mollifier and appropriate ...
- 1,221
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185
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Proof of converse of Young's convolution inequality
It is an exercise in Wheeden Zygmund Measure and integration($9.2$(b)). That is I need to show if the conclusion of Young's inequality hold ($||f*g||_r\leq ||f||_p ||g||_q$), show that
$$\frac{1}{p}+\...
- 11