# Tagged Questions

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### Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$Tf = f \ast \mu$$ ...
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### proof that$L^1 (G)$ is a subspace of $M(G)$

Let G be a locally compact group, and let $M(G)$ be the space of complex Radon measures on G. I identify the function f with the measure $f(x) \rm dx$ . but How do I prove this inclusion?؟ . .
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### How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
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### Applications of Young's convolution inequality

Recall that the convolution of two functions is given by $$f*g(y)=\int f(x)g(y-x)dx.$$ The well known inequality known as Young's inequality, say that $$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q$$ provided ...
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### Young's inequality for discrete convolution

Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have $$\|f\star g\|_r\le\|f\|_p\|g\|_q$$ for $p$, $q$, $r$ satisfying ...
If $G$ is a non-Abelian locally compact group, and $f$ is in $L^1{(G)}$ and $u$ is in $L^{\infty}(G)$, and $f\ast u=0$ can it be concluded that $u\ast f=0$?