Showing a convolution is well defined
Let $f: \Bbb{R}\to \Bbb{R}$ and $g: \Bbb{R}\to \Bbb{R}$ be continuous functions, where $g(x) = 0$ for all $x \notin [a,b]$ for some interval $[a,b]$.a) Show that the convolution...
View ArticleDoes the maximal function of a positive function say away from zero on...
I was reading the book "Fourier Analysis" by J. Duoandikoetxea and D. Cruz-Uribe. In one of the results, they want to prove the following weighted inequality:Theorem (Weighted Inequality): For...
View ArticleProving an upper bound of the resolvant operator
Given a self-adjoint operator $A$ in a Hilbert space, I need to prove $$\|(A-zI)^{-1} \|\leq \frac{1}{\text{Im}z}, z\in \mathbb{C} \setminus \mathbb{R}.$$ Can you provide some hint or a solution proposal?
View ArticleWhat is the meaning of differentiable?
Definition on my text book for differentiable is: for a point c, if$f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$ , then f is differentiable at cI'm confused that $f'(c)$ has a very similar...
View ArticleHow can we show that this integral is nonnegative?
Let$c_0>0$ and $\ell\in[0,1]$;$(E,\mathcal E,\lambda)$ be a measure space;$\mu$ be a probability measure on $(E,\mathcal E)$;$p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$p_\lambda:=\int...
View ArticleIf $T:H\to H$ is an isometric linear operator, then $T^* T=I$.
Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $H$ is an inner product space.I've been stuck on this for a while and don't really know where to start.
View ArticleHow to prove that $\left(\frac{r_1}{r_2}\right)^n{\rm...
Let $\Omega\subset \mathbb{R}^n$ be a bounded connected domain. Introduce the notation$$\Omega_r=\left\{x\in\Omega|d(x,\partial\Omega)<\frac{1}{r}\right\},\quad r>0,$$where $d(x,\partial\Omega)$...
View ArticleWhen does the image of a point lie in the relative interior of a convex set?
Suppose we have a continuous function $F : \mathbb{R}^n \to \mathbb{R}^n$ that is injective and such that the closure of the image of $F$, $\overline{F(\mathbb{R}^n)} \subset \mathbb{R}^n,$ is a...
View ArticleTwo "theorical" questions about Line Integrals
While the majority of exercises related to Line Integrals are more enfocated to calculous and the difficulty of finding a correct parametrization or so, i have found these two questions (which may be...
View ArticleIf $A$ has strictly positive reach, does the set $\{ x \in A \colon...
Let $A \subseteq \mathbb{R}^n$ with $\text{reach}(A) > 0$ (see https://en.wikipedia.org/wiki/Reach_(mathematics) ).Define for any $\epsilon>0$, the "removal of $\epsilon$-thick...
View Article$\phi$ is continuous and odd, $f$ is Lebesgue integrable, we define $T_nf =...
$\phi : \mathbb{R} \to \mathbb{R}$ is a continuous, odd function satisfying the conditions:$$\phi(0) = \phi(1) = 0, \ \ \ \ \ \ \phi(x + 2) = \phi(x) \ \forall x \in \mathbb{R}$$For a function $f$ that...
View Article$g: R \to R$ is integrable and has compact support. $f(y) =sin^2(y) ln|y|$....
$g : \mathbb{R} \to \mathbb{R}$ is integrable over $\mathbb{R}$ and has a compact support.$f(y) = \sin^2(y) \ln|y|$Show that the convolution $(f ∗ g)(x)$ is well defined for $x \in \mathbb{R}$ and that...
View ArticleIs a series of constant term $1-r$ Abel summable to 1?
Let $0\le r < 1$. For a given $r,$ clearly,$$\sum_{i=0}^{\infty} (1-r)$$diverges since $1-r>0$ is constant. On the other hand, letting $A(r) = \sum (1-r)r^i$. I thought$$\lim_{r\to 1} A(r) =...
View ArticleSymmetry of functions on the sphere
Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define$$S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}.$$Suppose there...
View ArticleThe convergence of the Flint Hills series vs the convergence of...
The Flint Hills series, is the series $$\sum_{n=1}^\infty\frac{1}{n^3\sin^2(n)},$$ and it's an open problem as to whether the series converges. From the proof of Corollary 4 of this paper, it seems...
View ArticleA sufficient condition for $f = 0$ when its integral is zero. [duplicate]
Suppose that $f$ is continuous on $[a, b]$ and $\int_{a}^{b} fg = 0$ for every continuous function $g$ on $[a, b]$. Prove that $f = 0$.My attempt: since $\int_{a}^{b} fg = 0$ for every continuous...
View ArticleClassifying a second order non-linear ODE
I am currently dealing with the following ODE as a stationary, special case version of a PDE model derived from Kuramoto-Sivashinsky.$$y'' y' = ay$$Where $a$ is a real (constant) parameter.I am going...
View ArticleFirst nonzero derivative of real analytic non-negative function is positive
Let $f:[0,1)\to \Bbb{R}_{\geq 0}$ be a real analytic, 1-periodic function. Moreover, assume that $f(x) = 0$ if and only if $x = 0$.$\textbf{Claim.}$ Let $n$ be the smallest non-negative integer such...
View ArticleMinimum of $\frac{f(x)}{x^{n+1}}$ is at end point?
Let $f:[0,1)\to \Bbb{R}_{\geq 0}$. Impose the following conditions on $f$:$1. f(x) = 0$ if and only if $x = 0$.$2. f'(x) = 0$ only for specific $x_1, x_2 \in [0,1)$ with $f(x_1)$ being a local maximum...
View ArticleUniform continuity inequality check
In this question: The confusion about the proof of if function is continuous on a closed interval $[a,b]$,then it is integrable, the OP derives the following inequality:$$\begin{equation} \...
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