$f_n \to f$ a.e. and $\| f_n\|_p \to \|f\|_p$. Is $\{f_n\}$ dominated by some...
Let $E\subset \mathcal{M}(\mathbb{R}^n)$ with $m(E)>0$, $\{f_j\}_{j\in \mathbb{N}}\subset \mathcal{L}^p(E)$ and $f\in \mathcal{L}^p(E).$ Let $1\leq p < + \infty$ and suppose that $f_j\to f$...
View ArticleDual of Fourier lebesgue space [closed]
What is the dual of Fourier Lebesgue space, $\mathcal{F}L_P$? with weight $\langle\xi\rangle$
View ArticleShow that $f(x):=\cos(x^2)$ is not periodic.
How can I proof that the following function $f(x):=\cos(x^2)$ is not periodic? I think that I should find the zero points of the function but I don't know how to calculate it.Thank you very much for...
View ArticleSum of Dirichlet kernel for angle differences over $n$ angles on unit circle
Let $$D(\theta_i-\theta_j):= \frac{\sin((n+1/2)(\theta_i-\theta_j))}{2\sin\frac{\theta_i-\theta_j}{2}}$$ being the Dirichlet type of kernel of angle difference between $\theta_i$ and $\theta_j$ where...
View ArticleHelp understanding smoothness classes. What exactly are the conditions to be...
I know that a $C^2$ surface means the 2nd partial derivatives exist and are smooth, but I'm a bit unfamiliar with math notation/colloquia. I want to make sure I understand correctly.Say we have an...
View ArticleWhat is the infimum/supremum of a set in the extended real line?
May be a simple question, but I still don't know what the definition of the supremum or infimum of a set in the extended real line is. For example when we define the Lebesgue measure, we define the...
View ArticleProve $|f(x)-f(a)-df(a)(x-a)|\le \frac{M}{2}\|x-a\|^2$ when $\|d^2f(x)\|$ is...
Suppose $a\in \mathbb{R}^p$ and $f$ is a real-valued function whose second-order partial derivatives all exist and are continuous on $B_r(a)$. Also, suppose that the operator norm $\|d^2f(x)\|$ of the...
View Articlehow to show that $1/(1+x^2)$ is a contraction?
I am trying to show that $1/(1+x^2)$ is a contraction but I cannot find the contraction factor. So far I got: $\lvert...
View ArticleIs a $\mathcal B(R_+) \otimes \mathcal F $-jointly measurable process...
Let $X: (\mathbb R_+ \times \Omega, \mathcal B(\mathbb R_+) \otimes \mathcal F ) \rightarrow (\mathbb R_+, \mathcal B(\mathbb R_+))$ be a jointly measurable map. I would like to prove that the...
View ArticleAbout a calculation in Grafakos' Classical Fourier Analysis.
I'm reading Grafakos' book on Fourier Analysis and at some point he says "There is an analogous calculation when $g$ is the characteristic function of the unit disk $B(0, 1)$ in $\mathbb{R}^2$. A...
View ArticleCalculate sum of series $\sum_{i=1}^{\infty} a_i a_{i + k}$ for any $k \in...
I'm trying to find closed form of the sum of the following series$$\sum_{n=1}^{\infty}\left[\rule{0pt}{5mm}\left(n + 1\right)^{\alpha} + \left(n - 1\right)^{\alpha} -...
View ArticleProve that the sequence $a_{1}= 1$, $a_{n+1} = \sqrt[n]{a_{1}+\dots+a_{n}}$...
So I have a sequence $a_{n+1} = \sqrt[n]{a_{1}+\dots+a_{n}}$, $n \in \mathbb{N}$ and where is $a_{1}= 1$.I have to prove that there is $c > 0$ such that for every $n \in \mathbb{N}$, $a_{n} \geq c$...
View ArticleProve/disprove that $a^{2m} + b^{2m} + c^{2m} > 2^{1-m}$ subject to $a + b +...
Problem. Let $a, b, c$ be reals with $abc\ne 0$, $a + b + c = 0$, and $a^2 + b^2 + c^2 = 1$. Prove or disprove that$a^{2m} + b^{2m} + c^{2m} > 2^{1-m}, \forall m\in \mathbb{Z}_{>2}$.Prior think....
View ArticleLet $f\colon (a,b)\to \mathbb{R}$ be nondecreasing and continuous. If...
I need help to understand the proof below of the following theorem.Let $f\colon (a,b)\to \mathbb{R}$ be an arbitrary function. If $E=\{x\in (a,b)\mid f'(x)\text{ exists and }f'(x)=0\}$, then...
View ArticleContinuity question on compact and connected domain
Let $\Omega$ be an open and bounded connected domain from $\mathbb{R}^N$. Consider a continuous function $f:\overline{\Omega}\to\mathbb{R}$. So $K=\overline{\Omega}$ is compact.My question is: Can we...
View ArticleConvergence rate of a special series
Consider a convex and non-increasing function $\varphi$, such that$$\sum_{n=0}^{\infty}\varphi(n)e^{n}\leq \infty, ~\ ~\ ~\ ~\ 0\leq \varphi(n)\leq 1$$What is convergence rate of $\varphi(n)e^{n}$? Can...
View ArticleProve convergence of $\limsup_{n\to\infty}$
I am new to Real Analysis, and I have found this problem hard to formalize.ProblemLet $(p_n)_{n\in\mathbb{N}}$ and $(q_n)_{n\in\mathbb{N}}$ sequences such that $(p_n)\to u$ and $(q_n)\to v$. Consider...
View ArticleSign permanence of locally Lipschitz functions calculated on a sequence
Suppose I have a sequence $a_k(m)>0$ with $m\in\mathbb{N}$ such that, given $k\in\mathbb{N}$ and $p\geq 1$, I can show that $$|a_{k+p}(m)-a_k(m) |\leq \frac{m^2}{k^2}$$ with $a_{k+p}<a_{k}$. I...
View ArticlePossible strategies to build a function with given expansions at two...
Consider two non-constant real polynomials $f(x)$ and $g(x)$:$$f=f_0 + f_1 (x-x_0) +...+f_N(x-x_0)^N $$$$g=g_0 + g_1(x-x_1) +...+g_M(x-x_1)^M $$where $f_0...f_N,g_0...g_M,x,x_0,x_1 \in \mathbb{R}$ and...
View ArticleMinimization of a function with exponential and 2nd degree polynomial
I would like to minimize the following function (from $\mathbb{R}$ to $\mathbb{R}$):$$f(x) = -2a(1+x)\exp(-\Vert xb-c \Vert^2) + (1+x)^2 d$$where $a,d \in \mathbb{R}$ and $b,c \in \mathbb{R}^d$So...
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