$L_2$ convergence of bivariate function
I have the following problem:Let $X,Y$ be random variables with distributions $P_X,P_Y$ and $f_0$ be a map from the support of X,Y to the reals. I define a new function $\chi_0(y) = E_X[f_0(X,y)]$. If...
View Articlelimit of integral with doubly period $1$ continuous function.
This is a problem that I recently saw in some lecture notes is proving a little more challenging that what I anticipated:Suppose $f$ is a continuous function on $\mathbb{R}^2$ and such...
View ArticleExponential type inequality
I found this problem in a set of old analysis prelims. I am looking for some hints after having spent some time working on the problem.Suppose $a_1,\ldots,a_n$ are real numbers such that $1+a_j>0$...
View ArticleIs there a theorem which provides conditions under which a power series...
Now asked on MO hereThis paper discusses how to prove that $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $. The first proof on this paper is Euler's original proof:$$\frac{\sin(\sqrt...
View ArticleLeast number of circles required to cover a continuous function on a closed...
Now asked on MO here.This question is a generalisation of a prior question. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of circles with radius $r$ required to cover...
View ArticleSet with elements of the form $๐_๐:[0,1] โโ$ defined as $๐_๐ (๐ฅ) = ๐ฅ^ ๐~...
Let $$๐ถ[0,1] = \{ ๐: [0, 1] โโโถ๐ ~\text{is}~\text{ continuous}\}$$ and$$d_{โ}(๐, ๐) = sup\{ |๐(๐ฅ) โ๐(๐ฅ)|: ๐ฅโ [0, 1]\} $$for $๐, ๐โ๐ถ[0,1].~\text{For}~\text{ each}~ ๐โโ$, define $๐_๐:[0,1] โโ$ by $๐_๐(๐ฅ)...
View ArticleConstruct/prove existence of a function with given expansions at two...
Consider two non-constant real polynomials $f(x)$ and $g(x)$:$$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 ArticleProve every Cauchy sequence in R is bounded
Here's my attempt to prove every Cauchy sequence in $\mathbb{R}$ is bounded, I would like to see if there are any flaws in it.Proof: Let ${(x_n)}$ be a Cauchy sequence in $\mathbb{R}$. Now let's...
View ArticleIf $G\in C([0,1])$ and strictly increasing, can we find a sequence $G_n\n...
Let $G:[0,1]\rightarrow [0,1]$ be a strictly increasing and continuous cdf with $G(1)=1$.I have proven some property for $G\in C^{\infty}([0,1])$ that relies on the continuity of $g(x)=G'(x)$. I hope...
View ArticleLandau Notation Problem
I have this function$$ K_{n} = \int_{1}^{+\infty}\frac{1}{(1+t^2)^n}dt$$$$ \text{Let }t\geq1,t^2+1\geq1+t\Leftrightarrow\frac{1}{1+t^2}\leq\frac{1}{1+t} \text{ and for } n \in {\mathbb{N^{*}}} :...
View ArticleFinishing the proof of the triangle inequality of Hausdorff metric
currently I am trying to show that a certain type of Hausdorff metric satisfies the following triangle equality and I am stuck.Setup:Take $(X,d)$ as metric space.Denote by $C(X)$ the set of closed sets...
View ArticleClosed form: show that $\sum_{n=1}^\infty\frac{a_n}{(n+1)4^n}=\zeta(2) $
Let $a_n$ be the sequence defined via$$ a_n=\sum_{k=1}^n{2n \choose {n-k}}\frac{1+(-1)^{k+1}}{k^2}$$then prove that$$\sum_{n=1}^\infty\frac{a_n}{(n+1)4^n}=\frac{\pi^2}{6} $$I tried simplifying $a_n$...
View Article$L^2$ vs $L^\infty$ projection
Let $\mathbb P_N$ the space polynomials of degree at most $N$ on $X=[-1,1]$. What is$$\sup_{f\in L^\infty(X)\setminus \mathbb P_N}\frac{\|f-P_2[f]\|_{\infty}}{d_\infty(f,\mathbb P_N)},$$where...
View ArticleIs there a function whose maximizers remain the same after any affine...
Let $f: \mathbb{R_+}^n\to \mathbb{R_+}$ be a function that is strictly increasing in each of its arguments. Let $M_f$ be the set of its maximizers on some fixed compact subset $D\subseteq...
View ArticleUniform convergence and differentiability, question about the proof
I have a question to the following proof of the well-known theorem.Theorem. Let $\{f_n \}$ be a sequence of functions converging to $f$ pointwise on $[a,b]$. If each $f_n$ is differentiable with...
View ArticleThe total variation and the integral of the derivative
I need a hint (not a complete solution) of the following problem:EDIT: when I was posting the question, I found I suddenly got it, so I need some verification.Suppose $F$ is a complex-valued function...
View ArticleLet $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup...
Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$Here's what I did so far:Let $\sup A, \sup B$, and $\sup (A \cup B)$ denote upper bounds of $A,B$, and $A \cup B$,...
View ArticleApproximation of a class of measurable functions by simple functions with...
It is well known that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $f \ge 0$ then there exist a sequence of simple non-negative measurable functions {$ S_n $} such that...
View ArticleFind a sequence function for combinatorial sequences
Iยดm trying to find a sequence function for the following sequence:$0, 1, 85, 419, 973, 1747, 2741, 3955, 5389, 7043, 8917, 11011, 13325, 15859, 18613, 21587, 24781, 28195$The first term is generated...
View ArticleIs this "continuous" function really continuous?
Let's assume $n \ge 2.$ Suppose $f:\Bbb R^n \times \Bbb R \to \Bbb R^{n+1}$ has the form$$f(x,t) = (\phi_t(x),t),$$where $\phi_{t_0}:\Bbb R^n \to \Bbb R^n$ is a continuous function for each fixed...
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