# Tagged Questions

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### How is equivalent to approximation property?

I'm trying to prove Lemma 4.8 of [1] online reading: Notation: $\tau_C(X)$ means the topology of uniform convergence on the compact subsets of $X$. Lemma 4.8. For a Banach space $X$ the following ...
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### Can we approximate $f(x) = \chi_{(0,\infty)}(x)$ by smooth monotone functions?

Given $f(x) = \text{sign}^+(x) = \chi_{(0,\infty)}(x)$, is it possible to a sequence $f_n$ such that $f_n$ is smooth, $f_n' \geq 0$ and $$f_n \to f?$$ In some sense of convergence? Preferably ...
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### Approximation of $C^\alpha(T^2)$ in $L^1$ by a $C^k$ function

Consider a Holder function $f \in C^\alpha(\mathbb{T}^2, \mathbb{R})$, $\alpha \in (0,1)$. I would like to approximate $f$ with $f_\epsilon \in C^k(\mathbb{T}^2, \mathbb{R})$, $k \in \mathbb{N}$, in ...
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### Explanation of a passage about a smooth approximation to $L^p$ function

I'm reading J.L Vazquez "Porous Medium Equation" book. In it, he says the following: We are given a function $a:\Omega \times (0,T) \to \mathbb{R}$ such that $a \geq 0$. We find a smooth ...
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### $C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
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### How to approximate L^1[0,1] functions?

Do functions on a uniform grid with n points in the interval $[0,1]$ approximate $L^1[0,1]$ functions, as $n \to \infty$? I want to sample functions in $L^1[0,1]$ space numerically and I want to be ...
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### Density of $C_0$ space in $L^p$ space for Hilbert-space-valued functions

The theory says that the space of continuous functions with compact support is dense in Lp space for functions taking value in finite dimensional space (or maybe just in real or complex space?) But, ...
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### A counter example of best approximation

Construct a point $f\in C[0,1]$ and a closed subspace $V\subset C[0,1]$ such that $f$ does not have a best approximation in $V$. Definition: $C[0,1]$ is the set of countinous function with the norm ...
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### Extending $W^{2,n}(\Omega)\cap C^2(\Omega)$-functions to $W^{2,n}(\Omega)$

I have the following theorem: Let $u\in C^2(\Omega)\cap W^{2,n}(\Omega)$ satisfy $Lu\geq f$ for $f\in L^n(\Omega)$. Then for any ball $B_{2R} = B_{2R}(y)\subset\Omega$ and $p>0$ where $u$ is ...
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### Approximation in $L^2$ of functions with values in a convex set

Here is my problem : Let $K$ be a convex set of $\mathbb{R}^m$ ($m\in \mathbb{N}^*$), such that $0$ belongs to the interior of K, I want to approximate (in $L^2(\mathbb{R}^m,\mathbb{R}^m)$) a function ...
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### Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t)$

I try to solve this problem all day, but can't reach any progress in it. Can you give me some hints? Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t) = \frac{1}{\sqrt[4]t}$ ...
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### Newton-Raphson in $\mathbb{R}^n$

Let $U\in\mathbb{R}^n$ be open and $f:U\to\mathbb{R}^n$ be a $\mathcal{C}^1$ map. $\exists p\in U$ such that $\;f(p) = 0$ and $Df_{|p}$ is invertible. Define $\phi(x)= x-(Df_{|x})^{-1}f(x)$, show ...
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### Complete normed vector space

I came across this question and was not sure how to proceed. Is the normed vector space of all polynomial functions on [0,1] complete with respect to the infinity norm? Thank you
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### Continuation of smooth functions on the bounded domain

Given a bounded domain $\Omega\subset \mathbb{R}^n$ and a smooth function $f$ with bounded derivatives on $\Omega$, is it possible to extend $f$ to $\tilde{f} : \mathbb{R}^n \to \mathbb{R}$ such that ...
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Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$... 1answer 70 views ### If f is in LipK[a, b], show that f can be uniformly approximated by polynomials in LipK[ a, b]. Question: If f \in LipK[a, b], show that f can be uniformly approximated by polynomials in LipK[ a, b]. Context: f \in LipK[a,b] then it is Lipschitz with constant K. The text I am currently ... 1answer 106 views ### How to show that linear span in C[0,1] need not be closed [duplicate] Possible Duplicate: Non-closed subspace of a Banach space Let X be an infinite dimensional normed space over \mathbb{R}. I want to find a set of vectors (x_k) such that the linear ... 0answers 136 views ### Orthogonal basis for waveform expansion I have many signals where each signal has a different waveform f(x). One example of such a waveform could be this f(x) sampled at 11 x positions: I am looking for a basis, Bi, for a series ... 1answer 110 views ### asymptotic limit of \int_0^{\infty}\left(1-\frac{t^2}{2(2k+3)}+\frac{t^4}{2\cdot 4\cdot(2k+3)\cdot(2k+5)}\right)^qdt Help me please with the following integral. I've asked this question before Asymptotic limit of the integral with polynomial, but it turns out that it was incorrect question. I should get an ... 1answer 75 views ### asymptotic limit at the integral I would like to get an asymptotic limit at the following integral: for p\ge 2, n \in N, t \ge 0$$ \int_{0}^{\frac 12 \sqrt{(n+1)!}}\left(1-\frac{t^2}{2^2(n+1)!}\right)^p \mathrm{d} t $$I think ... 0answers 226 views ### Comparing norms of a vector Let a be a vector in \mathbb R^m, such that \sum_{i=1}^{m}a_i=0. I would like to compare \sqrt{2m(2m−1)}\|a\|_{\infty} and \sqrt{2m}\|a\|_2, in the case when the vector a satisfies the ... 1answer 122 views ### inequality and equivalence for norms Let x \in R^m. It is known that \|x\|_{\infty}\leq \|x\|_2\leq \sqrt m\|x\|_{\infty}. What the difference between above inequality and if we are saying that \|x\|_2\sim \sqrt m\|x\|_{\infty}? ... 1answer 228 views ### A question about the coercivity of a lsc and convex function. I was doing a proof and I need to show a result to conclude it: X is a reflexive Banach space with a norm, \|\cdot\|, of class \mathcal{C}^1. f:X\to\overline{\mathbb{R}} is lower ... 1answer 327 views ### Differentiability of Moreau-Yosida approximation. I want to show that if X is a reflexive Banach space with norm of class \mathcal{C}^1 and f\colon X\to\mathbb{R}\cup \{+\infty\} is convex and lower semicontinuous, then f_{\lambda} is ... 1answer 77 views ### Density of polynomials having “additive-separation of their variables”? Let K be a compact subset of \mathbb{R}^2. Let P be a polynomial in the variables x and y. Given \epsilon>0, can we find two polynomials P_1=P_1(x) and P_2=P_2(y) such that$$ ...
While solving some exercise I came up with this problem: Assume that $f \in C^2[0,\infty]$ (i.e. $f$ has continuous second derivative on $[0,\infty)$ and there exists limit in $+\infty$ of $f$) be ...