# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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### Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational

Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius ...
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### Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?

Let $X := \mathbb R^n$, $C_b(X)$ the space of all real-valued bounded continuous, $C_c(X)$ the space of all real-valued continuous functions with compact supports, and $C_c^\infty(X)$ the space of ...
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### Is the smallest root of this quartic always the closest point on the Hyperbola? [closed]

Let $a>b>0$. Suppose we want to minimize $$f(x)=(x-a)^2+(1/x-b)^2,$$ over $x>0$. Equating $f'(x)=0$ leads to the quartic equation $$g(x)=x^4-ax^3+bx-1=0. \tag{1}$$ Question: Is the ...
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### An inequality for polynomials

I have been thinking about the validity of the following inequality: if $P(z)=\sum_{k=0}^na_kz^k, a_n\neq 0$ and $P(z)$ is non-zero in $|z|<1,$ then for $\theta \in [0, 2\pi],$ and $p>0$ \...
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### Regularity of the spherical mean of a compactly-supported function

The problem Consider a $C²$ function $f: X \to \mathbb{R}$, for some compact set $X \subset \mathbb{R}^d$ with $C^1$ boundary, say $\partial X$. I am only interested in $d\in \{2,3\}$. Then, consider ...
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### Boyd & Chua 1985: Is the proof of Lemma 2 correct?

$\newcommand\norm{\lVert#1\rVert}\newcommand\abs{\lvert#1\rvert}$I'm reading this article by Boyd and Chua , in which they prove the approximability of arbitrary time-invariant (TI) operators ...
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