# 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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### Help me find the antiderivative of $W(W(x))$ where $W$ denotes the Lambert W Function

Let $W$ denote the Lambert W Function. I must know the antiderivative of $W^2 = W(W(x))$. I'm already convinced this function is not elementary. This does nothing to settle up my curiosity, as I ...
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### Can we deduce $a_n\le Cn^2$ from $a_{n+1}\le Ca_n+Ca_n^{1/2}$? [closed]

Assume $\{a_n\}$ is a positive sequence satisfying $a_1\le C$ and $a_{n+1}\le C a_n+C a_n^{1/2}$. Can we deduce by induction $a_n\le Cn^2$? Here C's could be different constants.
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### Construct a continuous function $f(x)$ periodic with period $2\pi$ such that the Fourier series of $f(x)$ is divergent at $x = 0$

The following question was asked on Math Stack Exchange by me 15 days ago. I used a bounty, but still no response.So I am posting the question here Here is the link of the question here problem ...
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### Ratio of l1/l2 norm over n dimensional vector

According to here $$\frac{(|c_1|+|c_2|+\cdots+|c_n|)^2}{c_1^2+c_2^2+\cdots+c_n^2}\geq 1$$ The equality holds when for each $i\neq j\in[n]$, $|c_i||c_j|=0$. Could we improve the lower bound by ...
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### A function that maps every perfect set to $\mathbb{R}$

It's known that some real functions map every nonempty open subset onto $\mathbb{R}$. Is there any function from $\mathbb{R}$ to $\mathbb{R}$ that maps every nonempty perfect set onto $\mathbb{R}$?
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### Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
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### Continuous modification of tangent vector fields

Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
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### Etymology of “real numbers"

I would like to know why the real numbers are called “the real numbers.” I would also like to know the meaning of “real” in the phrase “real number.” Further questions and clarifications: I’d like to ...
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1 vote
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### Function whose derivatives eventually vanish almost everywhere

As a takeaway of this post we have the following result. P. Let $f:[0,1]\to\mathbb{R}$ be infinitely differentiable such that for all $x\in[0,1]$ the sequence $\{f^{(n)}(x)\}$ is eventually $0$. Then ...
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### Reference request: injectivity of CWT, density of dilations and translations in $L^p$

Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on ...
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### decimal expansion and distance [closed]

If we have a infinite sequence of decreasing irrational numbers say $$\{\gamma_1,\gamma_2,\gamma_3,\cdots \}, ~~~~~ \text{s.t}~~~~~0< \gamma_i<1$$ can we say $$\sum_{i=1}^{\infty}{\gamma_i}$$ ...
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### Multiplication with dilations of nonzero measurable function is injective

Denote $f_s(x):=f(sx)$ as the dilation of a function $f$. I want to know whether the following statement is true: Suppose $f$ and $g$ are measurable functions on $\mathbb{R}$, and $f$ is not almost ...
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### Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?

The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm $$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$ The space $L^2([a,b]\times S^2)$ ...
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### Antiderivatives via Taylor series and the FT of Calculus

If $f$ is a real function on an interval $[a,b]$ such that $f$ is computationally tractable on $[a,b]$: you can calculate $f(x)$ to $n$ bits of precision using an algorithm which is polynomial in $n$ ...
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### Does the Poincaré inequality hold on annular domains?

Does the following Poincaré inequality hold $$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$ where $B_r$ denotes a ball of radius ...
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