# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### Convolution notation

I refer to notations like $$f*K_\epsilon(0)$$ in Convolution with Gaussian question. Do they mean $(f*K_\epsilon)(0)$, i.e. the convolution evaluated at zero or $f*(K_\epsilon(0))$, i.e. $f$ ...
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### A very curious rational fraction that converges. What is the value?

Is there any closed form for the following limit? Define the sequence $$\begin{cases} a_{n+1} = b_n+2a_n + 14\\ b_{n+1} = 9b_n+ 2a_n+70 \end{cases}$$ with initial values $a_0 = b_0 = 1$. ...
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### Mean value theorem sum of $\frac{1}{f ' (x _k)}$ [duplicate]

Let $f$ be a function which is derivable in $[0,1]$ with $f(0)=0$ and $f(1)=1$. Show that for every $n$ in $\mathbb{N}$, there exists numbers $x_1,x_2,.....,x_n$ all in $[0,1]$ such as \begin{...
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### Introductory Topology Book Recommendation for Economics

Would you please share your 2 cent on book recommendation for introductory topology book to graduate student in Economics. Have exposure to the first half of the yearlong analysis course in the ...
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### $\succsim$ preorder on X being continuous imply lower contour set closed

$\succsim$ is preorder (i.e. preference relation) on X that is continuous. This implies the lower contour set is closed. Would you please share your 2 cent on my parenthesis explanation (e.g. line ...
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### Convolution with Gaussian question

Let $K_\epsilon(x):=\dfrac{e^{-x^2/\epsilon^2}}{\epsilon\sqrt\pi}$ for $x\in \mathbb{R}$, $\epsilon>0$. Let $f\in L^\infty(\mathbb{R})$ where $f$ is of bounded variation on any interval $[a,b]$. ...
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### Existence of sequence of polynomials such that $\lim_{n\to\infty} \int_0^1 |h(x) - p_n(x)|^2 dx = 0$

For a function $h:[0,1] \to \mathbb{R}$: $$h(x) = \begin{cases} 1~~\text{for}~~ x\in[0, \frac12] \\0 ~~\text{for}~~ x\in(\frac12, 1] \end{cases}$$ how could we prove the existence of sequence of ...