# 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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### “Oddity” of a log-Bessel sequence happening at powers of $2$

Define the sequence $b_1=1$ and $$b_n=\sum_{k=1}^{n-1}\binom{n-1}k\binom{n-1}{k-1}b_kb_{n-k}.$$ By now, there is enough in the literature that $C_n$ is odd iff $n=2^k-1$ for some $k$ where $C_n$ are ...
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### What are ways to compute polynomials that converge from above and below to a continuous and bounded function in $[0,1]$?

My interest is to take a coin of unknown bias $\lambda$ and use it to produce a coin of bias $f(\lambda)$. This is called the Bernoulli Factory problem, and only certain functions $f$ can be simulated ...
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### Error function of multivariate Gaussian

I'm trying to prove that a sequence of functions $(k_N)_{N\in\mathbb{N}}$ $$k_N(\vec{y}):=e^{-N^2r(\vec{y},Q\vec{y})}\sqrt{\frac{r}{\pi}}^kN^{k}$$ where $r>0$, $k>2$ and Edit: I have forgot to ...
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### On a monotonicity property of Fourier coefficients of truncated power functions

Is it true that $$a_{k,n}:=\int_0^{2\pi}x^k\cos(nx)\,dx$$ is nonincreasing in natural $n$ for each $k\in\{0,1,\dots\}$? This question is related to this previous one. Twice integrating by parts, one ...
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### Integrability of $\int \log(f(x)) f(x) dx$ for a probability density function $f$

I am looking for weak conditions when a probability density function $f$ on $\mathbb{R}^d$ has a finite integral $$\int_{\mathbb{R}^d} \log(f(x)) f(x) dx.$$ Any references would be appreciated.
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### Is there a generalization of these q-series identities?

Denote $(q;q)_n=(1-q)(1-q^2)\cdots(1-q^n)$. The below three identities are known. \begin{align*} \sum_{n=1}^{\infty}\frac{(-1)^{n-1}q^{\binom{n+1}2}}{(q;q)_n} &=1-\sum_{n\in\mathbb{Z}}(-1)^nq^{\...
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### Bounding supremum norm of Lipschitz function by L1 norm

Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1 := \int_{[0,1]^d} |f(x)|dx$ ? In ...
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### How can a divergent nested radical be regularized (analogously to Cesaro sum regularization of a divergent series)?

The infinite series $1-1+1-1+\cdots$ diverges because the sequence of partial sums, $1,0,1,0,\ldots$ has no limit. However, it is well know that we can get around this problem in a number of ways; ...
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### Asymptotic bound for $\sum_{x=0}^\infty \sum_{y=0}^\infty (x+y)^m e^{-\frac{x^2}{2i} - \frac{y^2}{2j}}$ for $i$ and $j$ large

Note: This question relates to two previous questions on math.stackexchange (1 and 2), neither of which had satisfactory answers after posting bounties. Whilst trying to count certain types of ...
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