All Questions
Tagged with integration limits-and-convergence
35 questions
24
votes
1
answer
1k
views
Integrating on $\mathbb{R}$ by summing on $\mathbb{Q}^+$
Does the following integration method hold for regular enough functions $F:\mathbb{R}\to\mathbb{R}$?
\begin{align}
&\zeta(2)\sum_{\frac{a}{b}\in\mathbb{Q}_n} \frac{F(\log \frac{a}{b})}{\sqrt{abn}...
- 634
12
votes
1
answer
633
views
Integrals of power towers
Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
- 6,819
7
votes
3
answers
909
views
Using the Stone-Weierstrass theorem to solve an integral limit
The following question was posted on math stack exchange here but it got no answers
Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
- 331
5
votes
3
answers
717
views
Speed of convergence for Weyl's Equidistribution theorem
If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that
$$\frac{1}{n}\sum_{k=0}^{n-1}f(ak)\rightarrow \int_0^1 f(x)dx.$$
Can we ...
- 65
4
votes
1
answer
205
views
Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$
Let $f = (f_0,f_1,\ldots,f_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum_{n\geq 0} z^n f_n$$ for its probability generating ...
- 730
4
votes
1
answer
2k
views
Exchanging series and integrals
I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...
- 91
4
votes
1
answer
282
views
How to estimate the order of this integral with parameter
Some introduction:
Given a homogeneous structure called "dilation" in $R^n$: For $t\geq 0$
$$D_t: R^n\rightarrow R^n$$
$$D_t(x)=(t^{a_1}x_1,...,t^{a_n}x_n)$$
where $1=a_1\leq...\leq a_n$, ...
- 561
4
votes
1
answer
339
views
Mikusiński's approach to Bochner integrals; replace absolute by unconditional?
In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions:
Defn. Let $X$ be a Banach space. ...
- 39k
3
votes
0
answers
238
views
Dominated convergence Theorem
I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with
Generalized Spatiotemporal Gaussian Process Models.
Theorem 2.1 in the page 33 uses ...
- 31
2
votes
1
answer
137
views
Approximating a limit of an integral
How can we prove the following asymptotic lower bound for the regularized Beta function when $n\rightarrow\infty$?
$$\int_0^{1} I_{2 t - t^2}\left(\frac{n - 1}{2}, \frac{1}{2}\right) dt=\Omega\left(\...
- 1,677
2
votes
1
answer
230
views
Does exist a variant of Frullani's theorem valid for $f(x)=\pi(x)/x$ or $f(x)=\psi(x)/x$, where the numerators are prime-counting functions?
Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia Frullani integral and other uses and contexts (see [2]). I wrote two imaginative examples of what can be ...
2
votes
2
answers
211
views
Limit of a integral whose integrand diverges under the limit
I am trying to simplify the following limit of integral where $\mu$ is given:
$$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^...
- 43
2
votes
2
answers
876
views
Monotone convergence theorem for stochastic integrals
I was wondering if there exists an equivalent of a monotone convergence theorem for stochastic integrals. I looked into plenty of books and papers, but I haven't found anything useful. I would expect ...
- 121
2
votes
0
answers
57
views
Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?
The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, ...
- 1,947
2
votes
0
answers
100
views
Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)
Under what conditions on $c_{r}^{m}$
does
$$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$
hold for
$$k(p,q)=\sum_{r=0}^{\infty}\sum_{l=...
- 1,256
1
vote
3
answers
281
views
limit of a singular integral
Denote $f_{\gamma}(x) =\frac { (1+\gamma)}{2} |x|^{\gamma}$. We consider:
$$I(\gamma) = \int_{-1}^1\int_{-1}^1 \ln (|x-y|) f_{\gamma}(x) f_{\gamma}(y) dx dy$$
I would like to know the limit of $I(\...
- 840
1
vote
1
answer
247
views
Convergence of a sum to the integral
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property :
$$\frac{1}{n}\sum_{\ell=0}^{n-1}f(a+b\...
- 65
1
vote
3
answers
579
views
Squeezing more convergence from the convergence in all $L^p$ spaces
Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
- 5,407
1
vote
0
answers
132
views
Numerical calculation of a double integral from the slowly-decaying oscillating function
Let us consider the following integral
$$
I = \int\limits_{0}^{+\infty}dx\int\limits_{-\infty}^{+\infty}dy \left[f(x,y) + g(x,y) \right].
$$
We know several properties of these functions.
There are ...
- 335
1
vote
0
answers
100
views
$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure [closed]
For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is
$$
\lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
- 11
1
vote
0
answers
62
views
When do quasiperiodic functions on $\mathbb{R}^2$ have an average over the plane?
For example, consider the following
\begin{equation}
\lim_{T\rightarrow\infty}\frac{1}{(2T)^2}\int^T_{-T}\int^T_{-T}\big[\cos(v/a)-\cos(u/b)\big]\cos(\sqrt{u^2+v^2})\ du\ dv,
\end{equation}
where $...
- 19
1
vote
0
answers
263
views
Does a growing manifold fixed at a point converge to its tangent plane?
Let $M$ be a smooth compact $(n-1)$ dimensional submanifold in $\mathbb{R}^n$. Let $H$ be the $(n-1)$ dimensional Hausdorff measure. Let $f(x,y,t)$ is a function for $x\in\mathbb{R}^n$, $y\in\mathbb{R}...
- 65
1
vote
0
answers
206
views
convergence of sets and limit of an integral
Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}$ be compact sets.
Let $f:X\times Y\rightarrow\mathbb{R}$ be a $C^{1}$ function.
Let $s:Y\rightarrow X$ be a function (not necessarily continuous).
...
- 11
1
vote
2
answers
151
views
Finding Kuramoto Model coupling strength with limits?
The following is an equation describing the coupled phases of N oscillators according to the Kuramoto model:
$$
1 = K \int_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\,
{\rm g}\left(KR\sin\left(\...
- 13
0
votes
1
answer
204
views
How to prove approximation for fresnel integral converges
I was looking at the fresnel integral $S(x)=\int^x_0\sin(s^2)ds$. From reading I learned that this integral approaches $\frac{1}{2} \sqrt{\frac{\pi}{2}}$ as $x \rightarrow \infty$. Through messing ...
0
votes
1
answer
126
views
Proof of $\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx<\infty$ for Schwartz function $f$
For a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$ do we have that
$$\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx$$
converges?
- 123
0
votes
1
answer
87
views
Properties of slowly-varying functions at two large points
Consider a slowly-varying function $L:(1,\infty) \mapsto (0,\infty)$, i.e. a function such that $L(cx)/L(x)\to1$ as $x\to\infty$ for all $c>0$. Assume that $\lim_{x \to \infty}L(x)=0$.
Question: is ...
- 195
0
votes
1
answer
127
views
Uniform convergence of differential quotients in $L^1$
I know that the question arose already in other contexts. However, I think this question might be different. If I have $f\in W^{1,1}$ then it is obvious that $\frac{f(x+t)-f(x)}{t}$ converges ...
0
votes
1
answer
161
views
Sufficient conditions for L1 convergence of exponentials
Suppose $(X,B,m)$ is a finite measure space and $(f_n)$ is a
sequence of functions converging almost surely and in $L^2(X,m)$.
Moreover, I know that for every $n$, $\int e^{f_n(x)} m(dx)<\infty$. ...
- 650
0
votes
0
answers
75
views
Dense subspace of square integrable functions on the complex disc
Denote by $L^{2}(D,(1-|z|^{2})^{a-1}|z|^{2b-2}dx dy)$ the set of square integrable functions on the complex disc $D= \lbrace z \in C, \; |z| <1 \rbrace$ with respect to the measure $(1-|z|^{2})^{a-...
0
votes
0
answers
146
views
Does the following sequence $\{g_n\}$ converge?
Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where
\begin{eqnarray}\label{eqn:constraint1}
f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
- 550
0
votes
0
answers
105
views
Is this integral finite and how does it decay to zero?
I would like to know if the following is convergent/finite (it represents a bound from a truncated Legendre series approximation)
\begin{equation}
\varepsilon_n \leq \int_{-1}^1 \left(\int_{n\gg 1}^\...
- 201
0
votes
0
answers
91
views
Does $L^1$ convergence preserve the regularity of this sequence of functions?
Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges
$...
- 143
-1
votes
1
answer
193
views
Limit of the convolution of derivative of Gaussian heat kernel
I'm looking for the following limit:
$$\lim_{\varepsilon\to 0^+}\int_{-\sqrt{\varepsilon}}^{\sqrt{\varepsilon}}\frac{1}{\sqrt{2\pi}\varepsilon^{3/2}}\left(-1+\frac{x^2}{\varepsilon}\right)e^{-\frac{x^...
-2
votes
1
answer
708
views
Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero [closed]
Edit: I got rid of my old definitions. Everything should be clear now
Since no one has answered my question on MSE, I’m hoping to get an answer here. I apologize if you dislike my writing. I am an ...
- 63