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

learn more… | top users | synonyms

1
vote
0answers
70 views

Expansion of a power series as integral of cosine functions

Suppose $$f(x)=\sum_{k=0}^\infty (-1)^k x^{2k} \int_0^1 p(\xi_{2k})\int_0^{\xi_{2k}}q(\xi_{2k-1})\cdots\int_0^{\xi_3}p(\xi_2)\int_0^{\xi_2}q(\xi_1) d\xi_1 \;d\xi_2\cdots d\xi_{2k-1}\;d\xi_{2k},$$ ...
2
votes
1answer
69 views

Regularized integral and asymptotic expansion

Let $f:(0, \infty) \longrightarrow (0, \infty)$ be a monotonously increasing function (in fact, a step function) and let $P$ be a polynomial of degree $N$. Suppose I know that for some $k$, the limit $...
10
votes
1answer
876 views

The origin of the Ramanujan's $\pi^4\approx 2143/22$ identity

What is the origin of the Ramanujan's approximate identity $$\pi^4\approx 2143/22,\;\;\tag 1$$ which is valid with $10^{-9}$ relative accuracy? For comparison, the relative accuracy of the well known $...
6
votes
1answer
197 views

Simplicity of eigenvalues

Consider the Sturm-Liouville operator$$Au = -(pu')' + qu \text{ on }I = (0, 1),$$where $p \in C([0, 1])$, $p \ge \alpha > 0$ on $I$, and $q \in C([0, 1])$. No further assumptions are made; in ...
6
votes
3answers
221 views

On what kind of condition of a compact set $K$ in the plane, $C(K)$ has a generator?

Let $K\subset \Bbb{C}$ be a compact subset of the complex plane, and let $C(K)$ be the space of all complex continuous functions on $K$. We say that $f\in C(K)$ is a generator of $C(K)$ when the set $...
1
vote
2answers
252 views

$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$

$c_0(C[0,1])$ is the $c_0$-direct sum of countably many $C[0,1]$.How to prove $C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$. Here,Banach-space isomorphism means a bounded invertible operator ...
10
votes
2answers
255 views

Asymptotic behavior of Sturm-Liouville eigenvalues

I have two questions. Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$. Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$...
3
votes
0answers
48 views

measure of an image under an argmax function

I am trying to find any techniques to analyze the measure of an image of a set under an argmax function. For example, let $\Omega\subset\mathbb{R}^n$ be compact and $\phi:\Omega\to\mathbb{R}$ be ...
0
votes
2answers
70 views

Level sets and integral of functions of two variables

Let $f_1,f_2$ be two positive functions on $\Omega_1, \Omega_2 \subset R^2$ with $f_1|_{\partial \Omega_1}=f_2|_{\partial \Omega_2}=0$. For every $\lambda>0$, denote the the area of the domain ...
0
votes
1answer
78 views

$C^{\infty}_{loc}$-convergence - right definition

Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
7
votes
1answer
100 views

If $u \in H^1(U)$, then $Du = 0$ almost everywhere on the set $\{u = 0\}$, auxiliary result

Let $\phi$ be a smooth, bounded and nondecreasing function, such that $\phi'$ is bounded and $\phi(z) = z$ if $|z| \le 1$. Set$$u^\epsilon(x) := \epsilon \phi(u/\epsilon).$$Do we necessarily have that$...
7
votes
0answers
48 views

Continuous inclusions Sobolev theorem, question [closed]

How do I see that if $f$, $g \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $fg \in H^s(\mathbb{R}^n)$ and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ...
4
votes
2answers
413 views

Generalized Jordan theorem and winding number

By the generalized Jordan theorem any continuous injective map $S^{n-1} \hookrightarrow R^n$ splits $R^n$ into two regions, one being bounded (interior) and the other one unbounded (exterior). It ...
3
votes
2answers
380 views

A problem on real valued functions in $\mathbb{R}^2$ with least variation

Let $\alpha(s) = (x(s),y(s))$ be the arc length parametrization of a plane, smooth, closed, convex curve, of length $L$. Let $J:(0,L)\to\mathbb{R}$ be a smooth and Bounded variation (BV introduced ...
1
vote
1answer
82 views

Quadratic Convergence in Fixed Point Iteration

Quadratic convergence is the hallmark of Newton's Method for root-solving. I'm looking for a result that implies the Newton result that looks like this: Theorem : Let $f:\mathbb{R}^n\rightarrow\...
0
votes
0answers
29 views

Representation of multi-variable functions as a composition of 1- or 2-variable functions [duplicate]

This is a re-post of my question from M.SE that remains unanswered for several months. I'm familiar with Kolmogorov–Arnold representation theorem, but AFAIK their construction makes essential use of ...
1
vote
0answers
122 views

Divergence Theorem for Distributions

I am interested in a generalization of the divergence theorem: Given an open subseteq $U \subseteq \mathbb{R}^n$, a compact set $G \subseteq U$ with smooth boundary $\partial G$ and a $C^1$-vector ...
0
votes
1answer
68 views

Approximating characteristic functions by cutting the real axis into smaller and smaller pieces

Let $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi ...
3
votes
2answers
208 views

Extremal problem for sequences

Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum ...
4
votes
1answer
116 views

Using wavelets to capture the $L^2$ norm of $f''$

I posted this question on MSE a couple of days ago. Someone gave some hints, which, besides the fact that I struggle to understand them, go in a numerical analysis direction, which I am not interested ...
3
votes
1answer
77 views

Urysohn type cut off function

I am looking for a cutoff function. The Urysohn's Lemma says Let $X$ be a $T_{4}$ space and $A,B \subset X$ be two closed and disjoint subsets of $X$. Then there exists a continuous function $f:X \...
3
votes
1answer
88 views

How to show monotonocity and the limit? [closed]

Let me reformulate my recent question. Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density: $$\phi(x) = C\left\{ \begin{array}{lcc} \sqrt{...
1
vote
3answers
147 views

Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? [closed]

Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1,...
3
votes
1answer
53 views

Does there exist any subsequence $(u_{n_k})$ converging strongly in $L^q(\mathbb{R})$, for any $1 \le q \le \infty$? [closed]

Fix a function $\varphi \in C_c^\infty(\mathbb{R})$, $\varphi \not\equiv 0$, and set $u_n(x) = \varphi(x + n)$. Let $1 \le p \le \infty$. Does there exist any subsequence $(u_{n_k})$ converging ...
13
votes
1answer
428 views

Does there exist some $C$ independent of $n$ and $f$ such that $ \|f''\|_p \geq Cn^2 \| f \|_p$, where $1 \leq p\leq \infty$?

Let $f$ be a trigonometric polynomial on the circle $\mathbb{T}$ with $\hat{f}(j) = 0$ for all $j \in \mathbb{Z}$ with $\lvert j \rvert < n$. Does there exist some $C$ independent of $n$ and $f$ ...
2
votes
1answer
137 views

Using $H^2$ to find a cyclic vector in $\ell^2$

Let us consider $\ell^p(\mathbb{Z})$. We know that the vector $e_1=(\dots,0,0,1,0,0,\dots)$ is a cyclic vector in sense that given the right shift operator $S:(\dots,x_0,x_1,x_2,\dots)\mapsto (\dots,...
4
votes
2answers
114 views

existence of a special conformal mapping

Sorry I don't know how to give an appropriate title. In the complex plane, suppose there is a graph $x+if(x)$ separating the plane into two unbounded components, where $f(x)$ is smooth and bounded, ...
4
votes
1answer
130 views

Continuous non-constant function with infinite intersections with horizontal line on a compact interval?

The title might be misleading, but whether such a function exists is what boggles me about the following problem: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that for all $...
0
votes
1answer
109 views

Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation: $i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields: $i(u_t,v) + \beta (u_x,v_x) = 0$ ...
3
votes
1answer
227 views

Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset $\...
4
votes
2answers
162 views

Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
3
votes
1answer
276 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that $$I=\sum_{...
2
votes
0answers
69 views

Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$. So $G\in\mathcal{G}$ can be written $G(x)=\...
4
votes
3answers
198 views

Measure of intersections in probability spaces

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$. ...
1
vote
1answer
115 views

A question about Borel sets on the unit interval

It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...
3
votes
1answer
87 views

Is the variation of two BV functions the same in the set in which they coincide?

Given two real $BV$ functions $u$ and $v$ in an open interval $(a,b)$ consider the set $A=\{x: \text{both } u \text{ and } v \text{ are continuous at } x \text{ and } u(x)=v(x)\}$ is it true that $|...
2
votes
2answers
166 views

Pointwise convergence for continuous functions

Let $f_n:[0,1]\rightarrow \mathbb R$ be a sequence of continuous functions converging pointwise, i.e. such that $\forall x\in [0,1]$, the sequence $(f_n(x))_{n\in \mathbb N}$ converges. We set $f(x)=\...
-3
votes
1answer
176 views

Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]

Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)? If so, please show me how to construct it.
0
votes
0answers
53 views

Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes". Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
2
votes
1answer
111 views

Existence of non-negative extensions of smooth functions on axes

I am struggling to solve an extension problem of smooth functions, and I would like someone to help me. The setting is as follows: Let $X_1$, $X_2$, and $T$ be either the real lines $\mathbb R$ or ...
1
vote
0answers
65 views

Inverse Laplace transform of a non-negative function

Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform, $$ f(s)=\int_0^\infty e^...
2
votes
0answers
170 views

How to analytically evaluate this n-dimensional iterated integral?

I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral $$\int_{-\infty}^{+\infty}dx_1\int_{-\...
0
votes
0answers
49 views

Approx the jump point of a $BV$ function from both hand side

Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as $$ u(x)= \begin{cases} 0,&\text{ if }x\in(-1,0)\\ 1,&\text{ if }x\in(0,1) \end{cases} $$ Clearly, we have $u\in ...
2
votes
1answer
252 views

Is this a log-concave function?

Let $(a_k)$ be a log-concave positive decreasing sequence. Is $\sum\limits_{k=1}^n a_k(1-e^x)^{k-1}$ log-concave in $x<0$, for each natural $n$?
3
votes
0answers
130 views

Extreme derivatives in Baire class 1

In the 1994 volume of "Differentiation of Real Functions" A. Bruckner poses the following problem (p.41): "Find necessary and sufficient conditions on a continuous function $F$ that its Dini ...
2
votes
1answer
94 views

Do we have independence if we let the indices of the events increase?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. $A_{m,1}...
1
vote
0answers
77 views

The real method of interpolation and operator ideals,

Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding) $K(\...
-2
votes
1answer
99 views

Is this intergral inequality valid? [closed]

Does the inequality $\int_2^{\infty} \dfrac{\sqrt x(\log x)^3 + (1+ \log x^2) x}{x(\log x)^2(x^2 - 1)} \,\mathrm {d}x > \ln \dfrac{17}{10}$ hold ?
2
votes
1answer
125 views

Approximation of the cumulative normal distribution

As is well known, there is no explicit formula for $\int_{-\infty}^\infty step(t−x)\cdot e^{−t^2/2}dt=\int_x^\infty e^{−t^2/2} dt$ for generic $x,$ where $step(z)$ is the step function, $step(z)=1$ ...
4
votes
1answer
634 views

When does this interesting sum diverge?

For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$ I don't know of any references or methods for this -- not even for $x=1$, for which the ...