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2
votes
1answer
112 views

A question on existence of solutions of a linear ODE system

I am working on a problem of harmonic functions on surfaces, and in one step I got the following system of ODEs with prescribed asymptotes. I was wondering what methods could give us the existence or ...
1
vote
0answers
104 views

Uniform bound for an alternating series of functions

I have mainly two questions, the first one being motivated by the second one. 1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
4
votes
1answer
59 views

Reference: Hardy space regularity of the Jacobian determinant

I'm looking for a reference, expository in nature, for the proof of the following theorem of Coifman, Lions, Meyer and Semmes. Theorem: For all $u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, ...
9
votes
0answers
185 views

A multiple integral

Let us consider the multiple integral $$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos ...
3
votes
1answer
123 views

For Every Measure Zero Set $E$ There Exists a Positive Measure with Lower Lebesgue Density 0 and Upper Lebesgue Density 1

This is related to a question asked on mathstackexchange http://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo. This question is ...
0
votes
0answers
51 views

Approximate of binary function by indicator functions of one variable

Let $(X,F,u)$ and $(Y,G,v)$ be two probability spaces. I know a result that the indicator functions of product measurable sets can be approximated by indicator functions of one variable. That is, for ...
0
votes
2answers
122 views

Markov-Bernstein like inequalities for monotone polynomials

Let $P$ be a polynomial with real coefficients, and $\deg P=d$. There is Markov-Berenstein inequality: $P′(x)\leq\frac{d\|P\|}{\sqrt{1-x^2}}$,where $\|P\|=\max_{|x|\le1} |P(x)|$ and $|x|\leq1$. Are ...
0
votes
0answers
57 views

About approximate eigenvalue

I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4. Suppose $X$ is a real Banach Space, $M$ is a ...
1
vote
0answers
130 views

Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$): $\min_{x\in U}\|x-y\|$. I believe the minimizer $x_{U}^{*}$ is ...
2
votes
1answer
213 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Definition Let $E$ and $F$ be two Banach spaces together with a plain subset ...
0
votes
1answer
135 views

Legendre differential equation with additional term

In an application I encountered the ODE $$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right) \right) \left( ...
7
votes
2answers
233 views

Cofinality of a $\sigma$-ideal of $\mathbb{R}$

The cofinality of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is [EDIT: cofinal] in $P$, i.e. for every element $p\in P$ ...
1
vote
0answers
79 views

Determining the exact form of a projection in a Hilbert space

Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$ where $\mathcal{L}^2[0,T]$ is the set of Lebesgue ...
6
votes
1answer
129 views

Estimates of Hausdorff dimension (and its derivatives)

For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...
2
votes
0answers
113 views

A strong form of implicit function theorem (what happens when the derivative is degenerate?)

(this can be considered as some ad) Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
3
votes
0answers
150 views

Boundary of an open, bounded and convex set in $\mathbb{R} ^n$

Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
9
votes
1answer
248 views

Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $x_n$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$. My ...
1
vote
1answer
108 views

Interpolation and embeddings for parabolic function spaces

I have a somewhat easy looking question on parabolic function spaces: Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...
4
votes
1answer
279 views

A question on null sequences

Is it true that a sequence of real numbers $\{a_n\}$ converges to zero if and only if the sequences $\{\sin^2(nh)a_n\}$ $(h \in \mathbb{R})$ all converge to zero? In case the answer is ...
0
votes
0answers
75 views

Is the exponential Mathieu operator trace-class?

Let $H \psi(x) = -\frac{d^2}{dx^2} \psi(x) - \alpha \cos(x) \psi(x)$ on $[0,2\pi]$ be the Mathieu operator ( according to Mathieu's ODE). My question is: Do we know whether $U(t):=e^{-tH}$ for some ...
0
votes
0answers
131 views

Natural integration constant for normal and discrete integration: is there a connection?

It is often assumed that integration, unlike differentiation is defined only to an arbitrary constant. So the antiderivative function is often left undefined or postulated to be zero in zero. But I ...
2
votes
1answer
229 views

What does this ODE have to do with the associated Legendre polynomials?

I am currently struggeling with the following differential equation: $$(t^2-1)f''(t)+tf'(t)(1-8a+8at^2)-4(a+a^2-2at^2+\phi (-a+2at^2))f(t)= 4\lambda f(t),$$ where $a \in \mathbb{R}$ constant, $\phi ...
0
votes
1answer
126 views

Uniform boundedness in $L^1[0,1]$ implies finite $\limsup$ almost everywhere for a subsequence? [closed]

Given a sequence of functions $f_k \in L^1([0,1])$ such that $||f_k||_{L^1(0,1)}\leq C$. Is there a subsequence $\{k_l\,|\,l\in \mathbb N\}\subseteq \mathbb{N}$ such that for $\mathcal{L}^1$-almost ...
2
votes
2answers
96 views

formula for repeated finite differences

I am looking for a proof of a well-known fact, whose proof must be very easy, though I've been struggling to find it. Let $\Delta$ be the map from real-valued functions of a real variable, given by ...
3
votes
1answer
86 views

Metric density theorem in most general setting?

It's a consequence of Lebesgue's theorem that every measurable $E\subset\mathbb{R}^n$ has a metric density that's $1$ a.e. on $E$ and $0$ a.e. on $\mathbb{R}^n\setminus E$. What are the most general ...
1
vote
2answers
355 views

Spectrum of Mathieu equation

I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = ...
2
votes
1answer
146 views

Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$?

I am currently the convergence of different processes. Doing this, I ended up with this expression and was wondering whether it is true that$$\lim_{n \rightarrow \infty}\sum_{k=0}^{n} ...
0
votes
1answer
127 views

Question about the derivative of a fuctional

I have this lemma+proof and i dont understand why it follows from $J'(u_n)\rightarrow 0$ that $-\Delta_p u_n- f(x,u_n)\rightarrow 0$ such that $J(u)=\frac1p\int_{\Omega} |\bigtriangledown u|^p ...
1
vote
0answers
77 views

Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
8
votes
2answers
589 views

Mathematical equivalent to ladder operators?

A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...
2
votes
1answer
167 views

Equicontinuity and $L^2$ convergence imply uniform convergence

I'm currently working through an old Paper of Garsia, Rodemich and Rumsey (A Real Variable Lemma) and theres one thing i don't get. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of continuous real ...
0
votes
0answers
58 views

Trying to solve for total derivatives at a stationary point (maybe using the implicit function theorem)

Suppose we have a function $F(q) \in \mathbb{R}$, where $q=(q_1, \dots, q_n) \in [0,1]^n$, at least thrice differentiable in $(0,1)^n$. We fix the value of one variable $q_i \in (0,1)$, then maximize ...
7
votes
2answers
337 views

Riemannian distance functions on the real line

A distance function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0,\infty)$ that is defined by a smooth Riemannian metric on the real line satisfies the following properties: $d$ is a length metric ...
27
votes
2answers
1k views

Everywhere differentiable function that is nowhere monotonic

It is well known that there are functions $f \colon \mathbb{R} \to \mathbb{R}$ that are everywhere continuous but nowhere monotonic (i.e. the restriction of $f$ to any non-trivial interval $[a,b]$ is ...
3
votes
3answers
282 views

An apparently simple question (behaviour at infinity of a power series)

Let $(a_n)$ be a sequence of real numbers, and suppose that the real power series (function) $S(x):=\sum_{n=0}^{\infty} a_n x^n$ converges for every $x\in\mathbb{R}$. $\mathbf{Question}$: Suppose ...
1
vote
2answers
137 views

Approximation of smooth compactly supported functions on $\mathbb{R}^2$ using sums of products of one variable functions

Let $f \in C^{\infty}(\mathbb{R}^2)$ be smooth and compactly supported. Can we approximate $f(x,y)$ by sums of the form $\sum_{i=1}^m g_i(x) h_i (y)$ where $g_i, h_i \in C^{\infty}(\mathbb{R})$ are ...
1
vote
0answers
79 views

convergence of supergradient

Let $\{g_n\}$ be a sequence of concave functions defined on $\mathbb{R}$ and set $$\lambda_n(x)=\lim_{\Delta x\to 0+}\frac{g_n(x+\Delta x)-g_n(x)}{\Delta x}$$ Assume there exists a concave function ...
0
votes
0answers
45 views

convergence of concave envelope

Let $\{f_n\}$ be a sequence of uniformly upper bounded functions defined on $\mathbb{R}$ s.t. for every $x\in\mathbb{R}$ $$f_n(x)\to f(x),~ n\to\infty$$ Define $g_n$ and $g$ as the concave envelope ...
2
votes
1answer
134 views

To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$

I wants to understand the integrals of the form $$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$ where $\mu(\alpha)$ is a non decreasing function ...
1
vote
1answer
117 views

Non-continuous higher differentiability, II

In a comment on this question, Tom Goodwillie proposed a notion of higher differentiability that I elaborate to something like the following: Let $f:\mathbb{R}^n \to \mathbb{R}$. Let's say that $f$ ...
8
votes
1answer
240 views

When is this multiple integral finite?

Consider the following integral: $$ I_k(\alpha)=\int_{[0,1]^k}|x_1-x_2|^{\alpha}|x_2-x_3|^{\alpha}\ldots|x_{k-1}-x_k|^{\alpha}|x_k-x_1|^{\alpha}d\mathbf{x}. $$ where $k=2,3,4,\ldots$ The question is ...
3
votes
0answers
85 views

Generalized family of Holder inequalities

Is the "only if" direction of the following fact known? For fixed sequences $(a)_i = a_1, \dots, a_r$, $(b)_i = b_1, \dots, b_r$ and $(c)_i = c_1, \dots, c_r$, the inequality $\prod_{i = 1}^r ...
3
votes
0answers
55 views

Limit of a hypergeometric integral

Let $n,N,T$ be positive integers, with $N=\binom{n}{2}$, and $3\leq n\leq T\leq N$. Define: $$P(z):=z^{N+1-T}\int_0^1\frac{(1-t^2)^{n-2}}{(1-(1-z)t)^{N+1}}{}_2F_1\left[-T,N+1,N+2-T; ...
5
votes
3answers
297 views

Non-continuous higher differentiability

The standard definition is that a function $f:\mathbb{R}^n\to \mathbb{R}$ is differentiable at a point $x$ if there exists a linear map $\mathrm{d}f_x: \mathbb{R}^n \to \mathbb{R}$ such that $$f(x+h) ...
-4
votes
1answer
154 views

Is there always a subspace of $L^p$ isomorphic to direct sums of $\ell^2,\ell^p$? [closed]

It is known that each $L^p$ (on a space with finite measure like $[0,1]$) $p>1$ space contains an isomorphic complemented copy of $\ell^2$ and $\ell^p$. I think this is the Kadets-Pelczynski ...
0
votes
1answer
152 views

Second order ODE

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+\left[ n \beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
3
votes
0answers
66 views

What are the criteria for an elementary function to be infinitely integrable in elementary functions?

What features of elementary functions define a class of functions whose consecutive indefinite integration also gives an elementary function? Is there a way to check whether a given elementary ...
18
votes
2answers
2k views

Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
1
vote
0answers
42 views

Looking for CDFs that I can integrate a particular transformation of

I need two CDFs $G$ and $\lambda$ with unbounded support such that I can integrate $$ \int_{-\infty}^t \lambda(a(x+b))dG(x), $$$a>0,b\in\Re$. As far as I can tell, there exist no functions that ...
-2
votes
1answer
111 views

a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...