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

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17
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1answer
510 views

Minimum value of $|p(1)|^2+|p(2)|^2 +…+ |p(n+3)|^2$ over all monic polynomials $p$

Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$. This question was proposed (problem ...
1
vote
2answers
156 views

Lower bound for $ \sum_{i=1}^n x_i f(x_i)$ when $\sum_{i=1}^{n}x_i = K$

Considering, the set of all n dim. vectors $\{x_i\}_{i=1,...,n} $ such that $x_i \geq 0 $ and $\sum_{i=1}^{n}x_i = K$ Any continuous and strictly increasing function $f^+(x)$ : $ \mathbb R^+ \to ...
0
votes
1answer
133 views

About weak derivatives [closed]

I have a question about weak derivatives. Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...
0
votes
1answer
78 views

Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, ...
1
vote
0answers
97 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
4
votes
2answers
207 views

Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$

A set $X\subset \mathbb{R}$ is called nice if for every $\epsilon > 0$ there are a positive integer $k$ and $k$ bounded intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup ...
2
votes
1answer
178 views

An elementary functional inequality

Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold $\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...
1
vote
1answer
76 views

Convergence of sequence of polynomials defined by boundary conditions

I'm sorry if my question sounds trivial, but analysis is not my field. Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials ...
1
vote
0answers
27 views

Condition for maximizer of convex combination to be expansion mapping

I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$ $$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$ such ...
3
votes
2answers
140 views

What is the identity of this shift operator-like infinite series?

I just ran across the following expression and would like to know if anyone can identify it: $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\frac{d^n f(x)}{dx^n}\frac{d^n g(y)}{dy^n}$. It almost looks ...
3
votes
0answers
35 views

Limit Behavior of Iterated Curvature-Function

What can happen, if one defines an infinite sequence of functions as follows $f_0\in C^\infty: x\in\mathbb{R}\mapsto y\in\mathbb{R}$ $f_{n+1}: \int_0^x ...
2
votes
2answers
129 views

Are all mixtures of these unimodal functions unimodal?

Let us say that a function $F\colon(0,\infty)\to\mathbb{R}$ is increasing-decreasing if, for some $c\in[0,\infty]$, $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. Is it true that ...
0
votes
1answer
197 views

Collection of graduate research projects in Real Analysis [closed]

While there are many open problems in Real Analysis like Khabibullin's conjecture or Lehmer's conjecture, those are big enough to take an expert's life for several years, let alone some graduate ...
0
votes
1answer
137 views

Sobolev type embedding

Consider a compact manifold $M$ and a point $q \in M$. Let us say that that the following inequality holds: $$ \Vert \varphi u\Vert_{L^p} \leq C\Vert \varphi u\Vert_{H^1},$$ where $\varphi \in ...
9
votes
0answers
57 views

Assymptotics of a Selberg type integral

Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral $$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} ...
0
votes
1answer
116 views

Unimodality of a certain parametric integral

Suppose $f: [0,1] \to [0,\infty)$ is a smooth, concave and strictly increasing function satisfying $f(0)=0$. Is it true that the map $$ F(y) = \int_0^1 \frac{y^{3/2}}{(y+f(x))^2} dx $$ has exactly one ...
1
vote
1answer
121 views

Extension of a smooth function from a convex set

Let $C\subset \mathbb{R}^{n}$, $C'\subset\mathbb{R}^{m}$ be two convex sets with a non-empty interior. A function $F\: : \: C\to C'$ is said to be differentiable at $x\in C$ if there exists a linear ...
0
votes
0answers
42 views

Is the following definition of a functional derivative natural?

if $\delta S = \int \sqrt g F[\phi] \delta \phi$ Then is it natural to define the functional derivative as follows, $\frac{\delta S}{\delta \phi} = F[\phi]$. In particular does this definition ...
7
votes
1answer
1k views

Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

Physical Motivation : Hear to these audio files S_f and P_f. S_f is Fourier partial sum and P_f is the new reconstruction, both use spectrum only in the region (0,4KHz) for reconstructing the ...
0
votes
0answers
95 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$

Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...
2
votes
1answer
185 views

$BMO$-property via a John-Nirenberg type estimate?

Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also $$ f_B:= \frac1{|B|}\int_B f \, dx. $$ Suppose $f \in L_{\rm loc}^p(\Omega)$ for all ...
4
votes
0answers
153 views

Evaluate a multiple integral

I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed. $$I_N=\int_{0\le r_n\le r_{n-1}\le\cdots\le r_1} e^{-(r_1^2+\cdots+r_n^2)} \prod_{i<j} \sinh(r_i-r_j) ...
1
vote
0answers
64 views

Does the Total variation of the Fourier partial sum of a bv function with jumps converge to TV of the function as $N\to\infty$

Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly, Let $f$ be a periodic ...
1
vote
1answer
79 views

First mean value theorem for integration and Lebesgue measureability

According to first mean value theorem for integration, if $G \ : \ [a,b] \to \mathbb{R}$ is a continuous function, there exists $x \in (a,b)$ such that $$\int_a^b G(t) dt = G(x)(b-a)$$ Assume $G$ is ...
8
votes
1answer
234 views

Exceptional values of real-valued functions on [0,1]

Given a continuous real-valued function $f$ from $[0,1]$ to itself with $f(0)=0$ and $f(1)=1$ such that $f^{-1}(c)$ is finite for all $c$ in $[0,1]$, let $E(f)$ be the set of $c$ in $[0,1]$ such that ...
1
vote
0answers
110 views

Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?

Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable). Also, let $f:D_1\cup ...
1
vote
0answers
47 views

Analogs of the paralleloram identity in higher degrees

I asked this two months ago in MSE, but nobody answered, so I hope it will be suitable here. A homogenious polynomial of degree $k\in{\Bbb N}$ on a finite dimensional vector space $X$ over $\Bbb R$ ...
1
vote
1answer
71 views

On the domain of functionals in measure with singular kernels

this post is concerned with functionals defined in measures. Consider the following functional $$\mathcal{W}[\mu]=-\int_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$ were we define ...
2
votes
1answer
112 views

Construct smooth functions with prescribed derivatives

To be specific, suppose we are given a sequence of smooth functions $\{f_k\}_{k\geq 0}$ on flat torus $\mathbf{T}^2$(you may think of it as doubly periodic functions on $\mathbf{R}^2$ and smooth). ...
1
vote
0answers
55 views

Heat equation inequality

There is an inequality that tells us that for some sufficiently smooth $f$ satisfying $(\partial_t - \Delta )f \le - \delta f^2 +K$ for $\delta,K >0$ that $f$ is bounded by some constant. ...
9
votes
1answer
310 views

A question involving e, floor, and all x > 0

Is $\lfloor(x+1/2)e\rfloor = \lfloor(x+1)(1+1/x)^x\rfloor$ for all $x > 0$? The question occurred in connection with (nonhomogeneous) Beatty sequences, $\lfloor nr+h\rfloor$, where irrational ...
9
votes
1answer
296 views

$L^1$ norm of exponential sum of $n^2 x$

What is the asymptotic order of $$ \int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx $$ as $N \to \infty$. This should be known, but I cannot find it in the literature.
1
vote
1answer
226 views

The existence of differential operator of the form $AB=0$

We define $\mathcal A$ is a differential operator of order $n$ with variable coefficients if $$ \mathcal A:=\sum_{|\alpha|\leq n}A_\alpha (x) D^\alpha $$ where $\alpha$ is an muti-index and ...
0
votes
0answers
42 views

convex function with distributional Hessian $D^2 f \leqslant \lambda$, $\lambda$-concave?

Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
1
vote
1answer
171 views

Is regularity closed under products?

Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - ...
8
votes
0answers
133 views

Intersection of connected components in $\mathbb{R}^n$

Let $n$ be a positive integer and let $K\subseteq \mathbb{R}^n$ be compact. Pick $x^* \in \mathbb{R}^n\setminus K$. Let $E$ be the connected component of $\mathbb{R}^n\setminus K$ that contains ...
7
votes
1answer
179 views

Poincaré inequality for curl-integrable functions

Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and $$ u_B := \frac1{|B|}\int_B u \, dx. $$ The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...
0
votes
1answer
135 views

When can two Cauchy transforms intersect?

Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = ...
4
votes
0answers
113 views

$H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao. They study the ...
5
votes
1answer
117 views

Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative

Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...
0
votes
0answers
119 views

Problem with operator and Fourier transform

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
7
votes
2answers
288 views

Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says $\sum_{n\geqslant 0}z^{2^n}$ doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...
1
vote
1answer
166 views

Finite trigonometric polynomial

I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that $T(x):= \sum_{n \in \mathbb{Z}} ...
0
votes
0answers
39 views

Is every $C^1$-domain which is homeomorphic to the unit ball in $\mathbb{R}^d$ Lipschitz equivalent to the unit ball?

Suppose we have a domain $\Omega\subset \mathbb{R}^n$ which is homeomorrphic to the unit ball $B(0,1)\subset \mathbb{R}^n$ and such that $\partial \Omega$ is of class $C^1$ (technically, this means ...
0
votes
1answer
63 views

Regular curve given implicitly

Let $F:D\subseteq\mathbb{R}^2\to\mathbb{R}$, $D$ open and connected set, be a $C^1 (D)$ application. What are the minimum requirements for $F$ such that the solutions of the equation $F(x,y)=0$ are ...
2
votes
0answers
41 views

Writing a function as a sum of functions of bounded diameter

This problem is distilled from one arising in a study of complex random variables, but I've removed as much baggage as I can without (I hope) making it trivial. Fix $D>0$. A function $f:\mathbb ...
4
votes
2answers
134 views

How to find an ODE with prescribed terminal values?

Let us consider an ODE $$\frac{dx_t^y}{dt}=g(x_t^y),$$ where y is the initial condition i.e. $x_0^y=y$. Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...
5
votes
2answers
188 views

Continuous map from connected subset of R^n to one of the real zero of an odd degree polynomial whose coefficients are polynoms of the variables

Let take a real multivariate polynomial $P(x_1, \ldots, x_n, y)$ such as the degree of P relatively to the variable $y$ is odd. Thus, for each $X = x_1,\ldots,x_n \in\mathbb{R}^n$, the univariate ...
8
votes
1answer
133 views

Specifying $L^p$ norms of derivatives

Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$? For ...
6
votes
1answer
182 views

Henstock, Differentiation under the integral sign

Does anyone know, where I can find the proof of necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here are the theorems but not all the ...