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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Let $X$ be a random variable which takes on values from $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and assume $p(x)&... 0answers 33 views Asymptotics for fractional Laplacian This question is motivated by Asymptotic formula for fractional Laplacian For the equation $$\begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\... 1answer 76 views Convergence of increasing rearrangment Let A\subset \mathbb{R} be measurable such that there are a,b\in \mathbb{R}, a<b fulfilling [b,\infty)\subset A\subset [a,\infty). The right rearrangement of A^{*} of A is defined as A^... 2answers 193 views Is radial part of a Schwartz class function also in Schwartz class? Let f\in\mathcal{S}(\mathbb{R}^n), Schwartz class. Consider the function g defined on [0,\infty) by$$g(r)=\int_{S^{n-1}}f(rw)d\mu(w),$$where d\mu is the normalised surface measure of S^{n-1}... 0answers 71 views L^2 convergence of a particular function I encounter the following problem when I study harmonic analysis by myself: Given a function f \in L^2([0,1]). Let's fix some irrational number \omega. For any N \in \mathbb{Z}^{+}, let's define ... 0answers 183 views Is the arithmetic-geometric mean of 1 and 2 rational? It is easy to show that, for two fixed real numbers \alpha, \beta > 0, the sequences given by a_ 1 = \frac{\alpha + \beta }{2}, g_1 = \sqrt{\alpha\beta}, and a_{n+1} = \frac{a_n + g_n}{2}, ... 1answer 110 views Topological characterization of invertible real matrices [closed] Let n\geq 2 be an integer. Consider the topological space M_n of n-by-n matrices with real entries. Can you give a short non-constructive proof of the existence of a continuous function M_n\... 0answers 31 views Monotonically increasing and bounded function is in BV_{loc}? For any n\in \mathbb{N} let f_n:\mathbb{R}\to [0,1] be monotonically increasing and \lim_{x\to -\infty} f_n(x)=0 and \lim_{x\to \infty} f_n(x)=1. It follows f_n is differentiable a.e.. I'm ... 0answers 73 views L^p estimate of a multiplier operator I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers \{m_{n}\}_{n \in \mathbb{Z}} that satisfies:$$\sum_{n \in \... 1answer 61 views Decide the order of of an integration involving the$\log$function Let $$A_n=\int_{n^{-\frac{1}{2}}}^{1}\frac{\log(nx)}{nx(\log\log(nx)-\log\log(1+x))}dx.$$ I want to discribe the order of$A_n$, by geting a progressive formula or a good lower bound for it. The order ... 0answers 21 views Strict Riesz's rearrangement inequality when function is not nonnegative The strict Riesz rearrangement inequality (Lieb- and Loss's book Analysis, Section 3, Theorem 3.9 ,page 93) says that if the functions$f,g,h,$are all nonnegative and$g$is strictly symmetric ... 0answers 60 views Explicit formula for$ (-\Delta)^s \left( \int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-2s} dz\right) $For$x \in \Omega \subset \mathbb R^N$, is it possible to compute explicitly $$(-\Delta)^s \left( \int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-2s} dz\right)$$ where$(-\Delta)^s$is the ... 1answer 48 views Integral bound for square of log derivative I am currently facing the following problem: Given a polynomial$f(x) = \sum_{s \in S_f} u_s x^s$,$f(0)\neq 0$,$\lvert S_f \rvert \leq t$(i.e.$f$is$t$-sparse) with$u_s$coming as samples from i.... 2answers 113 views Almost independence of$x^\top a$and$x^\top b$for$x$uniform on the sphere in$\mathbb R^d$and$a,b \in \mathbb R^d$with$a^\top b = 0$Let$d$be a large positive integer. Let$x$be uniformly distributed on the unit-sphere in$\mathbb R^d$and let$a$and$b$be perpendicular vectors in$\mathbb R^d$, i.e such that$a^\top b=0$. Let ... 0answers 46 views Stability of non-differentiability under modification on a small set Definitions: Let$\mathcal N$denote the set of continuous, nowhere differentiable real valued functions on$[0, 1]$. For$0 < \epsilon < 1$, we say$f \in \mathcal N$has$\epsilon$-robust ... 1answer 144 views Isoperimetric type inequality in$\mathbb{R}^2$Fix$L \in (0,\infty)$and consider$\mathcal{C}_Ldefined as follows: \begin{align*} \mathcal{C}_L := \{ \gamma:[0,1] \rightarrow \mathbb{R}^2 |~ \gamma \text{ is smooth and length(\gamma$)$=L$}\}.... 0answers 98 views Basel problem speed of convergence [closed] There is a Math.SE post arguing the Basel problem converges subgeometrically. But what is the exact speed of convergence, or are there any other bounds? 1answer 76 views Generalized Selberg integral I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions $$\int_0^1 \ldots \int_0^1 \prod_{i=1}^d u_i^{\frac{k_i-1}{2}} \prod_{m=1}^d (1-u_m)^{\... 1answer 237 views Malgrange preparation theorem with less regularity (This question was previously posted on MSE and I decided to post it here too.) I am studying the proof of the Malgrange preparation theorem given in the book "Stable mappings and their ... 0answers 18 views Upper derivative of the modified Bessel function of the first kind and order alpha j_alpha? [closed] I calculated the upper derivative of the modified Bessel function of the first kind and order alpha j_{\alpha} with respect to the variable with the maple program, but I could not show it for example ... 2answers 238 views Solution to simple non-autonomous ODE Consider the following ODE with parameters \alpha,\beta,\gamma \in \mathbb R$$f'(t)= \begin{pmatrix} \alpha-\beta t & \gamma t \\ \gamma t & -(\alpha-\beta t) \end{pmatrix} f(t).$$This ODE ... 0answers 41 views Variation of the sum of absolute values of coefficients for shifted Chebyshev polynomials Setting Let \rho \in ]0,1[, \varepsilon\in[0,\rho], k \in \mathbb{N}^* and$$P^\varepsilon_k(X) = \tfrac{T_k\left(\tfrac{2(X+\varepsilon)}{\rho+\varepsilon}-1 \right)}{\left|T_k\left(\tfrac{2(1+... 0answers 39 views Superharmonicity of the distance function Suppose$V$is a convex open proper subset of$\mathbb{R}^m$($m\geq2$). It is known that the function$u(x)=$dist$(x,\partial V)$is superharmonic on$V$. Is there a similar result without$V$being ... 0answers 111 views How does the area affect the integral? Let$\Omega\subset\mathbb{R}^n$a open bounded set. For any$r>0$consider the integral: $$J_\Omega(r)=\int_{\Omega}\frac{|x^s|dx}{r^c+\sum_{i=1}^m|x^{p_i}|r^{d_i}},$$ where$s,p_i\in\mathbb{N}^n$... 1answer 119 views Is there a description of the points of the Cantor set on which the Cantor function is differentiable? Let$C$be the usual ternary cantor set, and$f$the Cantor function, or Devil’s staircase associated to it. We know that$f$is differentiable a.e., and on every point of the complement$C^c$, the ... 1answer 67 views Maximal Hausdorff dimension of the set on which derivatives do not agree Let$f, g: [0, 1] \to \mathbb R$be functions that are differentiable a.e. with$f’ = g’$almost everywhere. What is the maximal Hausdorff dimension$d$(and corresponding Hausdorff$d$-measure) of ... 1answer 198 views A measurable set that acts as a speedometer Definitions and some motivation: Say a car is driving on a straight road. All we know is where it starts, and how much time it spends in certain stretches of the road. With just this much information, ... 0answers 75 views Weak-type inequality for the partial Fourier sum operator I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark: For any$N \in \mathbb{Z}^{+}$, let's use$S_{N}$to denote the partial ($N$terms) Fourier sum ... 1answer 76 views Analogous form of Hardy-Littlewood maximal inequality (weak/strong type) on affine subspaces I'm using some online notes (Professor Schlag, Yale University) to study harmonic analysis by myself. He introduced the following claim as an exercise: For any function$f \in L^{1}(\mathbb{R}^{d})$... 1answer 467 views Function whose sets of discontinuities and zeros are the rationals Question: Is there a real valued function$f:\mathbb{R}\to\mathbb{R}$such that its set of discontinuities is$\mathbb{Q}$and its set of zeros$\{x\in \mathbb{R}\mid f(x)=0\}$is also$\mathbb{Q}$? ... 0answers 58 views Extension of Liouville's Criterion for Liouville Numbers to Cantor Series [closed] Is this a valid way to extend Liouville's criterion for Liouville numbers to Cantor Series? $$0\le\sum_{k=n+1}^{\infty}\frac{a_k}{b_k!}\le\sum_{k=n+1}^{\infty}\frac{b_k-1}{b_k!}$$ $$a_k\ ,\ b_k\ \in\ \... 1answer 321 views Is there a dense planar rational point set within which the distance of any two points is an irrational number? i.e. could we find a subset X\subset \mathbb{Q}^2 such that \overline{X}=\mathbb{R}^2 and that for any x,y\in X the distance |x-y| is an irrational number? I'm considering the following ... 1answer 68 views Condition for the maximum to be non-increasing Let u\in\mathcal{C}^1(\mathbb{R}_+\times[0,1],\mathbb{R}) such that, for any t\geq 0, for all x_0\in[0,1] satisfying u(t,x_0)=\sup_{x\in[0,1]}u(t,x), we have$$\partial_t u(t,x_0)\leq 0.$$Is ... 1answer 133 views Real part of tail of logarithm Given a positive integer n, consider f_n = -\min_{|z|=1} \Re \sum_{i>n} \frac{z^i e^{-i/n}}{i}. What can be said about the growth of f_n? How large can it get? Taking maximum instead of ... 2answers 105 views A question on minimum principle Suppose D be an unbounded domain of \mathbb{R}^m for m\geq3, and u is superharmonic on D. We know that if \liminf_{x\to y}u(x)\geq0 for all y in \partial^\infty D (the boundary of D ... 0answers 109 views The square of a measure Notation: We denote by \mathcal L the usual Lebesgue measure on [0, 1]. We denote by \mathcal P = \{a_0, ..., a_n\} a partition of [0, 1] and \Delta \mathcal P := \max_{0 \leq i \leq n} |a_n -... 1answer 131 views Does the derivative of a BV function with no jump part vanish on level sets? Let u: \mathbb R^n \to \mathbb R be a BV function with no jump part, i.e., writing Du = D^a u + D^s u + D^j u for the decomposition of Du into absolutely continuous, Cantor, and jump part ... 0answers 49 views Multiplication of a Riesz basis Let {(\phi_n(.),\psi_n(.))}_{n\geq 1} be a Riesz basis in H^1_0(0,1) \times L^2(0,1). My question is the following: If we multiply the basis by the matrix e^{Mx}, x \in (0,1) where M is a ... 1answer 130 views Perron-Frobenius and Markov chains on countable state space The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer: Let A:\ell^1 \rightarrow \ell^1 be a contraction, i.... 1answer 99 views Riesz rearrangement inequality In the Lieb-Loss's book Analysis, they present the Riesz rearrangement in Section 3, Theorem 3.9 (page 93). Note that the functions f, g, h, are all nonnegative. I want to ask whether the ... 0answers 48 views Tail asymptotics of Durfee square identity This post is related to the problem Asymptotics of a combinatorial series According to the Durfee square identity:$$\sum_{k \ge 0} \frac{q^{k^2}}{(q;q)_k^2} (q;q)_{\infty} = 1,$$where (q;q)_k is ... 0answers 61 views Can a sequence of absolutely continuous functions be rescaled to be equicontinuous? Given a function f: \mathbb R \to \mathbb R, we say g: \mathbb R \to \mathbb R is a topological rescaling of f if g = fh for some orientation preserving homeomorphism h of \mathbb R. Given ... 6answers 690 views Elementary proof that an open subset of \Bbb{R}^n does not have measure zero? There is an elementary theory of subsets of \Bbb{R}^n of measure zero, namely one defines the volume of a cube in the obvious way and one says that a subset A has measure zero if given any \... 1answer 933 views Anti Arzela-Ascoli Notation: We say a sequence of real numbers diverges if it does not converge to a finite limit. We say a sequence f_n of real valued functions on [0, 1] are equibounded if \sup_{n \in \mathbb N}... 0answers 53 views Ekeland's standardness-property inheritable? Ekeland's inverse function theorem gives weak conditions under which a function f:E\rightarrow F between two graded Fréchet-spaces is locally surjective. The theorem requires the codomain F to be ... 4answers 206 views Integrals involving fractions of exponentials I am trying to calculate the average degree of a complex network, which requires me to solve for the following integral:$$\int \mathrm{d} x \frac{\exp{\left[-x -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\... 1answer 110 views Estimating singular double integral How can I estimate $$\int_{(0,1) \setminus B_{\delta}(1/2)} \int_{B_\delta(1/2)} \frac{u(y)v(y)}{|x-y|^{\alpha +1}} \, dy \, dx$$ in terms of a positive power of$\delta$and suitable norms of$u$... 1answer 36 views Fractional Laplacian and support Let$u:\mathbb [-1,1] \to \mathbb R$such that$\mathrm{supp}(u) \subset B_{1/2}(0)$. Under what assumptions on$u$does it hold $$\mathrm{supp}\Big((-\Delta)^s u\Big) \subset B_{1/2}(0),$$ where$(-\...
The following inequality appeared in the analysis of a random approximation algorithm: $$\int_u^{u+1} x^p\ \mathrm{dx} \leq \sqrt{u^p(u+1)^p}\text{, for } -1\leq p\leq 0, u\geq 1.$$ This resembles ...
It follows from Whitney extension theorem that for every closed set $C \subseteq \mathbb{R}^n$ and for every $k \geq 1$ there exists a function $f \in C^k(\mathbb{R}^n)$ such that \$ C = \{x : f(...