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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Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?

I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals) $$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\ g(x)=\frac{\...
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  • 61
1 vote
0 answers
159 views

Function uniquely determined by its values at integer arguments

A smooth enough, slow growing real-valued function $f(x)$, is uniquely determined by its values at integer arguments. I don't remember the name of the theorem and the conditions for this to be true. ...
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3 votes
1 answer
90 views

Question on the existence/uniqueness of the fixed point

Let $E$ a Banach space ($E$ is the space of continuous functions on $[0,T]$ for my case). Let $F, G: E\times E\to E$ be contraction maps of contraction constant $\epsilon>0$. Given $b\in\mathbb R$, ...
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1 vote
0 answers
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Help with a surface of delay differential equations

This question is difficult for me to phrase, as it's very much outside of my mathematical purview. This is a question which intersects directly with my research, but as I work predominantly in ...
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15 votes
1 answer
3k views

Did Euler know (unconsciously) to integrate by differentiating?

Considering a method to find the anti-derivative of an (sufficiently smooth) real function by differentiating published some years ago (equation (48) in Kempf et al., New Dirac Delta function based ...
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2 votes
0 answers
74 views

Maximal function to high power

Consider the following maximal function : in dimension $n$ consider $B(0,1)\subset \mathbb{R}^n$ the unit ball, if $f\in L^1(B(0,1))$, $\alpha\geq 0$ : $$ M_\alpha f : x \mapsto \sup \left\{ \frac{1}{...
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  • 111
0 votes
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Parseval identity extension?

I have stumbled upon the following three-dimensional series: $$\Lambda_p = \sum_{\underline{n}} \left(\frac{\left|n_1\right|}{\left|\left|\underline{n}\right|\right|_2}\right)^p \left|\hat{f}(\...
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  • 21
0 votes
1 answer
96 views

Construction of holomorphic function

I was trying to construct a holomorphic function $f$ on $\mathbb{C}$ such that $|f|^2(z)=e^{(|z|^2-\frac{1}{2})^2}$. I will be happy if someone can give me an idea how to do that. I would like also ...
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0 votes
1 answer
71 views

Functional relationship between two quantities

Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by $$ \alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{...
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-1 votes
0 answers
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Solving an equality [closed]

Assume that $\int_0^\infty f(t)\cos (xt) dt= \frac{1}{1+x^2}$, what is $f$?
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4 votes
0 answers
143 views

Generalized Jensen's inequality for positively homogeneous functions

The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
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0 answers
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Alternative to the Sampling Theorem / Invertible transform with sampling criteria

I seek a transform $T$ that operates on real-valued $x(t)$, that Is perfectly invertible Has discrete counterpart with continuous reconstructor Provides conditional reconstruction guarantees ...
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3 votes
1 answer
97 views

Expressing a vector valued function in terms of its derivatives

Consider a function $$ f:\mathbb{R}^n\rightarrow\mathbb{R}^m $$ given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$. Does there ...
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1 vote
1 answer
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Non-Fourier complete orthogonal basis?

The Fourier Transform (FT) Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero Is invertible: info-preserving, has inverse function Is energy-...
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-1 votes
0 answers
35 views

Solution existence for fraction equation [closed]

I'm wondering if there is a way to show the existence of a solution for $x$ in this equation $$ \beta = \frac{(1+e^{-x})(\alpha + e^x)}{(1+e^{-\gamma x})(\alpha +e^{\gamma x})} ,$$ where $\alpha,\beta,...
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3 votes
0 answers
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Examples of infinite dimensional involutions

Examples of infinite dimensional involutions I'm looking for more examples of involutions of the type portrayed below, in which two sets of indeterminates (real or complex) each can be transformed ...
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3 votes
1 answer
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More on the inequality $f'(x)/(1-f(x)^2)-1/(1-x^2)\ge0$

A previous question was as follows: Assume that $f\colon[0,1]\to[0,1]$ is a diffeomorphism so that $(f''(x)/f'(x))'<0$ and that $f''(0)=0$. It seems to me that $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ ...
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1 vote
1 answer
157 views

A condition on the inequality $f'(x)/(1-f(x)^2)-1/(1-x^2)\ge 0$

Assume that $f:[0,1]\to [0,1]$ is an diffeomorphism so that $(f''(x)/f'(x))'<0$ and that $f''(0)=0$. It seems to me that $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ on $[0,1]$. But no proof so far. The ...
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2 votes
0 answers
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The Laplace transform and the Lagrange compositional inversion formula

I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
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  • 7,920
2 votes
1 answer
263 views

Prove positivity of rational functions

We say a rational function $F(z)$ is positive if the coefficients of its Maclaurin expansion, in the variable $z$, are non-negative. In this context, let $$F_r(z):=\frac{1 - 2z + z^r - (1 - z)^r}{(1 - ...
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1 vote
0 answers
28 views

Reference for a general theory of spaces of one-directional rays?

There is a lot of work done on projective spaces, over real, complex numbers or over an abstract field. But I do not find a reference for similar theory where the vectors are projected to the same &...
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1 vote
1 answer
97 views

Does Newton-Leibnitz apply to Sobolev space

For a function $u\in W^{1,p}$, we always use a sequence of $C^1$ function to approach it and derive the consequences. However, can we just claim for a.e. x, y: $$u(x)-u(y)= \int_0^1 Du(y+t(x-y))\cdot (...
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7 votes
2 answers
488 views

A counterexample to: $\frac{1-f(x)^2}{1-x^2}\le f'(x)$ — revisited

Can we find a counterexample to the following assertion? Assume that $f:[-1,1]\to [-1,1]$ an odd function of class $C^3$, and assume thaht $f$ is a concave increasing diffeomorphism of $[0,1]$ onto ...
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  • 133
1 vote
0 answers
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distance between two orthogonal projection matrices and its covering number

Let $X, Y \in \mathbb{R}^{n\times p}$ such that $\Vert X- Y \Vert_{HS} \leq \delta$ (Hilbert-Schmidt norm). Also, assume that both $X, Y$ have full column rank. Let the orthogonal prpjection operator ...
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  • 187
0 votes
0 answers
28 views

How to find close form roots or at least good approximation of roots of such function?

I need to solve for $D$: $$ KD^{N-1} S + P = KD^N + \left(\frac{D}{N}\right)^N, $$ where $$ S = \sum_i x_i, \, P = \prod_i x_i, $$ $$ K_0 = P \left(\frac{N}{D}\right)^N, \, K = AK_0 \frac{\gamma^2}{(\...
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2 votes
0 answers
44 views

Smoothness of Radon transform

Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by $$ R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
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  • 5,650
1 vote
1 answer
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About the continuity of the integral on the boundary of a ball

I’m considering a $H^1$ function u on a open domain D. Is the integral: $$ \int_{\partial B_r(x)} u \hspace{2pt}dH^{n-1}$$ continuous with respect to x? I tried to prove that it’s differential by ...
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3 votes
1 answer
89 views

Estimate for an oscillatory integral of the first kind

I am confused in finding the right bound for the following oscillatory integral $$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$ Where $\psi(2^{-k} \xi)$ is a smooth ...
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3 votes
1 answer
108 views

Convex sphere in R^3

Is every convex sphere (in the sense of Alexandorff, which is the boundary of some convex body in $\mathbb{R}^3$) with Alexandorff curvature $\geq 1$, bi-Lipschitz to the unit round sphere in $\mathbb{...
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  • 729
4 votes
1 answer
313 views

Using Young's inequality to show elementary inequality?

Let $p, q\geq 2$, $s\geq p$ and $f,g$ be non-negative smooth enough functions. Then why does the following inequality hold: $$-f^{q-2}g^{s}|\nabla f|^{p}+f^{q-1}g^{s-1}|\nabla f|^{p-1}|\nabla g|\leq C(...
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  • 199
0 votes
0 answers
29 views

Minimum eigenvalue of covariance matrix under probability constraint

Consider a random (column) vector $X\in \mathbb{R}^d$. I am interested in the quantity $$ \Lambda(\alpha)=\inf_{E\in \mathcal{B}(\mathbb{R}^d), \ P(X\in E)\ge \alpha} \lambda_{\min}\left(E[XX'1\{X\in ...
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  • 377
1 vote
1 answer
80 views

Sequence of reals such that $x_{n+1}\leq ab^{n}x_{n}^{1+s}$ converges to $0$?

Let $\{x_{n}\}_{n=0}^{\infty}$ be decreasing sequence of non-negative reals. Suppose that there exist constants $a, s>0$ and $b>1$ such that $$x_{n+1}\leq ab^{n}x_{n}^{1+s}$$ and $$x_{0}\leq a^{-...
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  • 199
3 votes
0 answers
41 views

Maximal function in Orlicz space

Consider the maximal operator defined for a function $f\in L^1_{loc}$: $$ Mf : x\mapsto \sup_{r>0} \frac{1}{|B(x,r)|} \int\limits_{B(x,r)} f. $$ It is well know that $M : L^1 \to L^{(1,\infty)}$ ...
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  • 111
3 votes
1 answer
69 views

Hölder inequality between different Orlicz spaces

If we have a product of functions $fg$ with $f\in L^r$ and $g\in L^s$ for some $s,r>1$ satisfying $1/r+1/s=1$, then we know that $fg\in L^1$. But if $g$ is a little bit more than $L^s$, say $L^s \...
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  • 111
0 votes
0 answers
43 views

Critical points of function-curvature

As a side effect of the COVID-19 pandemic exponential growth became a buzz word that was "copy-pasted" a lot in public discussion. It may be assumed that the general public can't make sense ...
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1 vote
2 answers
136 views

Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?

Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral. $$ I = \int_{\mathbb{R}} \int_{\mathbb{...
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  • 2,463
2 votes
1 answer
123 views

Coefficients of certain Taylor series

For $t\in(-1,1)$, let $$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$ and $$g(t):=\frac1{f(t)}.$$ Note that the functions $f$ and $g$ are even. Question 1: Is ...
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1 vote
0 answers
109 views

examples of function difficult to prove to be $\geq0$?

I have often wondered whether there has ever come a point in your research, when you were confronted with an explicit real function $f(x_1,x_2,\ldots,x_n)$ and an explicitly defined compact set $S\...
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3 votes
1 answer
213 views

One series converges iff the other converges

In Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this sequence of partial sums converges $$ \begin{split} \sum_{1<n\leq N}\frac{a_{n}}{\...
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3 votes
2 answers
142 views

Power of a matrix, largest eigenvalue in absolute value, and convergence acceleration

I want $S^k$, with $S=I-\Lambda^{-1}M$, to tend to zero quite fast as $k\rightarrow \infty$, as this is what drives the convergence in a fixed-point algorithm. Here $M=X^TX$ is a fixed $m\times m$ ...
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1 vote
1 answer
37 views

Lipschitz continuity and quadratic growth in Loewner order

Consider the partial Loewner order $\le_L$ for symmetric matrices: let $A,B$ be symmetric matrices of the same dimension, we say $A\le_L B$ if $B-A$ is positive definite. Now let $f:\mathbb{R}^n\to \...
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  • 423
3 votes
2 answers
220 views

Probability of picking neighbors in $\{1,\ldots, n\}$

Motivation. Swiss license plates consist of $2$ letters indicating the region, followed by a number, such that the pairing (region, number) is unique by car. In the small town where I live, I saw two ...
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1 vote
1 answer
42 views

Weak lower semicontinuity of a sequence of Riemann sums

Let us have a sequence of functions $\{f^K\}_{K \in \mathbb{N}} \in C([0,1],\mathbb{R})$ which is uniformly bounded in $L^2((0,1))$. We observe a sequence of Riemann sums $$R^K=\frac{1}{K} \sum_{k=0}^{...
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1 vote
0 answers
21 views

How to relate this integration with the integral expansion of special functions?

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
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4 votes
1 answer
84 views

Can the integral inherit the Lipschitz continuity of its integrand?

Let $C$ be the set of continuous functions on $[0,T]$ taking values in $[0,1]$. Denote $\|f-g\|_t:=\max_{0\le u\le t}|f(u)-g(u)|$ for $f,g \in C$ and $t\in [0, T]$. Let $\phi: C\times C\times \{(s,t): ...
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  • 938
0 votes
0 answers
24 views

Wigner transform and logarithms

The Wigner transform $W$ of a density matrix $\rho(x,y) = \sum_{j=1}^{\infty} \lambda_j u_j(x) \overline{u_j(y)}$ is given by $$W[\rho] = \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d} e^{-i\xi\cdot y}\...
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15 votes
0 answers
420 views

Quantitative Skorokhod embedding

The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
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  • 651
5 votes
2 answers
124 views

Inverse Mellin transform of 3 gamma functions product

I want to calculate the inverse Mellin transform of products of 3 gamma functions. $$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$ Above contour integral has ...
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  • 59
0 votes
1 answer
44 views

Kernels of sequences of operators

Let $\left( S_{n}^{1}\right) $ and $\left( S_{n}^{2}\right) $ two sequences of operators in $\mathcal{L}(E_{1},F_{1})$ and $\mathcal{L}(E_{2},F_{2})$ where $E_{i},F_{i},i=1,2$ are Hilbert spaces such ...
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  • 525
2 votes
0 answers
280 views

Distribution of $\frac{(\sin(n))^2}{2^n}$ in dyadic intervals?

Good morning all, I was wondering what kind of methods could help in order to tackle the following problem : Define the set $A = \left\{ \frac{(\sin(n))^2}{2^n}\right\}$ for $n$ integer. So A is a ...
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