# Questions tagged [real-analysis]

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

4,328
questions

6
votes

1
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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{\...

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. ...

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 $b\in\mathbb R$, ...

1
vote

0
answers

33
views

### 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 ...

15
votes

1
answer

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### 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 ...

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}{...

0
votes

0
answers

66
views

### 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}(\...

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 ...

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{...

-1
votes

0
answers

57
views

### Solving an equality [closed]

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

4
votes

0
answers

143
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### 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}...

0
votes

0
answers

29
views

### 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
...

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 ...

1
vote

1
answer

91
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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-...

-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,...

3
votes

0
answers

142
views

### 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 ...

3
votes

1
answer

64
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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 $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ ...

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 ...

2
votes

0
answers

77
views

### 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 ...

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 - ...

1
vote

0
answers

28
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### 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 &...

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 (...

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 ...

1
vote

0
answers

37
views

### 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 ...

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}{(\...

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^\...

1
vote

1
answer

38
views

### 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 ...

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 ...

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{...

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(...

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 ...

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^{-...

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 $M : L^1 \to L^{(1,\infty)}$ ...

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 \...

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 ...

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{...

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 ...

1
vote

0
answers

109
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### 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\...

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}}{\...

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 $m\times m$ ...

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 \...

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 ...

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}^{...

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 $n,a,b$ ...

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): ...

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}\...

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 ...

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 ...

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 ...

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 ...