Questions tagged [functional-equations]

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Imaginary delay

I'm looking at a particular delay differential equation: Suppose we have $x'(t) = x(t - i \pi),$ with initial condition $\phi(t) = x_0(t),$ where $i \pi$ is imaginary. Well, uh, how do we derive a ...
CheeseBlues's user avatar
-1 votes
0 answers
77 views

What are the process and fundamentals of solving delay differential equations?

I have a lot of confusion around delay/difference differential equations, I'm hoping someone can clear it up. This is a very niche, specialized subject, so I need to ask here among people who have ...
CheeseBlues's user avatar
1 vote
0 answers
81 views

Invariant polynomials under a non-standard group action

There is a whole theory of finding the invariant polynomials for matrix groups $\Gamma$ acting on the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$. I would be interested in finding invariant ...
Jan-Willem van Ittersum's user avatar
0 votes
2 answers
306 views

How can I derive functional properties of (the solutions of) this simple functional differential equation?

I've not yet finished a course in functional analysis so I'm unsure how to go about this, but I've always been fascinated by a simple functional differential equation I concocted for almost no reason. ...
CheeseBlues's user avatar
1 vote
1 answer
162 views

Is there a systematic procedure to Solve Abel's, Böttcher's, or Schröder's Equation

I've been interested greatly in the study of functional equations for some time now, I've learnt many different techniques for their solution. Currently I have been studying superfunctions and ...
Anthony Corsi's user avatar
1 vote
0 answers
25 views

Hardy type inequality with singular weights

Recall that Hardy's inequality involving distance from the boundary of a convex set $\Omega \subsetneq \mathbb{R}^n ; n \geq 1$, asserts that $$ \int_{\Omega}|\nabla u|^p \, d x \geq\left(\frac{p-1}{p}...
Davidi Cone's user avatar
1 vote
0 answers
28 views

Rätz orthogonality and involution

In the Rätz’s sens of orhtoganality, can we find an exemple of an involution u(different to -Id)such that x orthogonal to y then x orthogonal to u(y)
MOHAMED TALLA's user avatar
11 votes
2 answers
802 views

What are the iterates of $x \mapsto 1 - \sqrt{1-x^2}$?

Let $f\colon [0,1]\to[0,1]$ be given by $f(x) = 1-\sqrt{1-x^2}$, i.e., the increasing auto-homeomorphism of $[0,1]$ whose graph is a quarter circle centered at $(0,1)$. I am interested in what can be ...
Gro-Tsen's user avatar
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4 votes
0 answers
110 views

$f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer

Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ : $$f(n) = \frac{n^2 + n + 4}{2}$$ so $$ \begin{split} f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\ f(...
mick's user avatar
  • 677
10 votes
2 answers
584 views

Proving the simple form of a function from statistical mechanics

I have discovered a pertinent solution to my problem in the article On the Kinetic Theory of Rarefied Gases by Harold Grad and the book Thermodynamik und Statistik by Arnold Sommerfeld, both of which ...
LuckyJollyMoments's user avatar
0 votes
1 answer
89 views

Finding minimal $\gamma$ that satisfies the integral equation

I have a function $f(z) \in [0,\infty]$ that satisfies $-1 < f(z) < 0$. I would like to find the minimal $\gamma$ that satisfies: $$ \int_0^{\gamma} f(z)dz = \log(1+f(0)).$$ Clearly, I cannot ...
nir's user avatar
  • 101
1 vote
0 answers
47 views

Can I apply $q$-Lagrange Inversion formula?

Now I have equation $F(x) = x \sum_{k\ge 0} g_k F(x) F(qx) \cdots F(q^{k-1} x)$, I need to get the coefficient of $x^n$ in $F(x)$, can I apply $q$-Lagrange Inversion formula to this? Moreover, I have ...
alpha1022's user avatar
8 votes
1 answer
461 views

$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?

While talking about tetration with my friend the following idea (re)occured. $$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$ or variations of it like the weaker $$f(f(f(f(z)))) = z ,\...
mick's user avatar
  • 677
0 votes
0 answers
98 views

Numerical approaches to functional equations

I'm interested in finding numerical approaches to solving functional equations such as f(xy)=f(x)+f(y), where the equations had no derivatives or integrals, and contains arguments involving x and y . ...
Doug Brunson's user avatar
4 votes
2 answers
223 views

Can every symmetric function be factorized through symmetric polynomials?

A symmetric function is a function $f:\mathbb R^n\to \mathbb R$ such that $f(x_1,\ldots,x_n)=f(\sigma(x_1,\ldots,x_n))$ for every permutation $\sigma\in S_n.$ The most commonly encountered symmetric ...
Nick Belane's user avatar
0 votes
0 answers
62 views

How to solve a system of polynomial equations which are mostly factored?

If I have a system of $k$ equations $$ \begin{cases} F_1(x) = 0,\\ \quad\vdots \\ F_k(x) = 0. \end{cases} $$ with $$ F_i(x) = c_i + \prod_{j=1}^n f^i_j(x)\quad i=1,\ldots,k $$ where $f^i_j: \mathbb{R}...
Allan Henriques's user avatar
2 votes
0 answers
58 views

Methods for holonomic recurrences

I wanted to ask if anyone knows of good texts/resources on methods for solving holonomic recurrence relations (if there are any general analytical approaches): $$p_1(n)a(n)+p_2(n)a(n-1)+\dotsb+p_k(n)a(...
Doug Brunson's user avatar
4 votes
2 answers
306 views

Solving the functional equation $2f(x)=f(x+a_n)+f(x-a_n)$

Let $a_n$ be a sequence of strictly positive real numbers such that $\lim_{n \to \infty}a_n=0$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ that admit primitives (i.e. there exists a function $F:...
Shthephathord23's user avatar
9 votes
4 answers
2k views

How may I find all continuous and bounded functions g with the following property?

Find all continuous and bounded functions $g$ with : $$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$ I have posted this question here, but received no answer.
Dattier's user avatar
  • 3,573
2 votes
2 answers
242 views

Non-exponential functions $f(x)$ satisfying $f(x+c)=\gamma(c)f(x)$

Question: what can be said about the existence of functions \begin{align} f:x\mapsto f(x)&\implies x+c\mapsto \gamma(c)f(x)\\ f(x)\ne f(y)&\iff\frac{f(x)}{\gamma(x)}\ne\frac{f(y)}{\gamma(y)} \...
Manfred Weis's user avatar
  • 12.3k
2 votes
0 answers
56 views

any ideas on how to solve matrix equation like this $X A_i Y = B_i$

the objective function is like $$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix I've tried gradient-descent, but as it turns out not well, I wonder if ...
Cup Y's user avatar
  • 21
0 votes
0 answers
68 views

How are eigenvalues of two psd kernels related?

Suppose $K(x,x')$ and $R(y,y')$ are two positive semi-definite kernels on $(x,x')\in \mathbb {X}\times\mathbb{X}$ and $(y,y')\in\mathbb{Y}\times\mathbb{Y}$, respectively, and satisfying the following ...
mathlover's user avatar
0 votes
1 answer
46 views

Symmetric and nearly additive bounded functions

Let $(y_n)_{n\ge 1}$ be a sequence with values in $(0,1)$ such that $\lim_n y_n=1$. Let also $f: [0,1]\to \mathbb{R}$ be a bounded function such that $f(0)=0$ and satisfies $$ \forall n\ge 1, \forall ...
Paolo Leonetti's user avatar
3 votes
1 answer
228 views

A pexiderization of the sine addition law on semigroups

Can we solve the follwing functional equation $$f(xy)=g(x)h(y)+g(y)h(x)$$ on semigroups for unknown complex valued functions $f,g,h$ ?
Aserrar Youssef's user avatar
2 votes
2 answers
249 views

One question about a specific first-order differential equation

Find the function-constant pairs $\langle f(x),c\rangle$ that satisfy the differential equation below: $$f'(x)=f(x+c),$$ where $c \in \mathbb{C}$ and $f(x) \in \mathbb{C}$. I found two families of ...
user369335's user avatar
4 votes
2 answers
368 views

Is there a recurrence for the coefficients of the Laurent series expansion of $\frac{1}{1-e^{e^x - 1}}$?

So there's an elementary (but in my opinion quite interesting!) result which is that the Laurent series expansion of $$\frac{1}{1-e^x} = -\frac{1}{x} + \frac{1}{2} - \frac{1}{12}x - \frac{1}{720}x^3 \...
Sidharth Ghoshal's user avatar
0 votes
1 answer
35 views

A set of equations for two (cumulative distribution) functions and their inverses

I am trying to characterize a set of distributions that satisfy two conditions. It is easy to characterize distributions fitting each of those conditions separately, but I am unable to make progress ...
Martin Modrák's user avatar
1 vote
0 answers
51 views

Functional approximation with derivatives

I am trying to solve a functional approximation problem. Consider a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm x}$ and ...
can't stop me now's user avatar
4 votes
2 answers
245 views

A functional inequality which calculates the limitation of human eyes

Find all pair of function $f^-,f^+:[0,1]\rightarrow[0,1]$ such that: (1)$f^-(x)\leq x\leq f^+(x)$. (2)$f^-(x)+f^+(1-x)=1$. (3)$f^-(x)f^-(y)\leq f^-(xy)\leq f^-(x)f^+(y)$. (4)$f^+(x)f^-(y)\leq f^+(xy)\...
Veronica Phan's user avatar
1 vote
1 answer
107 views

Spectral family associated with the Laplacian operator in $L^{2}(\mathbb{R}^{n})$ [closed]

Let $\lambda>0$ be given. Define $$G_{\lambda}(\xi) = \chi_{_{\lbrace |\xi|^{2} \leq \lambda \rbrace }}. $$ and $$ E_{0}(\lambda)f = \mathcal{F}^{-1}[G_{\lambda}(|\xi|^{2})\mathcal{F}(f)], \ \ f \...
user253963's user avatar
0 votes
0 answers
72 views

Solving the integral equation $y(x)=e^{-x^2/2}+\lambda\int^{+\infty}_{-\infty}e^{-i(x-z)}y(z)dz$

Could you please help me to solve the following integral equation? $$y(x)=e^{-x^2/2}+\lambda\int^{+\infty}_{-\infty}e^{-i(x-z)}y(z)dz$$ I tried to turn the exponentiential term into its trigonometric ...
MATAKA's user avatar
  • 1
8 votes
2 answers
642 views

Solving functional equation $f(xy)=f(x+y)$ and Diophantine equations

It is well known that the only solution is $f$ a constant function. However, by putting some restrictions on the functional equation, we might get other solutions, with potential implications to ...
Vincent Granville's user avatar
8 votes
1 answer
414 views

Functional equation with Fourier transform and $\frac{1}{x} f(\frac{1}{x}) $

What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$, they satisfy following functional equation: $$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$ $\...
Bertrand's user avatar
  • 1,101
10 votes
3 answers
822 views

Progress in robustifying mathematics - i.e. making mathematical theorems robust to small changes in hypotheses

The idea of making a mathematical theorem robust to small changes in its hypotheses has been known for some time. In areas such as group theory reasonable progress has been made leading to the theory ...
Ivan Meir's user avatar
  • 4,590
3 votes
0 answers
224 views

$f(x)>0$ and $f(y)>0$ implies $f(x+y)>0$, then there must exist an linear function $g$ such that $g(x)>0$ iff $f(x)>0$?

Background: Let $x,y\in\mathbb (0,+\infty)^n$. $f$ is a continuous function on $\mathbb R^n_+=(0,+\infty)^n$. Consider the following condition (1), the sign of $f(x+y)$ is dependent on the sign of $f(...
High GPA's user avatar
  • 263
5 votes
2 answers
286 views

A functional equation for a family of functions indexed by the symmetric group $S_3$

$\newcommand{\C}{\mathbb C}$A question asked recently was as follows: For the symmetric group $G:=S_3$, is it possible to construct functions $t_g\colon\C\to\C$ that satisfy the convolution identity ...
Iosif Pinelis's user avatar
3 votes
1 answer
156 views

A question about decompositions of rational functions

Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such ...
KhashF's user avatar
  • 2,227
2 votes
1 answer
199 views

Largest asymptotic growth for $2f(x)-f(2x)$

I am trying to find smooth functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that the quantity $$\Delta_f(x) := 2f(x)-f(2x)$$ is positive for $x$ large enough and has the greatest asymptotic growth. ...
cs89's user avatar
  • 948
6 votes
1 answer
565 views

Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?

Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via $$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...
Ali's user avatar
  • 3,883
3 votes
0 answers
185 views

Fermat's Last Theorem (FLT) in standard model space corresponding to an infinite Blaschke product

Let $u$ be an inner function and denote by $H^2$ the Hardy space on the open unit disc D. A model space $K_u$ associated to $u$ is a Hilbert space of the form $K_u=(uH^2)^⊥$ where ⊥ denotes the ...
Ridwane El Mellass's user avatar
0 votes
0 answers
102 views

What are the functions such that $ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^p$?

Let $1 \leq p \leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g \in L_p(\mathbb{R})$ such that $$ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^...
Goulifet's user avatar
  • 2,142
0 votes
1 answer
169 views

Is it possible to numericaly solve functional equation

Given a functional equation of form $f(f(x))=T(x)$ is there any good ways to solve it numerically? If not then at least approximate in some small region $x\in(-a;a)$. E.g. with the equation $f(f(x))=x+...
Warmist's user avatar
2 votes
0 answers
142 views

Square root of a function on a finite set

Let $S$ be a finite set and $f \colon S \to S$ be an arbitrary function. How can we find all functions $g \colon S \to S$ with $f = g \circ g$? If $f$ and $g$ are both required to be invertible, the ...
rimu's user avatar
  • 607
2 votes
2 answers
183 views

Functional equations and normal distribution

Let $\alpha \neq 1.$ If $X,Y$ are two independent random variable such that $U=X+Y$ and $V=X+\alpha Y$ are independent, then $X$ and $Y$ are normally distributed. In term of characteristic functions ...
Kurt.W.X's user avatar
  • 249
18 votes
3 answers
4k views

Is there a general solution for the differential equation $f''(x) = f(f(x))$?

I'm currently an undergraduate studying differential equations and I've been fixated on the differential equation $f''(x) = f(f(x))$ for the past 2 days. I can't seem to crack it but it feels like it ...
Trevor Krumrine's user avatar
10 votes
2 answers
452 views

A functional equation involving the inverse function

$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...
Iosif Pinelis's user avatar
0 votes
1 answer
140 views

Solve $(x-a)^{\alpha +1} - \lambda*(b-x)^{\alpha + 1} = C(\frac{a+b}2 - x)^{\alpha}$ over $\mathbb R$ [closed]

I have been having trouble solving the following equation. Any help would be appreciated. Let a,b,C,$\alpha$,$\lambda$ be real numbers with $C < 0$, $0 < \alpha < 1$, $\lambda > 1$. We ...
cata's user avatar
  • 337
0 votes
0 answers
192 views

Functions that satisfy a reverse triangle inequality: do they have a name?

Let $f : \mathbb{R}^n \to \mathbb{R}_+$ satisfy $$f(a) - f(b) \le C f(a - b)$$ $\forall a, b \in \mathbb{R}^n$ for some $C \ge 0$. Is there a name for such functions? (I would be happy to have a name ...
user27182's user avatar
  • 315
2 votes
1 answer
164 views

Existence of a distinguished continuous version of the logarithm of a continuous function

Let $E$ be a $\mathbb R$-Banach space and $\varphi\in C^0(E,\mathbb C\setminus\{0\})$ with $\varphi(0)=1$. I want to show that there is an unique $\psi\in C^0(E,\mathbb C)$ with $\psi(0)=0$ and $$\...
0xbadf00d's user avatar
3 votes
0 answers
358 views

Is the study of additive functions dead?

I am learning more about the theory of additive functions ($f(nm)=f(m)+f(n)$) and I am struck by how powerful the theorems given are. For example, we have a complete characterization of not only what ...
Milo Moses's user avatar
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