Questions tagged [functional-equations]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
23 views

Maximizing the integral of a transformation that depends on a neighborhood of values of the original function

I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one. We are working with non-negative real functions over a sufficiently nice region ...
1 vote
1 answer
58 views

Finding closed form roots for pseudo-trinomial

I have the below function: $$\pi(x) = \frac{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot r_1}{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) ...
2 votes
1 answer
528 views

Another functional inequality

Is there some general solution to the functional inequality: $$ f(xy) \leq y f(x) + x f(y)$$ Where $x,y\in[0,1]$? I can find many particular solutions but I just wonder if there is a more general ...
0 votes
2 answers
126 views

Cauchy's functional multiplicative equation on the unit interval

This question might be trivial, but I didn't find a clean reference and have not attempted to prove it myself yet: Let $f:[0,1]\rightarrow [0,1]$ be a continuous and monotonic function such that $f(0)=...
6 votes
1 answer
156 views

A second-order recursion (functional equation)

In a calculation of some momenta of random matrices (GOE), I encounter a functional equation, in the form of a second-order recursion, $$L(s+1)=L(s)+2s(2s+1)L(s-1).$$ Is it familiar to someone ? Is ...
1 vote
0 answers
112 views

Building representation of an arbitrary umbral calculus

Consider a set of integrable functions on the interval $(0,1)$. Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function). In such system the ...
1 vote
0 answers
54 views

Characterization of an integral operator with a Bessel kernel

I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$ I am ...
1 vote
1 answer
126 views

Functions for which $\lambda f(x)=f(\alpha_\lambda x + \beta_\lambda)$

How many functions $f:\mathbb{R}\to R\subseteq\mathbb{R}$ are there such that there is an $S\subseteq\mathbb{R}$ containing an interval containing $1$ such that for any $\lambda\in S$ there exist $\...
0 votes
0 answers
145 views

Do we have tetration uniqueness by $ A = \inf \sum_n a_n^2 $?

Let $f$ be a real analytic (on at least $|x|<2$) and real solution of the functional equation $f(0) = 1,f(x+1) = \exp(f(x))$. For the existence of such $f$, see here. Then $$ f(x) = \sum_n a_n x^n ;...
3 votes
2 answers
600 views

Are there any functions $f$ beyond trivial examples where $\int f(x +f(x + f(x +\dotsb)))\,dx$ $= F(x+F(x+F(x +\dotsb))) + C$ for some function $F$?

Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a ...
1 vote
2 answers
236 views

Recurrence relation with two variables

I am stuck on a recurrence relation with two variables. I'm familiar with techniques to solve recurrence relations with one variable and looked into ways to solve recurrence relations with multiple ...
4 votes
0 answers
77 views

A functional equation: Functional families that are "weakly" closed under product?

Suppose that for any real number $a$, we have a function $f_a:\mathbb R \to \mathbb R$ or such that $f_a(x)$ is monotonically strictly increasing in $x$ and hence invertible on its image. We also ...
0 votes
2 answers
334 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. ...
1 vote
0 answers
115 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 ...
1 vote
1 answer
197 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 ...
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 ...
1 vote
0 answers
34 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}...
8 votes
1 answer
481 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 ,\...
11 votes
2 answers
829 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 ...
1 vote
0 answers
29 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)
4 votes
0 answers
119 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(...
10 votes
2 answers
597 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 ...
3 votes
1 answer
235 views

A $GL_1$ Voronoi formula

I want a functional equation for the function defined by the Dirichlet series, $$ D(s,a/q)= \sum_{n=1}^\infty \frac{e^{2\pi i n a/q}}{n^s}. $$ which sends $s$ to $1-s$ and preferably sends $a$ to $\...
1 vote
0 answers
50 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 ...
0 votes
0 answers
158 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 . ...
4 votes
2 answers
247 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 ...
2 votes
0 answers
67 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(...
4 votes
2 answers
341 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:...
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.
2 votes
2 answers
247 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)} \...
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 ...
0 votes
1 answer
48 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 ...
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 $f,g,h$ ?
2 votes
2 answers
255 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 ...
4 votes
2 answers
388 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 \...
2 votes
1 answer
533 views

What kind of uniqueness can I conclude for solutions to a simple functional equation?

I'm going to ask a very vague question, and then give specifics for the version I particularly care about. I'm interested in answers at all levels of vagueness. At the most vague version, I am in ...
0 votes
1 answer
52 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 ...
1 vote
0 answers
54 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 ...
27 votes
1 answer
7k views

Are there any non-linear solutions of Cauchy's equation $f(x+y)=f(x)+f(y)$ without assuming the Axiom of Choice?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be s.t. $f(x+y) = f(x) + f(y), \ \forall x, y$ It is quite obvious that this implies $f(cx)=cx$ for all $c \in \mathbb{Z}$ and even further: $\forall c \in \...
4 votes
2 answers
246 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)\...
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 \...
0 votes
0 answers
87 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 ...
8 votes
2 answers
687 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 ...
28 votes
4 answers
2k views

For a continuous function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ does $(f(x)-f(y)) (f(\frac{x+y}{2}) - f(\sqrt{xy}))=0$ imply that $f$ is constant?

Suppose that $f: \mathbb{R}^+ \to \mathbb{R}^+$ is a continuous function such that for all positive real numbers $x,y$ the following is true : $$(f(x)-f(y)) \left ( f \left ( \frac{x+y}{2} \right ) - ...
8 votes
1 answer
454 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: $$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$ $\...
10 votes
3 answers
840 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 ...
5 votes
2 answers
296 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 ...
3 votes
0 answers
226 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(...
3 votes
1 answer
161 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 ...
2 votes
1 answer
202 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. ...

1
2 3 4 5