The functional-equations tag has no usage guidance.

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### Characterization of Rate Distortion Function (minimization of functional over probability mass function)

I am reading the book from T. Cover, Elements of Information Theory and found the equation 10.117 - Functional of the Rate Distortion Function. The functional is:
$J(q) = \sum_{x} \sum_{\hat x} ...

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**1**answer

164 views

### Cauchy-like functional equation $f(h(y)x+y)=g(y)f(x)+f(y)$

I am looking for the solution to the following two variable functional equation:
(*) $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$
where:
$h$ is some given continuous function,
$f, g,$ unknown functions on ...

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98 views

### Homomorphic Commutator? Equation

So I was considering the following functional equation:
Given $H :\Bbb{C}^2 \rightarrow \Bbb{C} $ find $\theta: \Bbb{C}^2 \rightarrow \Bbb{C}$ such that
$$ \theta(H(a,b), H(c,d)) = H(\theta(a,c), ...

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88 views

### Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form
$$
g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i ...

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56 views

### Existence of a solution to $xf(x) = \int_0^1 k(x,y) yf(y) dy$

Crosspost - I asked very similar question on math.sx.
Let $X = (0,1)\times (0,1)$ and $k\colon X \to \mathbb{R}$ be a Lebesgue measurable non-negative function such that
$$ \int_0^1 k(x,y) dy = ...

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72 views

### Energy Oscillations in a One Dimensional Crystal

Good day!
Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)?
article, that I have
Especially ...

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**1**answer

183 views

### Survey on functional equations and inequalities

Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both inside and outside mathematics)?
...

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109 views

### Distributive law between Kleisli triples

A distributive law of a monad $S$ over a monad $T$ is a natural transformation $l : T S \to S T$ such that:
$l \circ T \eta^S = \eta^S T$
$l \circ \eta^T S = S \eta^T$
$\mu^S T \circ S l \circ l S = ...

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**1**answer

70 views

### Functions with special separability

Suppose we have differentiable functions $F$, $f_1, \dots, f_n$, and $g_1, \dots, g_n$ satisfy the following relation
$$ F(x+y) = \sum_{i=1}^n f_i(x) g_i(y).$$
What are the possible forms of $F$?

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**1**answer

76 views

### Uniqueness of solutions of functional equations [closed]

A solution to $f(2x)=\alpha f(x)$ with a boundary condition $f(\beta) = \beta$ is
$$ f(x) = \left( \frac{\beta}{\alpha^{\log_2 \beta}} \right) \alpha^{\log_2 x}. $$
Do we know whether or not the ...

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**1**answer

150 views

### Existence of solution for this set of polynomial equations

We are given a number $n$ and a vector $p=(p_1,p_2,\ldots,p_r)$, where
$$p_1\geq p_2 \geq \ldots \geq p_r > 0 ; \ \ \ \ \sum_{i\in [r]} p_i \leq 1$$
I'm interested in proving that a solution for ...

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324 views

### Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)

We are given $n\in \mathbb N^+$ and $p\in[\frac{1}{2},\frac{n+1}{n+2}]$.
Our goal is to find $t\in[0,1]$ such that
$$(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i$$
Is there a closed-form ...

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325 views

### Consistent price index

This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem.
Let ...

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**1**answer

126 views

### Counterexample for the Generalized Associativity Equation

The Generalized Associativity Equation is given by
$$ F(G(x,y),z)=K(x,H(y,z)),$$
where the functions $F,G,H$ and $K$ are all from $\mathbb{R}^2$ to $\mathbb{R}$. In his book "Lectures on Functional ...

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68 views

### Archimedean $\varepsilon$-factors

Let $K$ be either $\bf R$ or $\bf C$. Let $p$ and $q$ be integers with $p \leq -1$, $q \geq 0$, and $p+q=-1$. Consider the Hodge structure $M = M(p,q)$ over $K$ with coefficients in $\bf R$, defined ...

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44 views

### Family of (Cumulative Distribution) Functions

I'm looking for a 2 (or more)-parameter family of functions $F$ with the following properties:
For each $f \in F$, $f(0)=0$, $f(1)=1$, and $f$ is (weakly) increasing.
$F$ is closed under products.
...

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166 views

### Solve this functional equation with respect to $f$

Let $v\not= 1$ be a real number. Let $f(s)$ be real analytic on an open interval containing $v$ and $1$, with a zero of order $m\ge 1$ at $s=1$.
My question is: Can we solve this functional equation ...

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**1**answer

335 views

### A functional inequality

$g:[0,1]\to[0,1]$ continuously differentiable and increasing such that for
all integers $t>0$ and for all $r\in(0,1)$, $g(r^{t+1})>g(r)\cdot g(r^t)$. Does this imply
that for all ...

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**2**answers

154 views

### Solving a functional equation

I would like to consider the following simple problem. I want to find two functions $f,g : \mathbb R \to \mathbb R$ such that, being given a collection $(h_v)_{v\in V}$ of real functions indexed by ...

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395 views

### solution of functional equation $f^{\circ k}(x) = x$

The equation $f^{\circ k}(x) = \mathrm{Id}$ for $x\in E$ is called the Babbage equation and the general solution is given in the following way [M. Kuczma, Functional equations in a single variable]:
...

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218 views

### General additive function of probability

Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that:
$H$ is continuous,
$H$ is symmetric w.r.t. the order of its arguments,
...

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123 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 ...

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**1**answer

266 views

### Is there any mathematical study about |a-b|=f(|g(a)-g(b)|)? or does there exist f() and g() satisfy this equation? [closed]

The problem is just as the title. It is clear that the linear function $f(x)=kx$ and $g(x)=(1/k)x$ can meet it. Is there any other function pairs f(x) and g(x) can meet this equation? or the equation ...

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615 views

### Relating the roots of polynomials to the solution sets of certain functional equations

Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...

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177 views

### Unusual Differential Equation for CDF

Consider the following differential equation
$$F(cx) = F(x) + x F'(x)$$
for $c>1$.
Does this differential equation belong to a some well known class?
Is there a way to find all the solutions ...

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**1**answer

180 views

### How to solve for the nonlinear functional equation? [closed]

I got a nonlinear functional equation like:
$f(x) = g(x) + h(f(Ax))$,
where $A$ is a constant, $x$ is a scalar, $g()$ and $h()$ are given. The objective is to solve for the expression of $f(x)$.
...

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**1**answer

115 views

### A Recursive Maximization Problem

Let $A\ge B>0$ be real constants. I say that a function $f:[0,1]\rightarrow[0,1]$ satisfies the $(A,B)$-condition if for all $p\in [0,1]$, the expression
$$q(A-Bp-Bf(q))$$
is maximized (not ...

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**1**answer

232 views

### An Integral Functional Equation

Let $f$ be a non-negative function supported and integrable on the positive real axis, such that
$$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$
where $c[p]$ a number (functional) dependent on function ...

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**1**answer

447 views

### A Differential Equation with Nested Functions

This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems.
I would like to know whether there exists a differentiable function from the ...

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111 views

### A Convolution Integral Equation

Is there any close-form solution for a function $f(t)$ satisfied the below equation:
$f(t)=g(t)+\frac{1}{t^2}(h(t)*f(t))$. Operator $*$ is convolution integral, and $g(t)$ and $h(t)$ are known ...

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**0**answers

327 views

### The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is ...

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367 views

### Functional equations

What are the general solutions of the functional equations?
$$
f(x,y)+f(y,z)=\frac{1}{f(x,z)}
$$
$$
f(x,y)f(y,z)f(x,z)=1
$$

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**1**answer

236 views

### Some functional equations in two variables

I have two questions.
i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation
$$
|f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|}
$$
has a solution ...

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**1**answer

160 views

### Is still it weakly continuous ？

If $\{u_n\}$ is bounded in $H$（real Hilbert space）with inner product such that $(\cdot,\cdot)$, then ${\|u_n\|^2u_n}$ is bounded also. Passing a subsequence, one has that $\{\|u_n\|^2u_n\}$ converges ...

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91 views

### Solving Multivariate and multi power Equations

There are 84 equations,
$r-A_DD^5-5A_DD^4D_i-(a_{d_i}+2b_{d_i}D_i)+(1-\alpha)\lambda_i=0,$
$A_L/L-r-(A_L/L^2)L_i-(a_{l_i}+2b_{l_i}L_i)-\lambda_i=0,$
$(1-\alpha)D_i-L_i=0$
where $i=1,\cdots,28$, ...

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488 views

### Series defined by a fixed-point functional equation

Description
I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here ...

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151 views

### Critical case linear autonomous functional differential equation

I am looking for asymptotic ($t\to\infty$) behavior of the general
solution $g(t)$ to a following linear functional differential equation
$$
\text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t)
$$
with ...

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**3**answers

936 views

### Rational functions with a common iterate

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...

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**1**answer

363 views

### Hahn-Banach theorem with real extended valued function

Hello to everyone,
My problem is the following: I have this version of the Hahn-Banach theorem:
Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any
convex function. Let $W$ be a vector ...

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**3**answers

184 views

### References for functional equations in more general settings than the reals

Hi there -
I'm Manny, a soon to be MSC thesist. I'm looking for a subject to write my thesis about - and recently I was caught by functional differential equations. Is there any neat reference for ...

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956 views

### Techniques to solve equations involving a definite integral [closed]

Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, ...

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896 views

### Which trigonometric identities involve trigonometric functions?

Another question that's getting no answers on stackexchange:
Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled ...

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1k views

### Finding f such that f(f(x))=g(x) given g

Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...

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368 views

### solvability of an elementary functional equation

Is there some other way to characterize the functions $f:\mathbb Z\times \mathbb Z\to \mathbb Z$ which are expressible as
$$f(x,y)=g(x)+g(y)-g(x+y)$$
for some $g:\mathbb Z\to\mathbb Z$?
Easy facts: ...

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**1**answer

3k 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 ...

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721 views

### Restriction of a linear functional equation to surface of a sphere

Let $f_i : R \rightarrow R$ and $g_j: R \rightarrow R$ be unknown functions, for $i = 1, \cdots, N$ and $j = 1, \cdots, K$. Let $A$ be a $K \times N$ matrix whose columns are unit-length vectors ...

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808 views

### Are there functions satisfying the following integral condition?

Can we find two functions $f$ and $g$ that are reasonably defined nontrivial(not everywhere zero, $f\neq g$, not linear polynomials) functions such that the following condition is satisfied?
$$ f( ...

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5k views

### Does any research mathematics involve solving functional equations?

This is a somewhat frivolous question, so I won't mind if it gets closed. One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those ...

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**10**answers

7k views

### The functional equation $f(f(x))=x+f(x)^2$

I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$
(so $c_0=0$ is imposed).
First things that ...

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**1**answer

522 views

### Is this method of “fractional sums” using a Fourier series viable?

Hi.
I have this idea about developing what I call a "continuum sum", that is, a method to "add up a non-integer number of terms", i.e. to see if there is a "natural" way to assign a meaning to the ...