# Tagged Questions

The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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### Sequence of *compact* operators that converges to a bounded linear operator $K_{\lambda}$

Proposition: Fix $1\leq p\leq \infty$ and a bounded sequence of real number $\lambda=(\lambda_i)_{i\in\mathbb{N}}$ and $e_i=\delta_{ij}\in l^p$. Defined the bounded linear operator ...
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### Find all solutions to $f\left(x^2+xf(y)\right)=xf(x+y)$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^2+xf(y)\right)=xf(x+y)$$ for all $x,y\in\mathbb{R}$. This is somewhat related to this question, but with an $xf(y)$ term instead ...
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### The Bi-linear Functional Equation $\sum_{i=1}^{n}f_i(x)g_i(y)=0$

Consider the following so-called bi-linear functional equation $$\sum_{i=1}^{n}f_i(x)g_i(y)=f_1(x)g_1(y)+f_2(x)g_2(y)+\cdot\cdot\cdot+f_n(x)g_n(y)=0 \tag{1}$$ where $f_i(x)$ and $g_i(y)$ are ...
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### How can I know how many real roots this equation has?

How many real roots does $2 \sin x-x=0$ have?
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### The function $f$ has property: $f(x)+f(1/x)=x$ and what is the largest set of real numbers?

Problem. $f(x)+f\left(\frac{1}{x}\right)=x$ and we need to figure out the largest set of real numbers that can be the domain of $f$. My steps: Step 1. I substituted $x$ with $1/x$ to replace the ...
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### Function that is both midpoint convex and concave

Which functions $f:\mathbb{R} \to \mathbb{R}$ do satisfy $$f\left(\frac{x+y}{2}\right) = \frac{f(x)+f(y)}{2}$$ for all $x,y \in \mathbb{R}$ I think the only ones are of type $f(x) = c$ for some ...
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### functional type equation

Let $f,g$ be two nonconstant positive functions on $I=[0,1]$ and we assume that : $$\sup_{x\in I}\sqrt{f^2(x)+g^2(x)}=\sqrt{\sup_{x\in I}f^2(x)+\sup_{x\in I}g^2(x)}$$ This implies that $(f,g)$ are ...
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### Can I calculate a fractional sum with functional equations and/or infinite series?

I was wondering if I can calculate fractional sums (non-integer sums) with functional equations in the following manner $$f(x):=\sum_{n=1}^xg(n)$$ $$f(x)=g(x)+f(x-1)\tag1$$ $(1)$ most certainly ...
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### I am looking for a mathematical equation to warp an image [closed]

Theoretically, I know that to warp an image, each pixel $(x,y)$ in the source image is transformed to $(x', y')$ using a function f (i.e. $x'=f(x,y)$ & $y'=f(x,y)$ ). But what mathematical ...
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### Fuzzy's Involution Generalization

As a consequence from my personal attempt to generalize the Fuzzy Logic's negation, I've got the following functional equation: \begin{align*} &f(x) + f(\alpha + \beta - x) = 2*f\left(\frac{\alpha+...
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### Increasing function with $f'(x)=f(f(x))$ [duplicate]

Is there a strictly increasing function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x$?
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### Proving a function to be a difference

I am concerned with a function $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ such that $$f(f(x_1,x_2),f(x_3,x_2))=f(x_1,x_3)$$ for any $x_1,x_2,x_3\in \mathbb{R}$, and $$f(x,0)=x$$ $$f(x,x)=0$$ ...