Intereting Posts

Does $\mathbb{R}^\mathbb{R}$ have a basis?
Show that $\det{(A + sxy^*)}$ is linear in $s$
Prove ten objects can be divided into two groups that balances each other when placed on the two pans of balance.
Evaluating Integrals using Lebesgue Integration
Set Theory- Generalized form of distributivity of unions over intersections
Subjects studied in number theory
How to find the limit of $\frac{1−\cos 5x}{x^2}$ as $x\to 0$?
Recurrence relation using substitution method
Complicated exercise on ODE
Prove that $gNg^{-1} \subseteq N$ iff $gNg^{-1} = N$
Refining my knowledge of the imaginary number
An induced matrix norm equal to the matrix $\infty$-norm
Spectrum of a Self-Adjoint Operator is Real
What is so special about $\alpha=-1$ in the integral of $x^\alpha$?
Definition of opposite category

I would like to know how to solve this equation :

$$f(x)^2 = f(\sqrt{2}x)$$

We assume that $f : \mathbb R \to \mathbb R$ is $\mathcal C^{2}$.

- Evaluation of $ \lim_{x\rightarrow \infty}\left\{2x-\left(\sqrt{x^3+x^2+1}+\sqrt{x^3-x^2+1}\right)\right\}$
- Why isn't $\int \frac{1}{x}~dx = \frac{x^0}{0}$?
- Integral of $\sqrt{1-x^2}$ using integration by parts
- $f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.
- Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
- Evaluate $\int \frac{\sec^{2}{x}}{(\sec{x} + \tan{x})^{5/2}} \ \mathrm dx$

The answer should be $f(x)=e^{-x^{2}/2}$, but I don’t know how to show this.

- Non-centered Gaussian moments
- Exponential Function as an Infinite Product
- Calculating derivative by definition vs not by definition
- Improper integral with log and absolute value: $\int^{\infty}_{0} \frac{\log |\tan x|}{1+x^{2}} \, dx$
- Is there an analytic solution for the equation $\log_{2}{x}+\log_{3}{x}+\log_{4}{x}=1$?
- Integral $\int \sqrt{x+\sqrt{x^2+2}}dx$
- Convergence/divergence of $\int_0^{\infty}\frac{x-\sin x}{x^{7/2}}\ dx$
- Interesting closed form for $\int_0^{\frac{\pi}{2}}\frac{1}{\left(\frac{1}{3}+\sin^2{\theta}\right)^{\frac{1}{3}}}\;d\theta$
- What does it mean to differentiate in calculus?
- Does $\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$ converge?

Hints :

- assume $f(x) > 0$ for all $x$
- write $g = \log f$
- apply the equality in $g$ twice to get a term $g(2x)$
- take the second derivative of the equality in $g$ and get $g(2x) = g(x)$
- conclude that $g”$ is constant

Additional notes :

- because $f(0)^2 =f(0)$, $f(0) \in \{0,1\}$
- if $\exists x \mid f(x)=0$, then $f(0)=0$ (because $f(x/\sqrt2) = \ldots = f(x/\sqrt2^n) = 0$ and $f$ is continuous in $0$
- as pointed here, if $\exists a \mid f(a)>0$, $f(a/\sqrt{2}^k) = f(a)^{\frac1{2^k}}$ and $f(0)=1$ by continuity of $f$ in $0$

So either:

- $f(0) = 1$, and then $f$ is strictly positive and $f(x) = e^{\lambda x^2}$,
- or $f(0)=0$ and $f = 0$.

PS: as Yves’ excellent post shows, relaxing the $\mathcal C^{\infty}$ assumption, even only in $0$, generates a wide class of additional solutions.

PPS: I’ve opened a new question to see what happens if we relax some of these conditions here: $f(\alpha x) = f(x)^{\beta}$ under different constraints

Setting $x=2^{t/2}$ and taking the logarithm twice,

$$(f(x))^2=f(\sqrt2x)$$

becomes

$$\log_2(\log_2(f(2^{t/2})))+1=\log_2(\log_2(f(2^{(t+1)/2})))$$

or

$$h(t)+1=h(t+1).$$

An obvious solution is $h(t)=t+c$, or $\log_2(\log_2(f(2^{t/2})))=t+c=2\log_2(x)+c$, $$f(x)=2^{Cx^2}.$$

More solutions are found by adding smooth periodic functions of period $1$, like

$$h(t)=t+A\sin(2\pi t)+c,$$

that yield

$$f(x)=2^{Cx^22^{A\sin(4\pi\log_2(x))}}.$$

Example with $C=-1,A=1$:

With starting point $(f(x))^2=f(\sqrt 2x)$, we can get to $(f(x))^4 = f(2x)$ and further $(f(x))^{16}=f(4x)$ and so on, such that

$$(f(x))^{2^n}=f(\sqrt{2^n}x)$$

thus showing that our function passes constants in an exponential manner. Then we take $f(x)=e^{g(x)}$ and get

$$e^{2g(x)}=e^{g(\sqrt 2x)}$$

from which we can say that $2g(x)=g(\sqrt 2x)$ or $4g(x)=g(2x)$. Taking the derivative here yields

$$4g'(x)=2g'(2x)\\

4g”(x)=4g”(2x)$$

At the point of this second derivative, we see that $g”(x)$ must be constant or periodic with period a multiple of $\sqrt 2$. Working backwards, we must have $g(x)=ax^2+bx+c$ (for non-periodic solutions), and with $2g(x)=g(\sqrt 2x)$ we must in fact have $g(x)=ax^2$ (non-periodic solutions). At this point in the process, there must be some other qualifier in order to get a single function $f(x)$ from the family

$$f(x)=e^{ax^2}$$

Caveat: $f(x)=0$ is also a possible solution not covered by the coefficient $a$ above.

Suppose that we consider $\exists x_0:f(x)\lt 0$. Then we must have $(f(x_0/\sqrt 2))^2=f(x_0)\lt 0$, which means that $f(x_0/\sqrt 2)=0+qi$ for some real $q$ and $i=\sqrt{-1}$. But now we also get that $(f(x_0))^2=f(\sqrt 2x_0)\gt 0$ which leads us into our previous solution set where none of the values are negative, which is a contradiction; therefore our assumption that there exists $x_0$ such that $f(x_0)$ is negative must be false or else our derivation of the solution set incorrectly constrains the resulting set. This would also mean that any solution containing non-real numbers cannot contain any real numbers. Since one of the tags in this particular question says “real analysis” I will take that as a cue to say, having any negative result of $f(x)$ brings the function into complex analysis, and therefore such possibilities for $f(x)$ are beyond the scope of this question.

- The complement of every countable set in the plane is path connected
- Which of the choices solution of the Cauchy problem?
- Spectral Measures: Scale Embeddings
- If Gal(K,Q) is abelian then |Gal(K,Q)|=n
- Can an odd perfect number be divisible by $101$?
- Irreducibility of an Affine Variety and its Projective Closure
- Intuitive explanation of variance and moment in Probability
- Finding root using Hensel's Lemma
- Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges
- Question about sets and classes
- Chromatic polynomial of a grid graph
- Finding the Roots of Unity
- Meaning of $\mathbb{Z}\left$?
- Morphism from a line bundle to a vector bundle
- Sum of digits of repunits