real analysis – Function without fundamental period

Proposition: If a periodic function hasn’t fundamental period then every open interval $)a, b($ contains a period.

My attemption: Let $G$ be the set of periods of $f$. Because of absence basic period $forall x in G$ there is $yin G$ such that $y<frac{x}{2}$. By fixing $x$ and repeating this procedure we can go further and construct other $periods$ less than $frac{x}{2^n} (nin mathbb{N}$) for arbitary higher $n$.

In my opinion choosing arbitary small period isn’t problem. We can do it also by “Dirichlet’s approximation” and may be there are some other ways. But what interests me that is how put $m$ multiple $(min mathbb{N})$ of this period in given interval $)a, b($. It seems a litle bit construction problem.

I would be grateful for help.

The intersection of zero loci of a family real analytic functions

If $(f_i)_{i in I}$ is a family of holomorphic functions on some open set $U subset mathbb C^n$ with zero loci $V_i$, then the intersection $bigcap_{i in I} V_i$ is complex analytic. Moreover, for every compact $K subset U$ there exists a finite subset $J subset I$ such that
$$K cap bigcap_{i in I} V_i = K cap bigcap_{i in J} V_i ,.$$
See Is intersection of zero set of any family of holomorphic functions an analytic set?


Now take the same setup but replace “analytic” by “real analytic” and $mathbb C$ by $mathbb R$.

The first statement is then trivially true, or at least if $I$ is countable: define $f = sum_{i in I} c_i f_i^2$ for very rapidly decreasing $c_i>0$, then the zero locus of $f$ (where it converges) is $bigcap_{i in I} V_i$.

Is the second statement also true? That is:

Does there exist for every compact $K subset U$ a finite subset $J subset I$ such that
$$K cap bigcap_{i in I} V_i = K cap bigcap_{i in J} V_i ,?$$

real analysis – Lipschitz hypersurface

I asked this already on Math SE. Maybe this definition is not quite common, but I’m asking myself what a Lipschitz hypersurface is. Intuitively this is a hypersurface which can locally be parametrized by Lipschitz functions. I hope this intuition is correct.
Nevertheless I didn’t find any precise definition. Does anybody know an exact definition of what a Lipschitz hyersurface is ?

real analysis – Attempt at portion of problem presented earlier

I was looking at this problem posted earlier Let $I$ be a generalized rectangle and let $f: I to mathbb{R}$. Show that $$lim_{ptoinfty}left(int_I|f|^pright)^{1/p} = max|f|$$

and struggling to prove LHS $leq$ RHS

Since the integral of a continuous function on a generalized rectangle is integrable, the integral is well defined. Take an archimedean sequence of partitions $textbf{P}_k$ of $f$. Since $|f|^p$ is integrable $int_{textbf{I}}|f|^p=limlimits_{k to infty}U(|f|^p,textbf{P}_k)$. $|f|$ is bounded above by $M=textbf{max}|f|$. so for each rectangle $textbf{J} in textbf{P}_k$ the largest value of $f$ say $mid M_textbf{J} mid leq M$. Can I say largest of $|f|^p=|M_{textbf{J}}|^p$?

Can someone help me with this. I trying the following
begin{equation}
begin{split}
limlimits_{p to infty} bigg{(}int_{textbf{I}}|f|^pbigg{)}^{frac{1}{p}}& =limlimits_{p to infty}(limlimits_{k to infty}U(|f|^p,textbf{P}_k))^{frac{1}{p}} \
& =limlimits_{p to infty}(limlimits_{k to infty}sumlimits_{textbf{J} text{in} textbf{P}_k}|M_textbf{J}|^p text{Vol} textbf{J})^frac{1}{p} \
&leq limlimits_{p to infty}(limlimits_{k to infty}sumlimits_{textbf{J} text{in} textbf{P}_k}M^p text{Vol} textbf{J})^frac{1}{p}\
& = M limlimits_{p to infty}(limlimits_{k to infty}(text{vol} textbf{I})^{frac{1}{p}}) \
& = M\
&= text{max}|f| \
end{split}
end{equation}

real analysis – Generalization of Bernstein’s inequality

I’m using Muscalu and Schlag’s textbook to study harmonic analysis and I encountered the following claim:

Given some function $f in mathcal{S}(mathbb{R}^{d})$, where $mathcal{S}(mathbb{R}^{d})$ denotes the Schwartz space of functions. Let $hat{f}$ denote the Fourier transform of $f$. Assume that there exists some measurable set $E$, such that $text{supp}(hat{f}) subset E subset mathbb{R}^d$. Then for any $1 leq p leq q leq infty$, we have the following inequality: ($|E|$ below denotes the Lebesgue measure of $E$)
$$||f||_{L^q} leq |E|^{frac{1}{p}-frac{1}{q}}||f||_{L^p}$$
I have managed to show the special case when $q=+infty$ and $p=2$ by using Young’s inequality and Plancherel identity. However, the hint says that we still need to use duality and interpolation to deduce the general conclusion. Any ideas on this?

Moreover, how might this estimate be related to the probability version of Bernstein inequality? Thanks in advance!

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Show that all coefficients of a complex power series with real only image vanish

Let $R>0$ and $f: K_R(0) rightarrow mathbb{C}, f(z) = sum_{k=0}^{infty} a_kz^k$ be a compley power series with convergence radius $R>0$ such that $f(K_R(0)) subset mathbb{R} $, where $K_R(0)$ is the disk around zero with radius R. I would like to show that all coefficients $a_k$ with $k in mathbb{N} $ vanish.

So far I have noticed that if the above statement is true, the power series must value the constant $a_0$ for any $ z in K_R(0) $ because otherwise the $a_k$ would not have to be zero for all $k>0$. My problem is that I do not see why there could not be any coefficients such that $sum_{k=1}^{infty} a_kz^k$ would be real as well and this is where I have come to a dead end. I thought this might have to do with linear independence of the powers of $z$. My other idea was to consider the coefficients in the form $a_k = frac{f^{(k)}(0)}{k} $, but I also could not figure out how this could reveal anything about the $a_k$ being zero.

Any hint would be highly appreciated.

real analysis – Do we have full control the oscillation of a function by modifying it on a small set?

Definitions and some motivation:

Let $mathcal B$ be the set of bounded measurable functions from $(0, 1)$ to $mathbb R$. Denote by $mathcal N$ the set of measurable subsets of $(0, 1)$ with Lebesgue measure $0$.

Given a function $f in mathcal B$, define the function $mathcal Of$ by

$mathcal Of(x) := inf_{N in mathcal N} lim_{delta to 0} sup_{y, z in B_delta (x) setminus N} |f(y) – f(z)|$.

Thanks to Lusin’s theorem, we know that we can modify $f$ on an arbitrarily small set and get a continuous function, and so we force the oscillation to be $0$ everywhere. But can we force it to be whatever we want?

Question:

Does there exist, for any $f, g in mathcal B$ and $epsilon, varepsilon > 0$, a function $f’ in mathcal B$ such that the following conditions are satisfied?

i) $f = f’$ everywhere except for a set of measure at most $epsilon$.

ii) $Of’ = Og$ everywhere.

iii) $int_{(0, 1)} |f’ – f’| dmu < int_{(0, 1)} |Of – Og| dmu + epsilon$

Note: All functions are genuine functions and not equivalence classes modulo null sets of such.