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10

These are both tantamount to known properties of the heat kernel $K_t(c)$ for $c$ in the circle ${\bf R} / {\bf Z}$. (In (1) the heat kernel is obtained by starting at $t=0$ with a row of delta functions $K_0(c) = \sum_{x \in \bf Z} \delta_x(c)$, and in (2) it's obtained by separation of variables.) Each of the properties can be proved in various ways, not ...

0

Looking at a related question on the side bar, it appears that Greg Kuperberg's answer to this MO question has a very simple version of induction on scales to show that the volume of the unit n-ball tends to 0 as n increases.

1

Some background on Kotelnikov's contribution is given in A G Vitushkin: Half a Century As One Day in the book Bolibruch, Osipov, Sinai (Editors): Mathematical Events of the Twentieth Century and also in the Russian original of this book.

-1

I do not understand what is unclear in my question, but I currently have an answer (possibly there could be a better one). $$\operatorname{odd}[f(x)]=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}g^{(n+1/2)}(0)$$ where $$g(x)=\sum_{k=0}^\infty \frac {x^k}{k!}f^{(2k)}(0)$$ $$\operatorname{even}[f(x)]=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}g^{(n-1/2)}(0)$$ ...

3

It is an elementary exercise in Banach space theory to show that there is NO operator from $L_p$ into $L_{q}$ that satisfies the inequality when $1\le p \not= 2$ with $p<q<\infty$. First, for such $p$ and $q$ every operator from $\ell_p$ into $L_q$ is strictly singular (because $\ell_p$ is not isomorphic to a subspace of $L_q$). Take any (bounded, ...

2

If $1<p<2$, then it is not possible to have the inequality $$\|f\|_p \lesssim \|\widehat{f}\|_{p'} \quad\quad\quad\quad\quad (1)$$ for all $f\ge 0$. This follows from the existence of (positive) purely singular measures $\mu$ with $\widehat{\mu}\in L^{p'}$ (in fact, $\widehat{\mu}$ can have power decay). (I used this fact also in my answer to this ...

2

Such an assertion is close enough to being an "exercise" that there may not be a really clear "reference" for it... but such a result can be explained easily and shortly enough, I think: Finite sums of (dilates of translates of) derivatives of $\cos x/2$ can be subtracted from a given (finitely-) piecewise smooth periodic function to give a $C^2$ periodic ...

0

This is a comment on the above answer, not an answer itself (I am not entitled) but which i feel should be made since the first paragraph has two misleading statements. Firstly, while it is true that the dual of the space of bounded, continuous functions can be identified with measures on the Stone-Čech compactification, it can also, perhaps more naturally, ...

2

The answer to the first question is unknown. It is even unknown if the fraction of graphs on $n$ vertices determined by their spectra converges to 0, or 1, or something between, or even converges at all. About the second question, it is not well posed. You have to decide whether you want weighted graphs (in which case you just have symmetric matrices and ...

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