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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 ...


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

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 ...


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 ...


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

As Denis Chaperon de Lauzières says, the precise answer to this question is given by the Beurling-Malliavin theorems. They give a very precise characterization of weights $w\geq 1$ for which there exists a Fourier transform $f$ of a function with bounded support, such that $f(x)w(x)$ is bounded. The necessary condition is that $$\int \frac{\log ...


1

The oldest reference I know which constructs compactly-supported functions with as good as possible decay of the Fourier transform at infinity is A. Ingham, "A note on Fourier transforms", J. London Math. Soc. s1–9 (1934), 29–32. See also Th. 1.3.5 in Hörmander's "The analysis of linear partial differential operators I: distribution theory and Fourier ...


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.



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