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Results tagged with ca.classical-analysis-and-odes
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user 153260
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
8
votes
A curious norm related to the L¹ norm
I think $C=2$ is the best constant. Consider $\varepsilon>0$ and let $f,g$ continuous in $[0,1]$ defined as follows.
\begin{equation*}
f(x) = \begin{cases} -1, & 0\leq x \leq \frac12 - \varepsilon \\
…
7
votes
3
answers
686
views
A generalization of discrete Hilbert's transform (Montgomery's inequality)
In the paper "Hilbert's inequality", Montgomery and Vaughan proved that a generalization of the discrete Hilbert transform is bounded in $\ell^2$. The inequality reads as follows
$$ \Big| \sum_{k\neq …
4
votes
1
answer
432
views
Reference or proof of a theorem of L. Fejér on summability of Fourier series
In the article "Ensembles exceptionnels" by A. Beurling, the author cites the following theorem of Fejér:
Suppose that a $2\pi$ periodic function $ f $, Lebesgue integrable in $(0,2\pi)$ satisfies the …
2
votes
0
answers
180
views
A sharp version of a Tauberian theorem
The following Tauberian theorem is true (see Theorem I.11.1 of ''Tauberian theory: A century of developments''). Let $ a_n $ a sequence of real numbers.
If $f(x) = \sum_{n=1}^\infty a_n x^n $ converge …
1
vote
0
answers
82
views
Weak maximum principle for a perturbation of the Laplacian
This question is motivated by similar considerations for the Kohn-Laplacian in the Heisenberg group, but it seems that I cannot even give an answer in the Euclidean case, so here we go.
Suppose that …