69 research outputs found
The periodic decomposition problem
If a function can be represented as the sum of
periodic functions as with
(), then it also satisfies a corresponding -order difference
equation . The periodic
decomposition problem asks for the converse implication, which may hold or fail
depending on the context (on the system of periods, on the function class in
which the problem is considered, etc.). The problem has natural extensions and
ramifications in various directions, and is related to several other problems
in real analysis, Fourier and functional analysis. We give a survey about the
available methods and results, and present a number of intriguing open
problems
Integral Concentration of idempotent trigonometric polynomials with gaps
We prove that for all p>1/2 there exists a constant such that,
for any symmetric measurable set of positive measure E\subset \TT and for any
, there is an idempotent trigonometrical polynomial f
satisfying \int_E |f|^p > \gamma \int_{\TT} |f|^p. This disproves a
conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence
of for p>1 and conjectured that it does not exists for p=1.
Furthermore, we prove that one can take when p>1 is not an even
integer, and that polynomials f can be chosen with arbitrarily large gaps when
. This shows striking differences with the case p=2, for which the
best constant is strictly smaller than 1/2, as it has been known for twenty
years, and for which having arbitrarily large gaps with such concentration of
the integral is not possible, according to a classical theorem of Wiener.
We find sharper results for when we restrict to open sets, or
when we enlarge the class of idempotent trigonometric polynomials to all
positive definite ones.Comment: 43 pages; to appear in Amer. J. Mat
On some Extremal Problems of Landau
2000 Mathematics Subject Classification: Primary: 42A05. Secondary: 42A82, 11N05.The prime number theorem with error term presents itself as &pi'(x) = ∫2x [dt/ logt] + O ( x e- K logL x). In 1909, Edmund Landau provided a systematic analysis of the proof seeking better values of L and K. At a key point of his 1899 proof de la Vallée Poussin made use of the nonnegative trigonometric polynomial 2/3 (1+cos x)2 = 1+4/3 cosx +1/3 cos2x. Landau considered more general positive definite nonnegative cosine polynomials 1+a1cos x+… + ancos nx ≥ 0, with a1> 1,ak ≥ 0 (k = 1,…,n), and deduced the above error term with L = 1/2 and any K< 1/(2V(a))½, where V(a): = (a1+a2+…+ an)/(( (a1)½-1)2). Thus the extremal problem of finding V: = minV(a) over all admissible coefficients, i.e. polynomials, arises.
The question was further studied by Landau and later on by many other eminent mathematicians. The present work surveys these works as well as current questions and ramifications of the theme, starting with a long unnoticed, but rather valuable Bulgarian publication of Lubomir Chakalov.Supported in part in the framework of the Hungarian-French Scientific and Technological
Governmental Cooperation, Project # F-10/04.
The author was supported in part by the Hungarian National Foundation for Scientific
Research, Project #s T-049301, T-049693 and K-61908.
This work was accomplished during the author’s stay in Paris under his Marie Curie
fellowship, contract # MEIF-CT-2005-022927
Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling
We prove three results on the density resp. local density and clustering of
zeros of the Beurling zeta function close to the one-line
. The analysis here brings about some news, sometimes even for
the classical case of the Riemann zeta function. Theorem 4 provides a zero
density estimate, which is a complement to known results for the Selberg class.
Note that density results for the Selberg class rely on use of the functional
equation of , which we do not assume in the Beurling context. In Theorem
5 we deduce a variant of a well-known theorem of Tur\'an, extending its range
of validity even for rectangles of height only . In Theorem 6 we will
extend a zero clustering result of Ramachandra from the Riemann zeta case. A
weaker result -- which, on the other hand, is a strong sharpening of the
average result from the classic book \cite{Mont} of Montgomery -- was worked
out by Diamond, Montgomery and Vorhauer. Here we show that the obscure
technicalities of the Ramachandra paper (like a polynomial with coefficients
like ) can be gotten rid of, providing a more transparent proof of the
validity of this clustering phenomenon
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