31,282 research outputs found
On the behavior at infinity of an integrable function
International audienceWe prove that, in a weak sense, any integrable function on the real line tends to zero at infinity : if f is an integrable function on R, then for almost all real number x, the sequence (f(nx)) tends to zero when n goes to infinity. Using Khinchin's metric theorem on Diophantine approximation, we establish that this convergence to zero can be arbitrarily slow
Abel-Tauber theorems for Hankel and Fourier transforms and a problem of Boas
We prove Abel-Tauber theorems for Hankel and Fourier transforms. For example, let f be a locally integrable function on [O, oo) which is eventually decreasing to zero at infinity. Let p = 3, 5, 7, · · · and £ be slowly varying at infinity. We characterize the asymptotic behavior f(t) l(t)t-P as t -+ oo in terms of the Fourier cosine transform of f. Similar results for sine and Hankel transforms are also obtained. As an application, we give an answer to a problem of R. P. Boas on Fourier series
Hilbert-Polya conjecture and Generalized Riemann Hypothesis
Extending a classical integral representation of Dirichlet L-functions
associated to a non trivial primitive character we define associated functions
B(y,z) which are eigenfunction of a Hermitian operator H. The eigenvalues are
the imaginary parts of the L-functions zeros. We prove that if s is a non
trivial zero of such a Dirichlet L-function with Re(s)<1/2, then: - the
associated eigenfunction B(z,y) is square integrable. - the operator H is
"Hermitian" for this function: =. We deduce from this (using the
idea of Hilbert-Polya and finding a contradiction) the Generalized Riemann
Hypothesis: the non trivial zeros of a Dirichlet L-function lie on the critical
line Re(s)=1/2. This results correspond to a weak form of the Hilbert-Polya
conjecture (as for Re(s)=1/2 the eigenfunctions presented here are not square
integrable).Comment: 16 Pages. Article withdraw as function Bs presented is not square
integrable as claimed. (Mistake is on one sub-domain considered: function I2
near y=0
On the order of summability of the Fourier inversion formula
In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems
A General Integral
We define an integral, the distributional integral of functions of one real
variable, that is more general than the Lebesgue and the
Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of
functions with distributional values everywhere or nearly everywhere.
Our integral has the property that if is locally distributionally
integrable over the real line and is a test
function, then is distributionally integrable, and the formula%
[ =(\mathfrak{dist}) \int_{-\infty}^{\infty}f(x) \psi(x)
\,\mathrm{d}% x\,,] defines a distribution
that has distributional point
values almost everywhere and actually almost everywhere.
The indefinite distributional integral corresponds to a distribution with point values
everywhere and whose distributional derivative has point values almost
everywhere equal to
The distributional integral is more general than the standard integrals, but
it still has many of the useful properties of those standard ones, including
integration by parts formulas, substitution formulas, even for infinite
intervals --in the Ces\`{a}ro sense--, mean value theorems, and convergence
theorems. The distributional integral satisfies a version of Hake's theorem.
Unlike general distributions, locally distributionally integrable functions can
be restricted to closed sets and can be multiplied by power functions with real
positive exponents.Comment: 59 pages, to appear in Dissertationes Mathematica
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