1,343 research outputs found
Discrete fractional Radon transforms and quadratic forms
We consider discrete analogues of fractional Radon transforms involving
integration over paraboloids defined by positive definite quadratic forms. We
prove that such discrete operators extend to bounded operators from to
for a certain family of kernels. The method involves an intricate
spectral decomposition according to major and minor arcs, motivated by ideas
from the circle method of Hardy and Littlewood. Techniques from harmonic
analysis, in particular Fourier transform methods and oscillatory integrals, as
well as the number theoretic structure of quadratic forms, exponential sums,
and theta functions, play key roles in the proof.Comment: The statements of Propositions 3, 6, 7, and Theorem 1 have been
corrected, and Corollary 1.1 has been adde
Criteria for selecting children for speech therapy in the public schools
Thesis (Ed.M.)--Boston Universit
On a discrete version of Tanaka's theorem for maximal functions
In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for
the Hardy-Littlewood maximal operator in dimension , both in the
non-centered and centered cases. For the discrete non-centered maximal operator
we prove that, given a function
of bounded variation,
where represents the total variation of . For the discrete
centered maximal operator we prove that, given a function such that , This provides a positive solution to a question
of Haj{\l}asz and Onninen \cite{HO} in the discrete one-dimensional case.Comment: V4 - Proof of Lemma 3 update
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