Discrete analogues in harmonic analysis: Spherical averages

By Á. Magyar, E. Stein, S. Wainger. 2004

188 citations recorded by the source corpus.

In this paper we prove an analogue in the discrete setting of d ,o fthe spherical maximal theorem for d . The methods used are two-fold: the application of certain “sampling” techniques, and ideas arising in the study of the number of representations of an integer as a sum of d squares, in particular, the “circle method”. The results we obtained are by necessity limited to d ≥ 5, and moreover the range of p for the L p estimates differs from its analogue in d .

Subfields: Applied Mathematics, Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a241816fcdf8bd96e31cf07d2040e3588c1a4c57

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Mariusz Mirek
2. L. Pierce
3. Isaac Z. Pesenson
4. Bartosz Trojan
5. H. Fuehr
6. Karlheinz Groechenig

In Computational Mechanics

1. Mariusz Mirek
2. L. Pierce
3. Isaac Z. Pesenson

In Numerical Analysis

1. Mariusz Mirek
2. L. Pierce
3. Isaac Z. Pesenson
4. Bartosz Trojan

Citations in the corpus

• Discrete maximal functions in higher dimensions and applications to ergodic theory (2014, background): “…extensive study both in pointwise ergodic theory along various arithmetic subsets of the integers (see e.g. [1, 9, 12, 16, 24, 25, 40]) and investigations of discrete analogues of classical operators with arithmetic features (see e.g. [12, 10, 11, 13, 21, 22, 26, 28, 31, 33, 32, 35, 36, 37, 38]).”
• On the endpoint regularity of discrete maximal operators (2013, unknown): “Another interesting variant would be to consider the spherical maximal operator [3, 16] and its discrete analog [15].”
• Sampling in paley-wiener spaces on combinatorial graphs (2008, result): “We know just three papers [13], [16], [21] in which authors consider sampling on Z and on ZN , but our approach to the problem and our results are very different from the methods of these papers.”
• Discrete Analogues in Harmonic Analysis (2009, background): “Such operators have been studied by Magyar [27], Magyar, Stein and Wainger [28], and Ionescu [21], resulting in the following optimal result:”