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This article is cited in 1 scientific paper (total in 1 paper)
On the cardinality of lower sets and universal discretization
F. Daia, A. Prymakb, A. Yu. Shadrinc, V. N. Temlyakovdefg, S. Yu. Tikhonovhij
a Department of Mathematical and Statistical Sciences, University of Alberta Edmonton, Alberta T6G 2G1,
Canada
b Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada
c Department of Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge
CB3 0WA, UK
d d University of South Carolina, 1523 Greene St., Columbia SC, 29208, USA
e Steklov Institute of Mathematics, Russian Federation
f Moscow Center for Fundamental and Applied Mathematics, Russian Federation
g Lomonosov Moscow State University, Russian Federation
h ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
i Universitat Autònoma de Barcelona, Spain
j Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain
Abstract:
A set $Q$ in $\mathbb{Z}^d_+$
is a lower set if $(k_1,\dots,k_d) \in Q$
implies $(l_1,\dots,l_d) \in Q $
whenever $0\le l_i \le k_i$
for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $\mathbb{Z}^d_+$.
Next we apply these results for universal discretization of the
$L_2$-norm of elements from $n$-dimensional subspaces of trigonometric polynomials generated by lower sets.
Keywords:
Lower sets; Downward closed sets; Integer partitions; Universal discretization; Multivariate trigonometric polynomials.
Received: 03.08.2022
Revised: 19.12.2022
Accepted: 22.12.2022
Linking options:
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