TY - EJOUR
DO - 10.1016/j.apal.2016.03.001
AB - In computability theory a variety of combinatorial systems are encountered (word problems, production systems) that exhibit undecidability properties. Here we seek such structures in the realm of Analysis, more specifically in the area of Fourier Analysis. The starting point is that sufficiently strongly convergent Fourier series give rise to predicates in the sense of first order predicate calculus by associating to any s-ary Fourier series the predicate "the Fourier coefficient with index (n(1), ... , n(s)) is non-zero". We introduce production systems, viewed as counterparts of the combinatorial ones, that generate all recursively enumerable predicates in this way using as tools only elementary operations and functions from classical Analysis. The problem arises how simple such a system may be. It turns out that there is a connection between this question and an as yet unproved conjecture by R. Bilchi. This is discussed in the second half of the paper. (C) 2016 Elsevier B.V. All rights reserved.
T1 - Undecidability through Fourier series
IS - 7
DA - 2016
AU - Buser, Peter
AU - Scarpellini, Bruno
JF - Annals Of Pure And Applied Logic
SP - 507-524
VL - 167
EP - 507-524
PB - Elsevier Science Bv
PP - Amsterdam
ID - 219538
KW - Recursively enumerable sets
KW - Fourier series
KW - Buchi's problem
KW - Jacobi theta functions
SN - 0168-0072
ER -