TY - CHAP
T1 - A Remark on Piecewise Linear Interpolation of Continuous Fourier Multipliers
AU - Karlovych, Oleksiy
AU - Shargorodsky, Eugene
N1 - info:eu-repo/grantAgreement/FCT/Concurso de avaliação no âmbito do Programa Plurianual de Financiamento de Unidades de I&D (2017%2F2018) - Financiamento Base/UIDB%2F00297%2F2020/PT#
info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDP%2F00297%2F2020/PT#
Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
PY - 2024/8/21
Y1 - 2024/8/21
N2 - Classical results by de Leeuw (Ann Math (2) 81:364–379, 1965) and Jodeit (Studia Math, 34:215–226, 1970) imply that, for every continuous Fourier multiplier a on Lp(ℝ), 1<p<∞, the piecewise linear function b, satisfying b(n)=a(n) for all n∈ℤ, is again a Fourier multiplier on Lp(ℝ). We observe that for every p∈(1,2)∪(2,∞), there exists a Lipschitz continuous periodic Fourier multiplier a on Lp(ℝ) and a set E⊂ℤ such that the piecewise linear function b with the nodes at the points of E, satisfying b(n)=a(n) for all n∈E, fails to be a Fourier multiplier on Lp(ℝ).
AB - Classical results by de Leeuw (Ann Math (2) 81:364–379, 1965) and Jodeit (Studia Math, 34:215–226, 1970) imply that, for every continuous Fourier multiplier a on Lp(ℝ), 1<p<∞, the piecewise linear function b, satisfying b(n)=a(n) for all n∈ℤ, is again a Fourier multiplier on Lp(ℝ). We observe that for every p∈(1,2)∪(2,∞), there exists a Lipschitz continuous periodic Fourier multiplier a on Lp(ℝ) and a set E⊂ℤ such that the piecewise linear function b with the nodes at the points of E, satisfying b(n)=a(n) for all n∈E, fails to be a Fourier multiplier on Lp(ℝ).
KW - Fourier multiplier
KW - Function of bounded variation
KW - Piecewise linear function
UR - http://www.scopus.com/inward/record.url?scp=85202528838&partnerID=8YFLogxK
U2 - 10.1007/978-3-031-62894-8_10
DO - 10.1007/978-3-031-62894-8_10
M3 - Chapter
AN - SCOPUS:85202528838
SN - 978-3-031-62893-1
T3 - Trends in Mathematics
SP - 99
EP - 107
BT - Tbilisi Analysis and PDE Seminar
A2 - Duduchava, Roland
A2 - Shargorodsky, Eugene
A2 - Tephnadze, George
PB - Springer
CY - Cham
ER -