TY - CHAP
T1 - A remark on piecewise linear interpolation of continuous Fourier multipliers
AU - Shargorodsky, Eugene
AU - Karlovych, Oleksiy
N1 - 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 L
p(ℝ), 1p(ℝ). We observe that for every p∈(1,2)∪(2,∞), there exists a Lipschitz continuous periodic Fourier multiplier a on L
p(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 L
p(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 L
p(ℝ), 1p(ℝ). We observe that for every p∈(1,2)∪(2,∞), there exists a Lipschitz continuous periodic Fourier multiplier a on L
p(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 L
p(ℝ).
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
T3 - Trends in Mathematics
SP - 99
EP - 107
BT - Tbilisi Analysis and PDE Seminar, TAPDES 2023
A2 - Duduchava, Roland
A2 - Shargorodsky, Eugene
A2 - Tephnadze, George
PB - Birkhäuser Cham
ER -