High-frequency coercivity loss for completely monotone memory on bounded time intervals
Abstract
We study Volterra memory terms with locally integrable completely monotone kernels on a finite time interval (0,Tend) and ask how much L2 coercivity they retain at high temporal frequencies. The main tool is an exact formula for the Rayleigh quotients of the cosine modes ψn(t)=(2/Tend)1/2cos(2πnt/Tend). It shows that these quotients lie between (1−κn/(2πn)) m(2πn/Tend) and m(2πn/Tend), where m is the real memory symbol and κn∈[1−e−2πn,1] is the best constant valid for all such kernels. Consequently, for non-constant kernels the algebraic decay rate of the quotients does not depend on Tend and coincides with the decay index ρ∈[0,2] of m; any value in [0,2] occurs. The Gaussian kernel e−t2, which is of positive type but not completely monotone, shows that this may fail otherwise: its quotients decay like n−4, while its symbol decays faster than any power. We also show that the largest Rayleigh quotient on (0,Tend) is a continuous and strictly increasing function of Tend, so that a kernel of total mass larger than one has exactly one critical horizon. Finally, because the memory operator on (0,Tend) is compact, it does not contribute to the uniform L2 coercivity constant, and for kn(t)=ne−nt the associated operators converge to the identity strongly but not in norm. The diffusion equation with memory serves as a model throughout.
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BibTeX
@misc{ishizaka2026highfrequency,
title = {{High-frequency coercivity loss for completely monotone memory on bounded time intervals}},
author = {Hiroki Ishizaka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.12482},
eprint = {2607.12482},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.12482},
}
RIS
TY - UNPB
TI - High-frequency coercivity loss for completely monotone memory on bounded time intervals
AU - Hiroki Ishizaka
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2607.12482
UR - https://arxiv.org/abs/2607.12482
AB - We study Volterra memory terms with locally integrable completely monotone kernels on a finite time interval (0,Tend) and ask how much L^2 coercivity they retain at high temporal frequencies. The main tool is an exact formula for the Rayleigh quotients of the cosine modes ψ_n(t)=(2/Tend)^1/2cos(2πnt/Tend). It shows that these quotients lie between (1−κ_n/(2πn)) m(2πn/Tend) and m(2πn/Tend), where m is the real memory symbol and κ_n∈[1−e^−2πn,1] is the best constant valid for all such kernels. Consequently, for non-constant kernels the algebraic decay rate of the quotients does not depend on Tend and coincides with the decay index ρ∈[0,2] of m; any value in [0,2] occurs. The Gaussian kernel e^−t^2, which is of positive type but not completely monotone, shows that this may fail otherwise: its quotients decay like n^−4, while its symbol decays faster than any power. We also show that the largest Rayleigh quotient on (0,Tend) is a continuous and strictly increasing function of Tend, so that a kernel of total mass larger than one has exactly one critical horizon. Finally, because the memory operator on (0,Tend) is compact, it does not contribute to the uniform L^2 coercivity constant, and for k_n(t)=ne^−nt the associated operators converge to the identity strongly but not in norm. The diffusion equation with memory serves as a model throughout.
ER -