PERIODIC ULTRA-DISTRIBUTIONS AND PERIODIC ELEMENTS INMODULATION SPACES by Joachim Toft(Linnaeus Uni)

In the present talk we characterize periodic elements in Gevrey classes, Gelfand-Shilov
distribution spaces and modulation spaces, in terms of estimates of involved Fourier coefficients, and by estimates of their short-time Fourier transforms. We show that such spaces
can be completely characterised in terms of formal Fourier series with suitable estimates
on their coefficients. For periodic Gelfand-Shilov distributions such characterisations can
be found in the literature in the case when the Gevrey parameter is strictly larger than
1. Our analysis is valid for all positive Gevrey parameters.
As a consequence, inverse problems for diffusion equations and similar equations on
certain bounded domains can be handled.
The proofs are based on new types of formulae of independent interests when evaluating
the Fourier coefficients and which involve short-time Fourier transforms.
The talk is based on a joint work with E. Nabizadeh.
References
[1] M. Reich Superposition in Modulation Spaces with Ultradifferentiable Weights, (preprint)
arXiv:1603.08723.
[2] M. Reich, M. Reissig, W. Sickel Non-analytic Superposition Results on Modulation Spaces with Subexponential Weights, J. Pseudo-Dier. Oper. Appl. 7 (2016), 365{409.
[3] J. Toft Periodicity, and the Zak transform on Gelfand-Shilov and modulation spaces, Complex Analysis
and Operator Theory (appeared online 2020).
[4] J. Toft, E. Nabizadeh Periodic distributions and periodic elements in modulation spaces, Adv. Math.
323 (2018), 193{225.

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