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21. Mai 2025

Joachim Toft (Linnaeus University)

Translation and Modulation Invariant Hilbert Spaces

In signal processing and time-frequency analysis, time-frequency shifts are important. The same is true for function or distribution spaces which are invariant under time-frequency shifts. For any time-frequency pair (x,ξ) ∈ R^2d, the time-frequency shift can be considered as a map which take any function or distribution f on Rd, into the function e^{i⟨·,ξ⟩f(·−x)}. In quantum mechanics, the time-frequency shift is called a phase shift.

In the talk we discuss Hilbert spaces H of distributions on R^d, which are invariant under time-frequency shifts, in the sense that for some constant C₀ >0, one has

||f(·−x)e^i⟨·,ξ⟩||_H≤C₀ ||f||_H,   f ∈H, x,ξ ∈R^d,

then H= L²(R^d) with equivalent norms.

The talk is based on collaborations with A. Gumber, R. Manna, P.K. Ratnakumar, J. Vindas.

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