The multiplier problem in the context of the Banach Gelfand Triple

04.12.2019 11:30 - 13:00

Hans G. Feichtinger (University of Vienna)

By definition multipliers are bounded linear operators between translation invariant (Banach) spaces of functions or distributions which commute with translations. They arise in electrical engineering as linear time-variant systems (LTIS), and most engineering books start with such systems in order to introduce convolution and the Fourier transform.

In fact, the natural expectation is that any such operator is a convolution operator, whose convolution kernel is called the "impulse response", or equivalently, that it is a multiplication operator on the Fourier transform side, i.e. a Fourier multiplier. 

Only very few cases allow a precise description of this situation. For multipliers of L1(G) or C0(G) (G LCA group) one has exactly the bounded measures as convolution operators and their Fourier transform is a well defined, continuous and bounded (transfer) function. 

On the other hand the case p=2 is easily solved via Plancherels Theorem, which implies that the space of Fourier multipliers is just L-infty over the dual group (frequency domain).

The main goal of this talk is to explain in which sense the setting of the Banach Gelfand Triple (resp. rigged Hilbert space) (S0,L2,S0*) is helpful in describing the situation in a much more general situation. This approach also includes a construction which is known as the Herz algebra Ap(G), for the characterization of Lp-multipliers, for p in (1,infty).

M. Ehler, K. Gröchenig
SR 11