Mean Dimension of Function Classes with Lebesgue Measurable Spectral Sets Original Research Article

The notion of mean dimension was introduced in the 1970s by Tikhomirov. It determines the mean number of linear dimensions required to identify an element of a given function class. Tikhomirov then posed the following problem: find the mean dimension of the unit ball BpE of the space of Lp-functions on Rn with spectra inside a given Lebesgue measurable bounded set E. In the language of signal analysis: determine the amount of linear information carried by generalized band-limited signals. In this paper Tikhomirov′s conjecture on mean dimension is confirmed in certain important cases and yet shown to fail in certain other cases.