Shorter Notes: The Nonexistence of Complex Haar Systems on Nonplanar Locally Connected Spaces

Henderson, George W.; Ummel, Brian R.

doi:10.2307/2039610

Cited by 2 publications

(2 citation statements)

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“…This argument is strictly for the real case of course but it has been extended [12] to the case of complex functions: there is a complex Chebyshev system for domains in C but not for domains in C 2 or in higher dimension. More specifically, it is proved in [6,12] that a complex continuous Chebyshev system exists on a locally connected set S if and only if S is homeomorphic to a closed subset of R 2 . This result allows us to prove that the situation in T 2 is strictly worse than in T.…”

Section: Lemmamentioning

confidence: 99%

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How many Fourier coefficients are needed?

Diederichs¹,

Kolountzakis²,

Papageorgiou³

2021

Preprint

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A. We are looking at families of functions or measures on the torus (in dimension one and two) which are specified by a finite number of parameters N. The task, for a given family, is to look at a small number of Fourier coefficients of the object, at a set of locations that is predetermined and may depend only on N, and determine the object. We look at (a) the indicator functions of at most N intervals of the torus and (b) at sums of at most N complex point masses on the two-dimensional torus. In the first case we reprove a theorem of Courtney which says that the Fourier coefficients at the locations 0, 1, . . . , N are sufficient to determine the function (the intervals). In the second case we produce a set of locations of size O(N log N) which suffices to determine the measure.

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Section: Lemmamentioning

confidence: 99%

“…Proof. If such an Ω had size N then, according to Theorem 3, the corresponding set of characters e ω (x) = e 2πiω•x , ω ∈ Ω, would be a continuous Chebyshev system on T 2 , According to [6,12] this would make T 2 embeddable into the plane, which it is not.…”

Section: Lemmamentioning

confidence: 99%