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Debashish Bose

Publications and source records attributed to Debashish Bose.

5 recordsLinked to original sources
arXiv2016

On The Rationality Of The Spectrum

Let $Ω\subset \mathbb{R}$ be a compact set with measure $1$. If there exists a subset $Λ\subset \mathbb{R}$ such that the set of exponential functions $E_Λ:=\{e_λ(x) = e^{2πi λx}|_Ω:λ\in Λ\}$ is an orthonormal basis for $L^2(Ω)$, then $Λ$ is called a spectrum for the set $Ω$. A set $Ω$ is said to tile $\mathbb{R}$ if there exists a set $\mathcal T$ such that $Ω+ \mathcal T = \mathbb{R}$. A conjecture of Fuglede suggests that Spectra and Tiling sets are related. Lagarias and Wang \cite {LW1} proved that Tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in \cite {BM1}, \cite {IK}. In this paper, we give some partial results to support the rationality of the spectrum.

math.CA↗
arXiv2011

"Spectral implies Tiling" for Three Intervals Revisited

In \cite{BCKM} it was shown that "Tiling implies Spectral" holds for a union of three intervals and the reverse implication was studied under certain restrictive hypotheses on the associated spectrum. In this paper, we reinvestigate the "Spectral implies Tiling" part of Fuglede's conjecture for the three interval case. We first show that the "Spectral implies Tiling" for two intervals follows from the simple fact that two distinct circles have at most two points of intersections. We then attempt this for the case of three intervals and except for one situation are able to prove "Spectral implies Tiling". Finally, for the exceptional case, we show a connection to a problem of generalized Vandermonde varieties.

math.CA↗
arXiv2010

Spectrum is periodic for n-Intervals

In this paper we study spectral sets which are unions of finitely many intervals in R. We show that any spectrum associated with such a spectral set is periodic, with the period an integral multiple of the measure of the set. As a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case.

math.CA↗
arXiv2010

On Fuglede's conjecture for three intervals

In this paper we prove the "Tiling implies Spectral" part of Fuglede's paper for the case of three intervals. Then we prove the "Spectral implies Tiling" part of the conjecture for the case of three equal intervals as also when the intervals have lengths 1/2, 1/4, 1/4. For the general case we change our approach to get information on the structure of the spectrum for the n-interval case. Finally, we use symbolic computations on Mathematica, and prove this part of the conjecture with an additional assumption on the spectrum.

math.CA↗