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Shobha Madan

Publications and source records attributed to Shobha Madan.

8 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↗
arXiv2002

Large classes of minimally supported frequency wavelets of L^2(\R) and H^2(\R)

We introduce a method to construct large classes of MSF wavelets of the Hardy space H^2(\R) and symmetric MSF wavelets of L^2(\R), and discuss the classification of such sets. As application, we show that there are uncountably many wavelet sets of L^2(\R) and H^2(\R). We also enumerate all symmetric wavelets of L^2(\R) with at most three intervals in the positive axis as well as 3-interval wavelet sets of H^2(\R). Finally, we construct families of MSF wavelets of L^2(\R) whose Fourier transform does not vanish in any neighbourhood of the origin.

math.FA↗
arXiv2001

A Class of non-MRA Band-limited Wavelets

We give a characterization of a class of band-limited wavelets of $L^2({\mathbb R})$ and show that none of these wavelets come from a multiresolution analysis (MRA). For each $n\geq 2$, we construct a subset $S_n$ of ${\mathbb R}$ which is symmetric with respect to the origin. We give necessary and sufficient conditions on a function $ψ\in L^2({\mathbb R})$ with supp $\hatψ\subseteq S_n$ to be an orthonormal wavelet. This result generalizes the characterization of a class of wavelets of E. Hernández and G. Weiss. The dimension functions associated with these wavelets are also computed explicitly. Starting from the wavelets we have constructed, we are able to construct examples of wavelets in each of the equivalence classes of wavelets defined by E. Weber.

math.FA↗