84
84

Jul 20, 2013
07/13

by
Debashish Bose; Shobha Madan

texts

#
eye 84

#
favorite 0

#
comment 0

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...

Source: http://arxiv.org/abs/1107.5209v1

2
2.0

Jun 29, 2018
06/18

by
Debashish Bose; Shobha Madan

texts

#
eye 2

#
favorite 0

#
comment 0

Let $\Omega \subset \mathbb{R}$ be a compact set with measure $1$. If there exists a subset $\Lambda \subset \mathbb{R}$ such that the set of exponential functions $E_{\Lambda}:=\{e_\lambda(x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in \Lambda\}$ is an orthonormal basis for $L^2(\Omega)$, then $\Lambda$ is called a spectrum for the set $\Omega$. A set $\Omega$ is said to tile $\mathbb{R}$ if there exists a set $\mathcal T$ such that $\Omega + \mathcal T = \mathbb{R}$. A conjecture of Fuglede...

Topics: Classical Analysis and ODEs, Mathematics

Source: http://arxiv.org/abs/1606.04814

42
42

Sep 20, 2013
09/13

by
Debashish Bose; Shobha Madan

texts

#
eye 42

#
favorite 0

#
comment 0

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.

Source: http://arxiv.org/abs/1002.4525v2

83
83

Sep 17, 2013
09/13

by
Biswaranjan Behera; Shobha Madan

texts

#
eye 83

#
favorite 0

#
comment 0

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 $\psi\in L^2({\mathbb R})$ with supp $\hat\psi\subseteq S_n$ to be an orthonormal wavelet. This result generalizes the characterization of a class of wavelets of...

Source: http://arxiv.org/abs/math/0108114v1

37
37

Sep 20, 2013
09/13

by
Biswaranjan Behera; Shobha Madan

texts

#
eye 37

#
favorite 0

#
comment 0

We study the action of translation operators on wavelet subspaces. This action gives rise to an equivalence relation on the set of all wavelets. We show by explicit construction that each of the associated equivalence classes is non-empty.

Source: http://arxiv.org/abs/math/0607695v1

37
37

Sep 17, 2013
09/13

by
Nicola Arcozzi; Biswaranjan Behera; Shobha Madan

texts

#
eye 37

#
favorite 0

#
comment 0

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...

Source: http://arxiv.org/abs/math/0207141v1

67
67

Sep 23, 2013
09/13

by
Debashish Bose; Shobha Madan; Parasar Mohanty; Saurabh Shrivastava

texts

#
eye 67

#
favorite 0

#
comment 0

In this paper we prove the bilinear analogue of de Leeuw's result for periodic bilinear multipliers and some Jodeit type extension results for bilinear multipliers.

Source: http://arxiv.org/abs/0903.4052v1

57
57

Sep 18, 2013
09/13

by
Debashish Bose; C. P. Anil Kumar; R. Krishnan; Shobha Madan

texts

#
eye 57

#
favorite 0

#
comment 0

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...

Source: http://arxiv.org/abs/0803.0049v2