Undersampled Windowed Exponentials and Their Applications

We characterize the completeness and frame/basis property of a union of under-sampled windowed exponentials of the form

$$ {\mathcal{F}}(g): =\bigl\{ e^{2\pi i n x}: n\ge 0\bigr\} \cup \bigl\{ g(x)e^{2\pi i nx}: n< 0\bigr\} $$

for \(L^{2}[-1/2,1/2]\) by the spectra of the Toeplitz operators with the symbol \(g\). Using this characterization, we classify all real-valued functions \(g\) such that \({\mathcal{F}}(g)\) is complete or forms a frame/basis. Conversely, we use the classical non-harmonic Fourier series theory to determine all \(\xi \) such that the Toeplitz operators with the symbol \(e^{2\pi i \xi x}\) is injective or invertible. These results demonstrate an elegant interaction between frame theory of windowed exponentials and Toeplitz operators. Finally, we use our results to answer some open questions in dynamical sampling, and derivative samplings on Paley-Wiener spaces of bandlimited functions.

     

Spectral Measures Associated with the Factorization of the Lebesgue Measure on a Set via Convolution

Let \(Q\) be a fundamental domain of some full-rank lattice in \({\mathbb {R}}^d\) and let \(\mu \) and \(\nu \) be two positive Borel measures on \({\mathbb {R}}^d\) such that the convolution \(\mu *\nu \) is a multiple of \(\chi _Q\) . We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated \(L^2\) space admits an orthogonal basis of exponentials) and we show that this is the case when \(Q = [0,1]^d\) . This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede’s Conjecture for spectral measures on \({\mathbb {R}}^1\) and we show that it implies the classical Fuglede’s Conjecture on \({\mathbb {R}}^1\) .     

Analysis of performance optimization in supercontinuum sources

We have determined the cause of performance variation as a function of system parameters in supercontinuum (SC) sources in a normal-dispersion fiber by numerically analyzing the noise evolution of the SC. High-order nonlinearity was considered in such an analysis for the first time to our knowledge. We found that the evolution of noise along the SC fiber has two stages: First, the noise is governed by dispersion, which is different from that of the signal, whereas self-steeping leads to an asymmetric noise distribution across the spectrum. Second, four-wave mixing generates new noise components, whereas Raman scattering augments the asymmetry. We evaluated the qualities of the spectrally sliced pulses at several stages to verify the analysis and found an asymmetric profile of the sliced-pulses' quality versus frequency.     

Enantioselective Brønsted base catalyzed [4+2] cycloaddition using novel amino-substituted tetraphenylene derivatives

This paper is concerned with the syntheses of 1,8,9,16-tetrahydroxytetraphenylene derivatives and their applications as Brønsted base organocatalysts for [4+2] cycloaddition between anthrone and maleimides. The structural modifications of the catalysts and their related catalytic properties are described and discussed in details.     

Robust optical signal-to-noise ratio monitoring scheme using a phase-modulator-embedded fiber loop mirror

We propose and experimentally demonstrate a novel in-band optical signal-to-noise ratio (OSNR) monitoring technique using a phase-modulator-embedded fiber loop mirror. This technique measures the in-band OSNR accurately by observing the output power of a fiber loop mirror filter, where the transmittance is adjusted by an embedded phase modulator driven by a low-frequency periodic signal. The measurement errors are less than 0.5 dB for an OSNR between 0 and 40 dB in a 10 Gbit/s non-return-to-zero system. This technique was also shown experimentally to have high robustness against various system impairments and high feasibility to be deployed in practical implementation.     

Oxadisilole-fused isobenzofurans. Synthesis and characterization of oxadisilole-substituted acenes

Mono- and bis-oxadisilole-fused isobenzofurans were first synthesized and isolated. A series of mono-, bis-, and tris-oxadisilole-substituted acenes were then synthesized by benzyne Diels-Alder reactions with the isobenzofurans. The photophysical, redox, and thermal properties of these new acenes were characterized.     

Translational absolute continuity and Fourier frames on a sum of singular measures

A finite Borel measure μ in ${\mathbb{R}}^d$ is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for $L^2(\mu )$. It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures ν and λ without atoms whose supports form a packing pair, then $\nu ?\lambda +{\delta }_t?\nu$ is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure ${\mu }_4+{\mu }_{16}$ does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping $R_{\theta }$ with $\theta \ne \pm \pi /2$, the two-dimensional measure ${\rho }_{\theta }(?):=({\mu }_4\times {\delta }_0)(?)+({\delta }_0\times {\mu }_{16})(R_{\theta }^{?1}?)$, supported on the union of x-axis and $y=(\cot ?\theta )x$, always admit a Fourier frame. Furthermore, we can find ${\{e^{2\pi i〈\lambda ,x〉}\}}_{\lambda \in {\mathrm{\Lambda }}_{\theta }}$ such that it forms a Fourier frame for ${\rho }_{\theta }$ with frame bounds independent of θ. Nonetheless, ${\rho }_{\pm \pi /2}$ does not admit any Fourier frame.     

On Fourier frame of absolutely continuous measures

Let μ be a compactly supported absolutely continuous probability measure on Rn, we show that L²(K,dμ) admits a Fourier frame if and only if its Radon-Nikodym derivative is bounded above and below almost everywhere on the support K. As a consequence, we prove that if μ is an equal weight absolutely continuous self-similar measure on R¹ and L²(K,dμ) admits a Fourier frame, then the density of μ must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere 1/2<λ<1, the L² space of the λ-Bernoulli convolutions cannot admit a Fourier frame.     

Tight wavelet frame sets in finite vector spaces

Let $q\ge 2$ be an integer, and ${\mathbb{F}}_q^d$, $d\ge 1$, be the vector space over the cyclic space ${\mathbb{F}}_q$. The purpose of this paper is two-fold. First, we obtain sufficient conditions on $E?{\mathbb{F}}_q^d$ such that the inverse Fourier transform of $1_E$ generates a tight wavelet frame in $L^2({\mathbb{F}}_q^d)$. We call these sets (tight) wavelet frame sets. The conditions are given in terms of multiplicative and translational tilings, which is analogous with Theorem 1.1 ([20]) by Wang in the setting of finite fields. In the second part of the paper, we exhibit a constructive method for obtaining tight wavelet frame sets in ${\mathbb{F}}_q^d$, $d\ge 2$, q an odd prime and $q\equiv 3$ (mod 4).