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We consider functional equations driven by linear fractional transformations, which are special cases of de Rham's functional equations. We consider Hausdorff dimension of the measure whose distribution function is the solution. We give a…
We revisit the concept of spectral averaging and point out its origin in connection with one-parameter subgroups of $SL_2(\bbR)$ and the corresponding M\"obius transformations. In particular, we identify exponential Herglotz representations…
In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of…
Convolutions or Hadamard products of analytic functions is a well explored area of research and many nice results are available in literature. On the other hand, very little is known in general about the convolutions of univalent harmonic…
We show that spectral synthesis thresholds are governed by a quantitative spectral complexity parameter, the Fourier Ratio, in addition to the geometric size of the Fourier support. In the Euclidean setting, we prove that if a compactly…
Spectral singularities are among generic mathematical features of complex scattering potentials. Physically they correspond to scattering states that behave like zero-width resonances. For a simple optical system, we show that a spectral…
We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n by n square matrix. The…
We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension.…
We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed…
We investigate the problem of reconstructing signals from a subsampled convolution of their modulated versions and a known filter. The problem is studied as applies to specific imaging systems relying on spatial phase modulation by randomly…
Speckle patterns are used in a broad range of applications including microscopy, imaging, and light-matter interactions. Tailoring speckles' statistics can dramatically enhance their performance in applications. We present an experimental…
We show that a spectrum of frequencies obtained by a random perturbation of the integers allows one to represent any measurable function on R by an almost everywhere converging sum of harmonics almost surely.
Measures generated by Iterated Function Systems composed of uncountably many one--dimensional affine maps are studied. We present numerical techniques as well as rigorous results that establish whether these measures are absolutely or…
Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we…
We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with…
Recent advances in spectroscopic instrumentation and calibration methods dramatically improve the quality of quasar spectra. Supercomputer calculations show that, at high spectral resolution, procedures used in some previous analyses of…
The free convolution is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant…
We investigate a family of Riesz products and show that they can be regarded as diffraction measures of generalized Thue-Morse sequences, possibly over an infinite alphabet. These measures are closely related to the dynamical system arising…
An important statistical test on the pseudo-random number generators is called the spectral test. The test is aimed at answering the question of distribution of the generated pseudo-random vectors in dimensions $d$ that are larger than the…
It has recently been established that, in a non-demolition measurement of an observable $\mathcal{N}$ with a finite point spectrum, the density matrix of the system approaches an eigenstate of $\mathcal{N}$, i.e., it "purifies" over the…