Ambiguities in one-dimensional phase retrieval of structured functions

Phase retrieval means that we wish to recover a complex-valued signal (discrete or continuous) from the magnitudes of its Fourier transform. Here we restrict ourselves to the recovery of structured functions, e.g. linear spline functions with equidistant knots. First, a complete characterization of the occurring ambiguities is presented. Moreover, we investigate additionally given moduli of the signal itself regarding their ability to reduce the set of ambiguities. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sparse phase retrieval of structured signals by Prony's method

The phase retrieval problem consists in the recovery of a complex-valued signal from the magnitudes of its Fourier transform. Restricting ourselves to the case of sparse structured signals f, which can be represented as a linear combination of N arbitrary translations of a given generator function, we show that almost all f can be recovered from 𝒪 (N²) intensity measurements |ℱ[f](ω)| up to trivial ambiguities. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums

Journal of Fourier Analysis and Applications - In this paper, we study the convergence of adaptive Fourier sums for real-valued $$2\pi $$ -periodic functions. For this purpose, we approximate the...     

Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two-scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m+1 and defectr, first considered by Lipow and Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degreem and defectr. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution.     

A deterministic sparse FFT algorithm for vectors with small support

In this paper we consider the special case where a signal x\({\in }\,\mathbb {C}^{N}\) is known to vanish outside a support interval of length m < N. If the support length m of x or a good bound of it is a-priori known we derive a sublinear deterministic algorithm to compute x from its discrete Fourier transform \(\widehat {\mathbf x}\,{\in }\,\mathbb {C}^{N}\). In case of exact Fourier measurements we require only \({\mathcal O}\)(m\(\log \)m) arithmetical operations. For noisy measurements, we propose a stable \({\mathcal O}\)(m\(\log \)N) algorithm.     

Catalysis and specificity in enzymatic glycoside hydrolysis: a 2,5B conformation for the glycosyl-enzyme intermediate revealed by the structure of the Bacillus agaradhaerens family 11 xylanase.

BACKGROUND: The enzymatic hydrolysis of glycosides involves the formation and subsequent breakdown of a covalent glycosyl-enzyme intermediate via oxocarbenium-ion-like transition states. The covalent intermediate may be trapped on-enzyme using 2-fluoro-substituted glycosides, which provide details of the intermediate conformation and noncovalent interactions between enzyme and oligosaccharide. Xylanases are important in industrial applications - in the pulp and paper industry, pretreating wood with xylanases decreases the amount of chlorine-containing chemicals used. Xylanases are structurally similar to cellulases but differ in their specificity for xylose-based, versus glucose-based, substrates. RESULTS: The structure of the family 11 xylanase, Xyl11, from Bacillus agaradhaerens has been solved using X-ray crystallography in both native and xylobiosyl-enzyme intermediate forms at 1.78 A and 2.0 A resolution, respectively. The covalent glycosyl-enzyme intermediate has been trapped using a 2-fluoro-2-deoxy substrate with a good leaving group. Unlike covalent intermediate structures for glycoside hydrolases from other families, the covalent glycosyl-enzyme intermediate in family 11 adopts an unusual 2,5B conformation. CONCLUSIONS: The 2,5B conformation found for the alpha-linked xylobiosyl-enzyme intermediate of Xyl11, unlike the 4C1 chair conformation observed for other systems, is consistent with the stereochemical constraints required of the oxocarbenium-ion-like transition state. Comparison of the Xyl11 covalent glycosyl-enzyme intermediate with the equivalent structure for the related family 12 endoglucanase, CelB, from Streptomyces lividans reveals the likely determinants for substrate specificity in this clan of glycoside hydrolases.     

An Area Preserving Projection from the Regular Octahedron to the Sphere

In this paper, we propose an area preserving bijective map from the regular octahedron to the unit sphere ${\mathbb{S}^2}$ , both centered at the origin. The construction scheme consists of two steps. First, each face F _i of the octahedron is mapped to a curved planar triangle ${\mathcal{T}_i}$ of the same area. Afterwards, each ${\mathcal{T}_i}$ is mapped onto the sphere using the inverse Lambert azimuthal equal area projection with respect to a certain point of ${\mathbb{S}^2}$ . The proposed map is then used to construct uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the triangular faces of the octahedron.     

Sparse fast DCT for vectors with one-block support

Numerical Algorithms - In this paper, we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II that reconstructs the vector $\mathbf {x}\in \mathbb...     

Optimal shift parameters for periodic spline interpolation

Using the exponential Euler spline, restricted on the unit circle, we sketch a unified approach to the periodic spline interpolation with shifted interpolation nodes. Mainly we are interested in the optimal choice of the shift parameter such that the corresponding interpolatory matrix possesses minimal condition or such that the related interpolation operator has minimal norm. We show that =0 is optimal in both cases. This improves known results of Merz, Reimer-Siepmann and Richards.