Linear Combinations Of Shifts Research Articles

Some properties of convolution in symmetric spaces and approximate identity

This paper deals with the symmetric space of functions and its subspace where continuous functions are dense is considered. Main properties of convolution which plays a vital role in harmonic analysis, as in other areas of mathematics are established in this space. Following the classical case, it is proved that the convolution can be approximated by linear combinations of shifts in a subspace of the considered space. An approximate identity for the convolution is also considered in that subspace.

On Application of Gaussian Functions for Numerical Solution of Optimal Control Problems

It is proved that the linear combinations of shifts and contractions of the Gaussian function can be used for an arbitrarily accurate approximation in the space of continuous functions of one variable on any fixed intervals. On the example of the soft lunar landing problem, a method for the numerical solution of optimal control problems based on this approximation procedure of the control function is described. Within the framework of the same example, the sensitivity of constraint functionals to the specification error of optimal parameters is investigated using three approaches as follows: 1) Pontryagin’s maximum principle (both numerically and theoretically); 2) the control parametrization technique in combination with the method of sliding nodes; 3) the newly proposed method. A comparative analysis is performed that confirms the effectiveness of the third method.

Zeros and fixed points of the linear combination of shifts of a meromorphic function

Let f be transcendental and meromorphic in the complex plane. In this article, we investigate the existences of zeros and fixed points of the linear combination and quotients of shifts of $f(z)$ when $f(z)$ is of order one. We also prove a result concerning the linear combination which extends a result of Bergweiler and Langley. Some results concerning the order of $f(z) <1$ are also obtained.

The Pompeiu property and approximation in L p by linear combinations of shifts

The Pompeiu property and approximation in L p by linear combinations of shifts

Beurling’s theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations

The results of the paper are obtained for functions from homogeneous spaces of functions defined on a locally compact Abelian group. The notion of Beurling spectrum, or essential spectrum, of functions is introduced. If a continuous unitary character is an essential point of the spectrum of a function, then it is the c-limit of a linear combination of shifts of the function in question. The notion of a slowly varying function at infinity is introduced, and the properties of such functions are considered. For a parabolic equation with initial function from a homogeneous space, it is proved that the weak solution as a function of the first argument is a slowly varying function at infinity.

Expansion in the system of shifts of a B-spline

A method of approximating functions by linear combinations of shifts of a fixed function, a B-spline of the first, second, or third degree, is proposed. This method is exact for the classes of splines of the corresponding degree with deficiency 1. We obtain estimates of strong type for the norms of approximating operators in certain spaces, from which it follows immediately that the approximation in these spaces coincides in order with the best approximation by splines.

CSSI-PRO: a method for secondary structure type editing, assignment and estimation in proteins using linear combination of backbone chemical shifts

Estimation of secondary structure in polypeptides is important for studying their structure, folding and dynamics. In NMR spectroscopy, such information is generally obtained after sequence specific resonance assignments are completed. We present here a new methodology for assignment of secondary structure type to spin systems in proteins directly from NMR spectra, without prior knowledge of resonance assignments. The methodology, named Combination of Shifts for Secondary Structure Identification in Proteins (CSSI-PRO), involves detection of specific linear combination of backbone (1)H(alpha) and (13)C' chemical shifts in a two-dimensional (2D) NMR experiment based on G-matrix Fourier transform (GFT) NMR spectroscopy. Such linear combinations of shifts facilitate editing of residues belonging to alpha-helical/beta-strand regions into distinct spectral regions nearly independent of the amino acid type, thereby allowing the estimation of overall secondary structure content of the protein. Comparison of the predicted secondary structure content with those estimated based on their respective 3D structures and/or the method of Chemical Shift Index for 237 proteins gives a correlation of more than 90% and an overall rmsd of 7.0%, which is comparable to other biophysical techniques used for structural characterization of proteins. Taken together, this methodology has a wide range of applications in NMR spectroscopy such as rapid protein structure determination, monitoring conformational changes in protein-folding/ligand-binding studies and automated resonance assignment.

Symmetric multi-box splines for higher degree

We consider the space Sn=Sn(v0,…,vn+r) of compactly supported Cn−1 piecewise polynomials on a mesh M of lines through ℤ2 in directions v0,…,vn+r. A sequence ψ=(ψ1,…,ψr) of elements of Sn is called a multi-box spline if every element of Sn is a finite linear combination of shifts of (the components of) ψ. For the case n=2, 3 we give some examples for multi-box splines and show that they are not always stable. It is further shown that any Cn−1 piecewise polynomial of degree n≥2 on M, is possibly a symmetric multi-box spline.

A Method to Selectively Observe a Desired Linear Combination of Chemical Shifts in GFT Projection NMR Spectroscopy

In recent years, several new methods have emerged for rapid NMR data collection. One class of methods is projection NMR spectroscopy, which involves phase sensitive joint sampling of two or more chemical shifts in an indirect dimension of a multidimensional NMR experiment. In the current implementations of this method, cosine/sine modulation of all chemical shifts involved in the joint sampling are collected and stored as separate FIDs. A post-acquisition data processing step (application of G-matrix) then separates the different inter-modulations of chemical shifts. Thus, joint sampling of K+1 spins results in 2 K combination of chemical shifts (also representing 2 K projection angles). One limitation of this approach is that even if only a few of the 2 K components of the multiplet (or projection angles) is desired, an entire data set containing information for all 2 K shift combinations is collected. We propose here a simple method which releases this restriction and allows one to selectively detect only the desired linear combination of chemical shifts/projection an- gles out of 2 K combinations in a phase sensitive manner. The method involves selecting the appropriate cosine/sine modu- lations of chemical shifts and forming the desired linear combination by phase cycling of the radiofrequency pulses and receiver. This will benefit applications where only certain linear combination of shifts are desired or/and are sufficient. Further, G-matrix transformation required for forming the linear combination is performed within the pulse sequence. This avoids the need for any post-acquisition data processing. Taken together, this mode of data acquisition will foster new applications in projection NMR spectroscopy for rapid resonance assignment and structure determination.

GFT NMR Experiments for Polypeptide Backbone and13CβChemical Shift Assignment

(4,3)D, (5,3)D and (5,2)D GFT triple resonance NMR experiments are presented for polypeptide backbone and (13)C(beta) resonance assignment of (15)N/(13)C labeled proteins. The joint sampling of m = 2, 3 or 4 indirect chemical shift evolution periods of 4D and 5D NMR experiments yields the measurement of 2(m) - 1 linear combinations of shifts. To obtain sequential assignments, these are matched in corresponding experiments delineating either intra or interresidue correlations. Hence, an increased set of matches is registered when compared to conventional approaches, and the 4D or 5D information allows one to efficiently break chemical shift degeneracy. Moreover, comparison of single-quantum chemical shifts obtained after a least squares fit using either the intra or the interresidue data demonstrates that GFT NMR warrants highly accurate shift measurements. The new features of GFT NMR based resonance assignment strategies promise to be of particular value for establishing automated protocols.

Approximation in the mean of linear combinations of shifts of certain functions

Approximation in the mean of linear combinations of shifts of certain functions