Orthonormal System Research Articles

On representing rotations by Rodrigues parameters in non-orthonormal reference systems.

A Rodrigues vector is a triplet of real numbers used for parameterizing rotations or orientations in three-dimensional space. Because of its properties (e.g. simplicity of fundamental regions for misorientations) this parameterization is frequently applied in analysis of orientation maps of polycrystalline materials. By conventional definition, the Rodrigues parameters are specified in orthonormal coordinate systems, whereas the bases of crystal lattices are generally non-orthogonal. Therefore, the definition of Rodrigues parameters is extended so they can be directly linked to non-Cartesian bases of a crystal. The new parameters are co- or contravariant components of vectors specified with respect to the same basis as atomic positions in a unit cell. The generalized formalism allows for redundant crystallographic axes. The formulas for rotation composition and the relationship to the rotation matrix are similar to those used in the Cartesian case, but they have a wider range of applicability: calculations can be performed with an arbitrary metric tensor of the crystal lattice. The parameterization in oblique coordinate frames of lattices is convenient for crystallographic applications because the generalized parameters are directly related to indices of rotation-invariant lattice directions and rotation-invariant lattice planes.

Dilation-and-modulation systems on the half real line

Translation, dilation, and modulation are fundamental operations in wavelet analysis. Affine frames based on translation-and-dilation operation and Gabor frames based on translation-and-modulation operation have been extensively studied and seen great achievements. But dilation-and-modulation frames have not. This paper addresses a class of dilation-and-modulation systems in $L^{2}(\mathbb {R}_{+})$ . We characterize frames, dual frames, and Parseval frames in $L^{2}(\mathbb {R}_{+})$ generated by such systems. Interestingly, it turns out that, for such systems, Parseval frames, orthonormal bases, and orthonormal systems are mutually equivalent to each other, while this is not the case for affine systems and Gabor systems.

A Note on Time Operators

We construct a time operator of a self-adjoint operator H with finite dimensional CCR-domain. As corollaries, we show that there exists a time operator of H with infinite dimensional CCR-domain. Moreover, we construct a time operator of H with dense CCR-domain if there is a complete orthonormal systems which consists of eigenvectors of H.

Fractal Jacobi Systems and Convergence of Fourier–Jacobi Expansions of Fractal Interpolation Functions

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of $${\mathbb{R}}$$ interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in $${\mathcal{L}^{2}(-1,1)}$$ with respect to the weight function $${\rho^{(r,s)}(x)=(1-x)^{r} (1+x)^{s}}$$ , $${r > -1}$$ and $${s > -1}$$ . In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of $${\alpha}$$ -fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions $${\mathcal{L}_{\rho}^{2}(-1,1)}$$ is found. The Fourier–Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier–Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier–Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.

Fitted Reproducing Kernel Method for Solving a Class of Third-Order Periodic Boundary Value Problems

In this article, the reproducing kernel Hilbert space [0, 1] is employed for solving a class of third-order periodic boundary value problem by using fitted reproducing kernel algorithm. The reproducing kernel function is built to get fast accurately and efficiently series solutions with easily computable coefficients throughout evolution the algorithm under constraint periodic conditions within required grid points. The analytic solution is formulated in a finite series form whilst the truncated series solution is given to converge uniformly to analytic solution. The reproducing kernel procedure is based upon generating orthonormal basis system over a compact dense interval in Sobolev space to construct a suitable analytical-numerical solution. Furthermore, experiments result of some numerical examples are presented to illustrate the good performance of the presented algorithm. The results indicate that the reproducing kernel procedure is powerful tool for solving other problems of ordinary and partial differential equations arising in physics, computer and engineering fields.

Absolute convergence of Fourier series in bounded complete double orthonormal systems

Absolute convergence of Fourier series in bounded complete double orthonormal systems

The Fourier coefficients of continuous functions with respect to certain orthonormal systems

The properties of the Fourier coefficients for some classes of functions with respect to both the Haar system and general orthonormal systems are studied. It is established that the results obtained in the paper cannot be strengthened.

Convergence of Fourier series with respect to general orthonormal systems

Convergence of Fourier series with respect to general orthonormal systems

On a system of piecewise linear functions with vanishing integrals on R

For an admissible sequence T we define an orthonormal system consisting of piecewise linear functions with vanishing integrals on R. Necessary and sufficient conditions on T are found for the corresponding system to be a basis in H 1(R).

Polynomial approximation via compressed sensing of high-dimensional functions on lower sets

This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. Of particular interest is the setting where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we present an innovative weighted $\ell_1$ minimization procedure with a precise choice of weights, and a new iterative hard thresholding method, for imposing the downward closed preference. Theoretical results reveal that our computational approaches possess a provably reduced sample complexity compared to existing compressed sensing techniques presented in the literature. In addition, the recovery of the corresponding best approximation using these methods is established through an improved bound for the restricted isometry property. Our analysis represents an extension of the approach for Hadamard matrices in [5] to the general case of continuous bounded orthonormal systems, quantifies the dependence of sample complexity on the successful recovery probability, and provides an estimate on the number of measurements with explicit constants. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the novel weighted $\ell_1$ minimization strategy.

The Fourier Coefficients of Continuous Functions with Respect to Certain Orthonormal Systems

В работе изучены свойства коэффициентов Фурье некоторых классов функций как относительно системы Хаара, так и для общих ортонормированных систем (ОНС). Установлено, что полученные результаты неусиляемы. Библиография: 9 названий.

Convergence of Fourier Series with Respect to General Orthonormal Systems

Convergence of Fourier Series with Respect to General Orthonormal Systems

Absolute convergence of Fourier series in bounded complete double orthonormal systems

Absolute convergence of Fourier series in bounded complete double orthonormal systems

Summability of general orthonormal Fourier series

S. Banach in [1] proved that for any function f ∈ L2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α > 0. D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α > 0. In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φn(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.

Étude de la normalité du bandeau orbito-naso-frontal

Craniosynostoses are cranio-facial malformations affecting about 1/2100 newborns in France. The involvement of anterior sutures (coronal and metopic) leads to orbito-frontal deformities. The treatment calls upon surgery the goal being, on an esthetic point of view, to restore a normal anatomy. The purpose of our work was to establish if some facial and/or frontal measures easy to perform on a CT are correlated to the global shape of the normal orbito-naso-frontal bandeau (ONFB). Cranial CTs of 123 consecutive non-malformed children aged between 4 and 12 months were selected in the database of Department of Radiology of the University Hospital of Besançon - France. The CTs were all relocated by rigid transformation in an orthonormal coordinate system. On each of the 123 CTs, 21 reproducible measures representative of the global shape of the ONFB were made. Statistical analyses of these measures were achieved, considering age and gender, in order to determine the correlation between each measure and the ONFB shape. The only measure statistically correlated to the ONFB shape was the distance between the fronto-zygomatic sutures (FZD). The FZD was independent from age (in an interval of 4 months) and from gender. The 20 other measures did not show any correlation with age or gender. The FZD allows in itself to determine the ONFB global shape. This measure, easily available on a CT, can help the surgeon to perform a customized reshaping of the ONFB. The development of a surgical template using this measure is ongoing.

On the existence of RUC systems in rearrangement invariant spaces

We characterize rearrangement invariant spaces X on [0, 1] with the property that each orthonormal system in X which is uniformly bounded in some Marcinkiewicz space , for equivalent to , , is a system of Random Unconditional Convergence (RUC system).

Uniformly bounded orthonormal polynomials on the sphere

Given any $\varepsilon>0$, we construct an orthonormal system of $n_k$ uniformly bounded polynomials of degree at most $k$ on the unit sphere in $\mathbb R^{m+1}$ where $n_k$ is bigger than $1-\varepsilon$ times the dimension of the space of polynomials of degree at most $k$. Similarly we construct an orthonormal system of sections of powers $L^k$ of a positive holomorphic line bundle on a compact K\"ahler manifold with cardinality bigger than $1-\varepsilon$ times the dimension of the space of global holomorphic sections to $L^k$.

Ore crushing process dynamics modeling using the laguerre model

The problem of ore crushing process dynamics modeling using the orthonormal Laguerre functions was considered. The analysis has shown that the maximum sampling interval that will allow to reconstruct the transition process by the discrete sample is 1.3 seconds. As expected, the structure that includes only the first-order Laguerre filter has the worst accuracy. Increasing the number of functions in the orthonormal system leads to the higher modeling quality of the crushing process output. It was also determined that the relationship between the scale factor and mean square error of identification is unimodal in nature, therefore, it is advisable to use optimization methods for finding the optimal value for this parameter. The research has allowed to reveal the structure and the scale factor of the Laguerre model, and the sampling interval, which allow to ensure the minimum mean square error when using the least squares method for estimating the model parameters.

On absolute convergence of multiple Fourier series

We consider absolute convergence of multiple series of Fourier-Haar coefficients for functions of many variables with partial derivatives of higher order. We show that the obtained results are best possible for general orthonormal systems.

Frames and Riesz bases of twisted shift-invariant spaces in [formula omitted

In this paper, twisted shift-invariant spaces in L2(R2n) are introduced. Characterizations of orthonormal system and Bessel sequence of twisted translates in L2(R2n) are obtained in terms of Weyl transform. These results appear to be totally different from the classical case of characterizations of orthonormal systems and Bessel sequences of translates in L2(Rn). Further, characterizations of frames, Riesz basis in twisted shift-invariant spaces in L2(R2n) are also obtained.