Trigonometric Polynomials Research Articles

Shape Preserving Approximation of Periodic Functions: Conclusion

AbstractWe give here the final results about the validity of Jackson-type estimates in comonotone approximation of $$2\pi $$ 2 π -periodic functions by trigonometric polynomials ($$q=1$$ q = 1 ). For coconvex ($$q=2$$ q = 2 ) and the so called co-q-monotone, $$q>2$$ q > 2 , approximations, everything is known by now. Thus, this paper concludes the research on Jackson type estimates of Shape Preserving Approximation of periodic functions by trigonometric polynomials. It is interesting to point out that the results for comonotone approximation of a periodic function are substantially different than the analogous results for comonotone approximation of a continuous function on a finite interval, by algebraic polynomials.

Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs

AbstractWe provide a new approach to the optimization of trigonometric polynomials with crystallographic symmetry. This approach widens the bridge between trigonometric and polynomial optimization. The trigonometric polynomials considered are supported on weight lattices associated to crystallographic root systems and are assumed invariant under the associated reflection group. On one hand the invariance allows us to rewrite the objective function in terms of generalized Chebyshev polynomials of the generalized cosines; On the other hand the generalized cosines parameterize a compact basic semi algebraic set, this latter being given by an explicit polynomial matrix inequality. The initial problem thus boils down to a polynomial optimization problem that is straightforwardly written in terms of generalized Chebyshev polynomials. The minimum is to be computed by a converging sequence of lower bounds as given by a hierarchy of relaxations based on the Hol–Scherer Positivstellensatz and indexed by the weighted degree associated to the root system. This new method for trigonometric optimization was motivated by its application to estimate the spectral bound on the chromatic number of set avoiding graphs. We examine cases of the literature where the avoided set affords crystallographic symmetry. In some cases we obtain new analytic proofs for sharp bounds on the chromatic number while in others we compute new lower bounds numerically.

A frequency-independent bound on trigonometric polynomials of Gaussians and applications

We prove a frequency-independent bound on trigonometric functions of a class of singular Gaussian random fields, which arise naturally from weak universality problems for singular stochastic PDEs. This enables us to reduce the regularity assumption on the nonlinearity of the microscopic models (for pathwise convergence) in KPZ and dynamical Φ34 in the previous works of Hairer-Xu and Furlan-Gubinelli to heuristically optimal thresholds required by PDE structures.

Найкраще наближення в інтегральній метриці лінійної комбінації періодичних функцій різної парності

In many areas of mathematics, extremal problems often arise related to approximation characteristics of both the approximating functions and the properties of the elements being approximated. For example, in the case of polynomials, the problem can involve the number of points where the values of the function coincide with the values of the polynomial used to replace the function on the studied interval. In practice, the problem of approximating a function from a given set R is reduced to replacing it, according to a defined algorithm, with a function (polynomial) of a fixed degree that is close to it in a certain sense. In uniform and integral metrics, the problem of finding the exact values of the best approximations for classes of r-times differentiable functions, where r is a natural number, has been explored in the works of J. Favard [1], N. I. Akhiezer, M. G. Krein [2], B. Nadi, S. M. Nikolsky [3], V. K. Dzyadyk [4], S. B. Stechkin, Sun Yun-shen, and others. The final results concerning the solution of the best approximation problem on Weyl-Nagy classes for arbitrary values of the parameters defining these classes belong to the Ukrainian mathematician V. K. Dzyadyk [4]. In his works on function classes generated by the well-known Bernoulli kernels, it was established that the number of coincidence points between the kernel and the approximating polynomial of degree n − 1, including their multiplicities, does not exceed 2n, which allowed for obtaining final results. The work [5] presents cases of such linear combinations of even or odd kernels for which the number of uniformly distributed interpolation points equals 2n + 2 for a polynomial of degree n − 1 that deviates the least in the metric of the L – space from the studied linear combination. The idea of studying composite kernels expressed as a linear combination of component terms belongs to O. I. Stepanyets [6], and it was implemented in problems of joint approximation of functions and their derivatives. In the 1980s and 1990s, O. I. Stepanets developed a new approach to classifying periodic functions, which allowed for a fine classification of extremely broad sets of periodic functions. The results obtained for these classes are, on one hand, of a general nature, and on the other, they provide a whole series of new, previously unknown results that could not be achieved with previously known classes. Following the approaches to function classification, we can consider a linear combination of function classes as a certain single class of a more complex nature. Then, the problem of finding the exact values of the upper bounds of the best approximations reduces to the problem of the best approximation for this composite class, which corresponds to convolutions with the generating composite kernel. In this work, we investigate linear combinations of three continuous 2π-periodic kernels of different parity, and it is established that there exist such composite kernels that their trigonometric polynomial of the best approximation of order n − 1 in the integral metric interpolates the kernel at 2n + 2 uniformly distributed points

Best one sided approximation of unbounded functions trigonometrical polynomials for one dimensional weighted-space

The aim of studying this research is to find the best one sided trigonometrical approximation of unbounded function of one variable in weighted space. Moreover we estimated the degree of the best one-sided trigonometrical approximation by averaged modulus of smoothness in same space.

A novel hybrid function-based Ritz method for static and free vibration of axially loaded bi-directional functionally graded porous beams

This paper introduces a novel hybrid function, synthesizing trigonometric and Hermitian polynomials, to analyze the structural responses of bi-directional functionally graded porous (2D-FGP) beams. The displacement field of the beam is based on the higher-order shear deformation theory. A power-law relation governs the material variation along the x- and z-directions, with the porosity manifesting in three distinct distributions. The governing equations are deduced through the application of Lagrange’s equation. The proposed hybrid function is developed to delineate the displacement field. The deflections, stresses, buckling loads, and natural frequencies of 2D-FGP beams are computed for diverse gradation exponents in the x- and z-directions, boundary conditions, porosity type, porosity coefficient, and slenderness ratio. A comparative assessment with finite element method results indicates that the findings agree with available data. It substantiates the accuracy and efficiency of the proposed methodology.

Theoretical velocity of an object frictionally coupled to a two-mode vibrating plate.

Net velocity has been demonstrated for objects frictionally coupled to a flat plate that oscillates periodically in-plane with two frequencies, provided plate displacement is nonantiperiodic: the ratio of frequencies γ cannot be the ratio of two odd integers. We give a mathematical derivation of the experimentally determined dependence of mean velocity on the relative amplitudes of the two frequency modes, and the phase lag between the modes, when γ=2, and when the magnitude of plate acceleration is much larger than the magnitude of acceleration by static friction. The approach uses an analysis of the symmetry properties of the roots of trigonometric polynomials, without explicit determination of those roots. The behavior when γ=1/2, and specific phase lags that inhibit net velocity for general γ, are also determined.

On symmetry adapted bases in trigonometric optimization

The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow an established relaxation strategy in order to obtain a converging hierarchy of lower bounds. Those bounds are obtained by numerically solving semi-definite programs (SDPs) on the cone of positive semi-definite Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu Dumitrescu (2007). To exploit the invariance, we show that the group has an induced action on the Toeplitz matrices and prove that the feasible region of the SDP can be restricted to the invariant matrices, whilst retaining the same solution. Then we construct a symmetry adapted basis tailored to this group action, which allows us to block-diagonalize invariant matrices and thus reduce the computational complexity to solve the SDP.The approach is in its generality novel for trigonometric optimization and complements the one that was proposed as a poster at the ISSAC 2022 conference Hubert et al. (2022) and later extended to Hubert et al. (2024). In the previous work, we first used the invariance of the trigonometric polynomial to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. Now, we first make the relaxation and then exploit invariance.Partial results of this article have been presented as a poster at the ISSAC 2023 conference Metzlaff (2023).

A fifth order block methods for solving second-order stiff ordinary differential equations using trigonometric functions and polynomial function as the basis function

A fifth order block methods for solving second-order stiff ordinary differential equations using trigonometric functions and polynomial function as the basis function

Factoring non-negative operator valued trigonometric polynomials in two variables

AbstractIt is shown using Schur complement techniques that on finite dimensional Hilbert spaces, a non-negative operator valued trigonometric polynomial in two variables with degree $$(d_1,d_2)$$ ( d 1 , d 2 ) can be written as a sum of hermitian squares of at most $$2d_2$$ 2 d 2 analytic polynomials.

NESTED POLYNOMIALS TRIGONOMETRIC AND HYPERBOLIC TYPE WITH APPLICATIONS

<p class="ABSTRACT">In this paper we introduce nested trigonometric and hyperbolic polynomials of rank n by employing nested trigonometric and hyperbolic functions, respectively. Our investigation delves into some interesting properties of nested polynomials, establishing some recurrence formulas and deriving some useful summation formulas. Additionally, we establish a connection between nested trigonometric polynomials and one specific class of special polynomials that satisfy a second-order Sturm-Liouville differential equation. Finally, we outline several applications of nested polynomials. Among others, it is shown that nested trigonometric polynomials can be used for the linear static analysis of curved Bernoulli–Euler beam.</p>

On the number of limit cycles in piecewise smooth generalized Abel equations with many separation lines

This paper investigates generalized Abel equations of the form dx/dθ=A(θ)xp+B(θ)xq, where p, q∈Z≥2, p≠q, and A(θ) and B(θ) are piecewise trigonometrical polynomials of degree m with n−1∈N+ separation lines 0<θ1<θ2<⋯<θn−1<2π. The main objective is to obtain the maximum number of non-zero limit cycles (i.e., non-zero isolated periodic solutions) that the equation can have, denoted by Hθ1,θ2,…,θn−1(m), and to analyze how the number and location of separation lines {θi}i=1n−1 affect Hθ1,θ2,…,θn−1(m). By using the theories of Melnikov functions and ECT-systems, we obtain lower bounds for Hθ1,θ2,…,θn−1(m). Our result extend those of Huang et al. who studied the special case of n=2, and reveal that the lower bounds decrease in the presence of pairs of symmetrical separation lines.

Ritz Variational Method for the Analysis of Thin Rectangular Plate Bending Problems with Adjacent Edges Clamped and Simply Supported Using the Super position of Trigonometric Series and Polynomial Basis Functions

Ritz Variational Method for the Analysis of Thin Rectangular Plate Bending Problems with Adjacent Edges Clamped and Simply Supported Using the Super position of Trigonometric Series and Polynomial Basis Functions

Exponential concentration for the number of roots of random trigonometric polynomials

Exponential concentration for the number of roots of random trigonometric polynomials

Fast Decreasing Trigonometric Polynomials and Applications

We construct trigonometric polynomials that fast decrease towards ±π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pm \\pi $$\\end{document}. We apply them to construct a trigonometric polynomial the derivative of which interpolates the derivative of a given 2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\pi $$\\end{document}-periodic function, at some prescribed distinct points in [-π,π)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-\\pi ,\\pi )$$\\end{document}, and vanishes at some other prescribed points in that interval. The construction requires that the function possesses derivatives where the interpolation is supposed to take place. Still, we are able to apply the result to trigonometric approximation of a 2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\pi $$\\end{document}-periodic piecewise algebraic polynomial which is merely continuous, while interpolating its derivative at some points (that, obviously, are not knots).

A simple method to construct multivariate dual framelets with high-order vanishing moments

When constructing multivariate framelets, it is often unavoidable to work with matrices of multivariate trigonometric polynomials and complicated matrix decomposition problems. These problems become even harder when good properties such as high-order vanishing moments are required on the framelets. In this paper, we establish a new method for constructing multivariate dual framelets with high-order vanishing moments. The underlying scheme of our algorithm is the famous Mixed Extension Principle that allows us to derive the high-pass filters (or framelet generators) from a given pair of refinable filters with high-order linear-phase moments. Our method only involves two steps: (1) directly constructing the first few pairs of high-pass filters by using the linear-phase moment conditions of the refinement filters; (2) solving a system of linear equations to obtain the rest of the high-pass filters. Both are easy to implement for scientific computation, regardless of what dimension or dilation matrix we work with. Apart from high-order vanishing moments, we will see that if the refinement filters take coefficients from some subfield [Formula: see text] of [Formula: see text] that is closed under complex conjugation, so do the high-pass filters. Furthermore, our algorithm gives the upper bounds for the number of high-pass filters in arbitrary dimensions. At the end of the paper, we will give several illustrative examples, from which we can also see that the support sizes of the high-pass filters are comparable with those of the refinement filters.

New pipe element based on Hermite-Jackson interpolation

A new pipe finite element is proposed for piping analysis within the framework of linear shell theory. The approach involves the use of a mixed interpolation of classical polynomial and trigonometric polynomial spaces, with trigonometric interpolation performed via Hermite-Jackson polynomials along the mid-surface section. Error estimates are provided and a convergence analysis is performed under specific assumptions on the regularity of the solution. The proposed element is validated through several numerical examples, which demonstrate its accuracy and efficiency in terms of computational cost. This method represents a promising approach for addressing the challenges of piping analysis.

Approximation and Interpolation of Singular Measures by Trigonometric Polynomials

AbstractComplex valued measures of finite total variation are a powerful signal model in many applications. Restricting to the d-dimensional torus, finitely supported measures can be exactly recovered from their trigonometric moments up to some order if this order is large enough. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the 1-Wasserstein distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the indicator function on the support of the measure and to converge to zero outside.

Trigonometric Polynomials with Frequencies in the Set of Cubes

Trigonometric Polynomials with Frequencies in the Set of Cubes

Fejér–Riesz factorization in the QRC-subalgebra and circularity of the quaternionic numerical range

We provide a characterization when the quaternionic numerical range of a matrix is a closed ball with center 0. The proof makes use of Fejér–Riesz factorization of matrix-valued trigonometric polynomials within the algebra of complex matrices associated with quaternion matrices.