Trigonometric Polynomials Research Articles

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.

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.

Hyponormality of Toeplitz Operators with Trigonometric Polynomial Symbols

For a trigonometric polynomial of the form Where and are nonzero. In this paper, we show that the Toeplitz operator is hyponormal if and only if and

On Best Multiplier Approximation of k-Monotone by Trigonometric Polynomial

The main goal of this paper is to study the degree of the best multiplier approximation of monotone unbounded functions in L_(p,λ_n)-space on the closed interval [-π,π] by means of K-functional, which we represented with, K(f,L_(p,λ_n ),W_(p,λ_n)^1,W ̃_(p,λ_n)^1), defined by the W_(p,λ_n)^1 and W ̃_(p,λ_n)^1 that are referred to during this research. In addition, we have established a set of definitions, concepts and some useful lemmas that are needed in our work.

Multi-antenna dual-blind deconvolution for joint radar-communications via SoMAN minimization

In joint radar-communications (JRC) applications such as secure military receivers, often the radar and communications signals are overlaid in the received signal. In these passive listening outposts, the signals and channels of both radar and communications are unknown to the receiver. The ill-posed problem of recovering all signal and channel parameters from the overlaid signal is termed as dual-blind deconvolution (DBD). In this work, we investigate DBD for a multi-antenna receiver. We model the radar and communications channels with a few (sparse) continuous-valued parameters such as time delays, Doppler velocities, and directions-of-arrival (DoAs). To solve this highly ill-posed DBD, we propose to minimize the sum of multivariate atomic norms (SoMAN) that depend on unknown parameters. To this end, we devise an exact semidefinite program using theories of positive hyperoctant trigonometric polynomials (PhTP). Our theoretical analyses show that the minimum number of samples and antennas required for perfect recovery is logarithmically dependent on the maximum of the number of radar targets and communications paths rather than their sum. We show that our approach is easily generalized to include several practical issues such as gain/phase errors and additive noise. Numerical experiments show the exact parameter recovery for different JRC scenarios.

Analytical Design of Gaussian Anisotropic 2D FIR Filters and Their Implementation Using the Block Filtering Approach

This work proposes an analytical design procedure for a particular class of 2D filters, namely anisotropic Gaussian FIR filters. The design is achieved in the frequency domain and starts from a low-pass Gaussian 1D prototype with imposed specifications, whose frequency response is efficiently approximated by a factored trigonometric polynomial using the Chebyshev series. Then, using specific 1D to 2D frequency mappings applied to the prototype, the frequency response for a 2D anisotropic filter with a specified orientation angle is directly derived in two versions, namely with a straight or elliptical shape in the frequency plane. The resulting filters have an accurate shape with low distortion. Several design examples for specified parameters (angle and selectivity) are provided. Then, simulations of directional filtering on various test images are given, which show their capability of extracting oriented lines or other various oriented objects from synthetic or real-life images. Finally, a computationally efficient implementation at the system level is proposed, based on a polyphase decomposition and block-filtering approach, which yields a 2D filter with a high degree of parallelism and low arithmetic complexity.