Trigonometric Polynomials Of Degree Research Articles

One-Sided Integral Approximations of the Generalized Poisson Kernel by Trigonometric Polynomials

We consider the generalized Poisson kernel Π q,α = cos(απ/2)P + sin(απ/2)Q with q ∈ (−1, 1) and α ∈ ℝ, which is a linear combination of the Poisson kernel $$P(t) = 1/2 + \sum\nolimits_{k = 1}^\infty {{q^k}} \cos kt$$ and the conjugate Poisson kernel $$Q(t) = \sum\nolimits_{k = 1}^\infty {{q^k}} \sin kt$$ . The values of the best integral approximation to the kernel Π q,α from below and from above by trigonometric polynomials of degree not exceeding a given number are found. The corresponding polynomials of the best one-sided approximation are obtained.

Critical lengths of cycloidal spaces are zeros of Bessel functions

Cycloidal spaces are generated by the trigonometric polynomials of degree one and algebraic polynomials. The critical length of a cycloidal space is the supremum of the lengths of the intervals on which the Hermite interpolation problems are unisolvent. The critical length is related with the critical length for design purposes in computer-aided geometric design. This paper shows an unexpected connection of critical lengths with the zeros of Bessel functions. We prove that the half of the critical length of a cycloidal space is the first positive zero of a Bessel function of the first kind.

Estimates for mean-square norms of functions with lacunary Fourier series

We consider the properties of functions f from the space L 2(T) on the period T = [−π, π) with lacunary Fourier series such that the size of each gap is not less than a given positive integer q − 1. We find two-sided estimates of the L 2 norms of such functions on T in terms of similar norms (more exactly, seminorms) on intervals I of length |I| = 2h < 2π. The estimates are obtained in terms of best one-sided integral approximations to the characteristic function of the interval (−h, h) by trigonometric polynomials of degree at most q−1. The issue considered in this paper appeared first in N. Wiener’s studies (1934). Important results in this area were obtained by A.E. Ingham (1936) and by A. Selberg in the 1970s.

Semi-quasitriangularity of Toeplitz operators

In this paper we give a necessary and sufficient condition, in terms of the coefficients of ?, in order for the Toeplitz operator T? to be semi-quasitriangular when ? is a trigonometric polynomial of degree two and has real coefficients.

The number of polynomial solutions of polynomial Riccati equations

Consider real or complex polynomial Riccati differential equations a(x)y˙=b0(x)+b1(x)y+b2(x)y2 with all the involved functions being polynomials of degree at most η. We prove that the maximum number of polynomial solutions is η+1 (resp. 2) when η≥1 (resp. η=0) and that these bounds are sharp.For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most η≥1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2η (resp. 3) when η≥2 (resp. η=1) and, again, these bounds are sharp.Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.

On the number of limit cycles for perturbed pendulum equations

We consider perturbed pendulum-like equations on the cylinder of the form x¨+sin⁡(x)=ε∑s=0mQn,s(x)x˙s where Qn,s are trigonometric polynomials of degree n, and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case ε=0 in terms of m and n. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. Some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be Chebyshev systems.

On the trigonometric polynomial Ansatz for gravity water waves

We show that Gerstner’s wave exhausts all solutions to the gravity water wave problem that have particle paths prescribed by trigonometric polynomials of degree one. We further show that any Ansatz of degree two reduces to Gerstner’s wave.

The Bernstein-Szegö inequality for fractional derivatives of trigonometric polynomials

On the set F n of trigonometric polynomials of degree n ≥ 1 with complex coefficients, we consider the Szego operator \(D_\theta ^\alpha \) defined by the relation \(D_\theta ^\alpha f_n (t) = \cos \theta D^\alpha f_n (t) - \sin \theta D^\alpha \tilde f_n (t)\) for α, θ ∈ ℝ, where α ≥ 0. Here, \(D^\alpha f_n \) and \(D^\alpha \tilde f_n \) are the Weyl fractional derivatives of (real) order α of the polynomial f n and of its conjugate \(\tilde f_n \). In particular, we prove that, if α ≥ n ln 2n, then, for any θ ∈ ℝ, the sharp inequality \(\left\| {\cos \theta D^\alpha f_n - \sin \theta D^\alpha f_n } \right\|_{L_p } \leqslant n^\alpha \left\| {f_n } \right\|_{L_p } \) holds on the set F n in the spaces L p for all p ≥ 0. For classical derivatives (of integer order α ≥ 1), this inequality was obtained by Szego in the uniform norm (p = ∞) in 1928 and by Zygmund for 1 ≤ p < ∞ in 1931–1935. For fractional derivatives of (real) order α ≥ 1 and 1 ≤ p ≤ ∞, the inequality was proved by Kozko in 1998.

Analytic Monotone Pseudospectral Interpolation

In the paper we construct an analytic pseudospectral interpolant to a monotone interpolating table. The approximant is a sum of a linear function and a trigonometric polynomial of degree depending on the number and variation of the interpolating conditions. In the case when the interpolating cites are equidistant exact interpolation is obtained. In all of the other cases the error of approximation is shown to decrease geometrically fast with respect to the degree of the trigonometric polynomial. The least degree of the interpolating polynomial is discussed. Applications in signal processing including numerical examples are considered.

Trigonometric approximation of functions belonging to certain Lipschitz classes by C1⋅ T operator

The study of approximation properties of the periodic functions in Lp(p ≥ 1)-spaces, in general and in Lipschitz classes Lip α, Lip (α, p), Lip (ξ(t), p) and weighted Lipschitz class W(Lp, ω(t), β), in particular, through trigonometric Fourier series, although is an old problem and known as Fourier approximation in the existing literature, has been of a growing interests over the last four decades due to its application in filters and signals [E. Z. Psarakis and G. V. Moustakides, An L2-based method for the design of 1-D zero phase FIR digital filters, IEEE Trans. Circuits Systems I Fundam. Theory Appl., 44(7) (1997) 551–601]. The most common methods used for the determination of the degree of approximation of periodic functions are based on the minimization of the Lp-norm of f(x) - Tn(x), where Tn(x) is a trigonometric polynomial of degree n and called approximant of the function f. In this paper, we discuss the approximation properties of the periodic functions in the Lipschitz classes Lip α and W(Lp, ω(t), β), p ≥ 1 by a trigonometric polynomial generated by the product matrix means of the Fourier series associated with the function. The degree of approximation obtained in our theorems of this paper is free from p and sharper than earlier results.

Bases of trigonometric polynomials consisting of shifts of Dirichlet kernels

The system of shifts of Dirichlet kernel on \(\frac{{2k\pi }} {{2n + 1}} \), k = 0, ± 1, …, ± n, and the system of such shifts of conjugate Dirichlet kernels with \(\frac{1} {2} \) are orthogonal bases in the space of trigonometric polynomials of degree n. The system of shifts of the kernels \(\Sigma _{k = m}^n \) cos kx and \(\Sigma _{k = m}^n \) sin kx on \(\frac{{2k\pi }} {{n - m + 1}} \), k = 0, 1, …, n−m, is an orthogonal basis in the space of trigonometric polynomials with the components from m ⩾ 1 to n. There is no orthogonal basis of shifts of any function in this space for 0 < m < n.

On the Critical Lengths of Cycloidal Spaces

The cycloidal spaces C n , generated by the trigonometric polynomials of degree 1 and algebraic polynomials of degree n−2, have important applications in computer-aided geometric design. The critical length of C n is the supremum of the lengths of the intervals on which C n is an extended Chebyshev space. We provide a constructive procedure to obtain the critical lengths of the spaces C n .

A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving

We show that the zeros of a trigonometric polynomial of degree N with the usual (2N + 1) terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements Mjk. This matrix is a multiplication matrix in the sense that, after first defining a vector whose elements are the first 2N basis functions, . This relationship is the eigenproblem; the zeros tk are the arccosine function of λk/2 where the λk are the eigenvalues of . We dub this the “Fourier Division Companion Matrix”, or FDCM for short, because it is derived using trigonometric polynomial division. We show through examples that the algorithm computes both real and complex-valued roots, even double roots, to near machine precision accuracy.

On a result by geronimus

In 1935, Ya.L. Geronimus found the best integral approximation on the period [−π,π) of the function sin(n + 1)t − 2q sin nt, q ∈ ℝ, by the subspace of trigonometric polynomials of degree at most n − 1. This result is an integral analog of the known theorem by E.I. Zolotarev (1868). At present, there are several methods of proving this fact. We propose one more variant of the proof. In the case |q| ≥ 1, we apply the (2π/n)-periodization and the fact that the function | sin nt| is orthogonal to the harmonic cos t on the period. In the case |q| < 1, we use the duality relations for Chebyshev’s theorem (1859) on a rational function least deviating from zero on a closed interval with respect to the uniform metric.

The non-equivalence between the trigonometric system and the system of functions with pointwise restrictions on values in the uniform and L1 norms

Let n denote the space of trigonometric polynomials of degree n i.e. n = span(e−ikt : |k| ≤ n) ⊂ Lp() and let (Ω, dx) be any mesurable space with finite measure. In this paper we use the quantitative version of the Helson-Rudin-Cohen idempotent theorem due to Green and Sanders (cf. [3]) to prove the following.

Constructing lattice rules based on weighted degree of exactness and worst case error

Recall that an integration rule is said to have a trigonometric degree of exactness m if it integrates exactly all trigonometric polynomials of degree ≤ m. In this paper we focus on high dimensions, say, d ≫ 6. We introduce three notions of weighted degree of exactness, where we use weights to characterize the anisotropicness of the integrand with respect to successive coordinate directions. Unlike in the classical unweighted setting, the minimal number of integration points needed to achieve a prescribed weighted degree of exactness no longer grows exponentially with d provided that the weights decay sufficiently fast. We present a component-by-component algorithm for the construction of a rank-1 lattice rule such that (i) it has a prescribed weighted degree of exactness, and (ii) its worst case error achieves the optimal rate of convergence in a weighted Korobov space. Then we introduce a modified, more practical, version of this algorithm which maximizes the weighted degree of exactness in each step of the construction. Both algorithms are illustrated by numerical results.

LIMIT CYCLES FOR SOME ABEL EQUATIONS HAVING COEFFICIENTS WITHOUT FIXED SIGNS

We prove that some 2π-periodic generalized Abel equations of the form x′ = A(t)xn+ B(t)xm+ C(t)x, with n ≠ m and n, m ≥ 2 have at most three limit cycles. The novelty of our result is that, in contrast with other results of the literature, our hypotheses allow the functions A,B, and C to change sign. Finally we study in more detail the Abel equation x′ = A(t)x3+ B(t)x2, where the functions A and B are trigonometric polynomials of degree one.

Integral approximation of the characteristic function of an interval by trigonometric polynomials

We prove that the value En−1(χh)L of the best integral approximation of the characteristic function χh of an interval (−h, h) on the period [−π,π) by trigonometric polynomials of degree at most n − 1 is expressed in terms of zeros of the Bernstein function cos {nt − arccos[(2q − (1 + q2) cost)/(1 + q2 − 2q cost)]}, t ∈ [0, π], q ∈ (−1,1). Here, the parameters q, h, and n are connected in a special way; in particular, q = sech − tanh for h = π/n.

On the Exact Constant in the Jackson–Stechkin Inequality for the Uniform Metric

The classical Jackson–Stechkin inequality estimates the value of the best uniform approximation of a 2π-periodic function f by trigonometric polynomials of degree ≤n−1 in terms of its r-th modulus of smoothness ωr(f,δ). It reads $$E_{n-1}(f)\leq c_{r}\omega_{r}\biggl(f,\frac{2\pi}{n}\biggr),$$ where cr is some constant that depends only on r. It has been known that cr admits the estimate cr<rar and, basically, nothing else has been proved.

Lower bounds for the number of limit cycles of trigonometric Abel equations

We consider the Abel equation x ˙ = A ( t ) x 3 + B ( t ) x 2 , where A ( t ) and B ( t ) are trigonometric polynomials of degree n and m, respectively, and we give lower bounds for its number of isolated periodic orbits for some values of n and m. These lower bounds are obtained by two different methods: the study of the perturbations of some Abel equations having a continuum of periodic orbits and the Hopf-type bifurcation of periodic orbits from the solution x = 0 .