This note is a continuation of our papers [1,2], devoted to $L$-approximation of characteristic function of $(-h, h)$ by trigonometric polynomials. In the paper [1] the sharp values of the best approximation for the special values of $h$ were found. In [2] we gave the complete solution of the problem for arbitrary values of $h$. In general case [2] the situation is more deep and results are not so simple as in [1]. For applications to the problem of optimal constants in the Jackson-type inequalities we need, however, results on $L$-approximation of $B$-splines and linear combinations of $B$-splines. Here we present some simple results about $L$-approximation of $B$-splines as well as give the the proof of its sharpness for the special values of $h$.
As an application of Bézout's theorem from algebraic geometry, we show that the standard notion of a trigonometric polynomial does not agree with a more naive, but reasonable notion of trigonometric polynomial.
In this paper, we extend the large sieve type estimates to sums involving pth powers of trigonometric polynomials. An approach to such estimates that does not rely on the usual L^2-technique is given. Our method is based on comparing the norm and the spectral radius of convolution operators on a normed space of trigonometric polynomials.
An example of trigonometric polynomials with extremely small uniform norm is given. This example demonstrates the potential limits for extension of Sidon's inequality for lacunary polynomials in a certain direction.
In this article the solution of the special problem of the conditional extremum for the conjugate trigonometric polynomials is given. A possibility to apply this result to the problems of optimal stabilization of quasidynamic chaos in discrete systems is mentioned.
In an earlier article [3], we presented an algorithm that can be used to rigorously check whether a specific cosine or sine polynomial is nonnegative in a given interval or not. The algorithm proves to be an indispensable tool in establishing some recent results on nonnegative trigonometric polynomials. See, for example, [2], [4] and [5]. It continues to play an essential role in several ongoing projects. The algorithm, however, cannot handle general trigonometric polynomials that involve both cosine and sine terms. Some ad hoc methods to deal with such polynomials have been suggested in [3], …
The trigonometric moment problem arises from the study of one-parameter families of centers in polynomial vector fields. It asks for the classification of the trigonometric polynomials $Q$ which are orthogonal to all powers of a trigonometric polynomial $P$. We show that this problem has a simple and natural solution under certain conditions on the monodromy group of the Laurent polynomial associated to $P$. In the case of real trigonometric polynomials, which is the primary motivation of the problem, our conditions are shown to hold for all trigonometric polynomials of degree 15 or less. In t…
We study the problem of reconstructing a multivariate trigonometric polynomial having only few non-zero coefficients from few random samples. Inspired by recent work of Candes, Romberg and Tao we propose to recover the polynomial by Basis Pursuit, i.e., by $\ell^1$-minimization. Numerical experiments show that in many cases the trigonometric polynomial can be recovered exactly provided the number $N$ of samples is high enough compared to the ``sparsity'' -- the number of non-vanishing coefficients. However, $N$ can be chosen small compared to the assumed maximal degree of the trigonometric pol…
We show that for every finite symetric set S of integer vectors, every real trigonometric polynomial on the d dimensional torus with spectrum in S has a zero in every closed ball of diameter D, where D is the sum over S of 1 over 4 times the L2 norm of the vector. We investigate tightness in some special cases.
Using the multilinear theory of oscillatory integral operators, we establish a square function inequality with implications to moment inequalities for discrete exponential sums which frequencies lie on curved hyper-surfaces. In particular a new distributional estimate for toral eigenfunctions is established.
We give a solution of the problem on trigonometric polynomials $f_n$ with the given leading harmonic $y\cos nt$ that deviate the least from zero in measure, more precisely, with respect to the functional $μ(f_n)=mes\{t\in[0,2π]: |f_n(t)|\ge 1\}$. For trigonometric polynomials with a fixed leading harmonic, we consider the least uniform deviation from zero on a compact set and find the minimal value of the deviation over compact subsets of the torus that have a given measure. We give a solution of a similar problem on the unit circle for algebraic polynomials with zeros on the circle.
We obtain optimal trigonometric polynomials of a given degree $N$ that majorize, minorize and approximate in $L^1(\mathbb{R}/\mathbb{Z})$ the Bernoulli periodic functions. These are the periodic analogues of two works of F. Littmann that generalize a paper of J. Vaaler. As applications we provide the corresponding Erdös-Turán-type inequalities, approximations to other periodic functions and bounds for certain Hermitian forms.
Extraction of harmonics of a given order from real trigonometric polynomials (signals) is one of the main problems in harmonic analysis. It has many applications in physics, radio and electrical engineering, in particular, in filtration of harmonic signals of different nature. There exist many methods (mainly, approximative) for solution of this problem. The most common ones are spectral methods based on Fourier transform and other resonance principles. In this paper we propose a new method for extracting harmonics by amplitude and phase transformation of trigonometric polynomials. These trans…
In this article we consider a method of proving a class of inequalities of the form (1). The method is based on the precise approximations of the sine and cosine functions by Maclaurin polynomials of given order. By using this method we present new proofs of some inequalities from the articles C.-P. Chen, W.-S. Cheung [J. Inequal. Appl. 2012:72 (2012)] and Z.-J. Sun, L. Zhu [ISRN Math. Anal. (2011)].
We obtain the estimates for the best approximation in the uniform metric of the classes of $2π$-periodic functions whose $(ψ,β)$-derivatives have a given majorant $ω$ of the modulus of continuity. It is shown that the estimates obtained here are asymptotically exact under some natural conditions on the parameters $ψ,$ $ω$ and $β$ defining the classes
The paper is related to the following question of P.~L.~Ul'yanov: is it true that for any $2π$-periodic continuous function $f$ there is a uniformly convergent rearrangement of its trigonometric Fourier series? In particular, we give an affirmative answer if the absolute values of Fourier coefficients of $f$ decrease. Also, we study a problem how to choose $m$ terms of a trigonometric polynomial of degree $n$ to make the uniform norm of their sum as small as possible.
In this paper we study the asymptotic analysis of the orthogonal trigonometric polynomials by the Riemann-Hilbert problem for the periodic analytic functions.
We obtain the asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with the equidistant nodes $x_k^{(n-1)}=\frac{2kπ}{2n-1},\ k\in\mathbb{Z},$ in metrics of the spaces $L_p$ on classes of $2π$-periodic functions, representable as convolutions of functions $φ, \ φ\perp1,$ which belongs to the unit ball of the space $L_1$, and fixed generating kernels in the case where modules of their Fourier coefficients $ψ(k)$ satisfy the condition $\lim\limits_{k\rightarrow\infty} ψ(k+1)/ψ(k)=0.$ We obtain similar estimates on the classes of $r$-differ…
We prove that $$\max_{t \in [-π,π]}{|Q(t)|} \leq T_{2n}(\sec(s/4)) = \frac 12 ((\sec(s/4) + \tan(s/4))^{2n} + (\sec(s/4) - \tan(s/4))^{2n})$$ for every even trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\{t \in [-π,π]: |Q(t)| \leq 1\}) \geq 2π-s\,, \qquad s \in (0,2π)\,,$$ where $m(A)$ denotes the Lebesgue measure of a measurable set $A \subset {\Bbb R}$ and $T_{2n}$ is the Chebysev polynomial of degree $2n$ on $[-1,1]$ defined by $T_{2n}(\cos t) = \cos(2nt)$ for $t \in {\Bbb R}$. This inequality is sharp. We also prove that $$\max_{t \in [-π,π]}{|…
We prove that for all p>1/2 there exists a constant $γ_p>0$ such that, for any symmetric measurable set of positive measure $E\subset \TT$ and for any $γ<γ_p$, there is an idempotent trigonometrical polynomial f satisfying $\int_E |f|^p > γ\int_{\TT} |f|^p$. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of $γ_p>0$ for p>1 and conjectured that it does not exists for p=1. Furthermore, we prove that one can take $γ_p=1$ when p>1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when $p\neq 2$. This shows str…
In this paper, we give a characterization of Nikol'ski\uı-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to a such a class are given. In order to prove our results, we make use of certain recent reverse Copson- and Leindler-type inequalities.
We give new lower asymptotical estimate of constant \[ C_n=\sup\biggl\{\frac{\|t_n\|_{C(\mathbb T)}}{\|t_n\|_{L(\mathbb T)}}:t_n\text{are real trigonometric polynomials}, \operatorname{deg}t_n<n\biggr\} \] as $n\to\infty$. This estimate improves known bound of L.V.Taykov (1965).
Ramanujan's trigonometric sum $c_q(n)$ can be interpreted as a set of trigonometric moments of a finite measure concentrated at primitive $q$-th roots of unity with equal masses. This gives rise to sets of corresponding polynomials orthogonal on the unit circle. We present explicit expressions of these polynomials for special values of $q$, e.g. when $q=p$ or $q=2p$ or $q=p^k$, where $p$ is a prime number. We generalize this procedure taking the Kronecker polynomial instead of cyclotomic one. In this case the moments are expressed as finite sums of $c_q(n)$ with different $q$.
In this paper we construct an injection from the linear space of trigonometric polynomials defined on $\mathbb{T}^d$ with bounded degrees with respect to each variable to a suitable linear subspace $L^1_E\subset L^1(\mathbb{T})$. We give such a quantitative condition on $L^1_E$ that this injection is a isomorphism of a Banach spaces equipped with $L^1$ norm and the norm of the isomorphism is independent on the dimension $d$.
In this work, some theorems are established for orthogonal trigonometric polynomials (OTP) including Favard, Baxter, Geronimus, Rakhmanov, Szegö and the strong Szegö theorems which are important in the theory of orthogonal polynomials on the unit circle (OPUC). All these results are based on the mutual representation theorem of OPUC and OTP which deduced by a Riemann-Hilbert problem simultaneously characterizing them. In addition, Szegö recursions, four-term recurrences and some new identities for OPUC and OTP are also obtained by using their Riemann-Hilbert characterizations respectively.
In this work, a Vietoris type theorem for the positivity of sine and cosine sum for a particular sequence of real numbers is provided. In this connection, the positivity of a particular type of sine sum involving ratio of some parameters is given, which is new in the literature. Various new results that follow from the Vietoris type theorem include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums for orthogonal polynomials. An open problem is also provided for the partial sums of the generalized polylogarithm.
The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities by reducing to polynomial inequalities. Finally, we show the great applicability of this algorithm and as examples, we use it to analyze some new rational (Pade) approximations of the function $\cos^2(x)$, and to improve a class of inequalities by Z.-H. Yang. The results of our analysis could be implemented by means of an automated proof assistant, so our work is a contribution to the library of automatic support tools for proving various analytic inequalities.
Here we introduce a way to construct generalized trigonometric functions associated with any complex polynomials, and the well known trigonometric functions can be seen to associate with polynomial $x^2-1$. We will show that those generalized trigonometric functions have algebraic identities which generalizes the well known $\sin^2(x)+\cos^2(x)=1$. One application of the generalized trigonometric functions is evaluating infinite series of rational functions.
We give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal $n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with the help of orthogonal polynomials on the unit circle. In particular it is shown that the nodes polynomial can be generated by a simple recurrence relation. As a byproduct interlacing properties of zeros of para-orthogonal polynomials are obtained. Finally, asymptotics for the quadrature weights are presented.
Consider real or complex polynomial Riccati differential equations $a(x) \dot y=b_0(x)+b_1(x)y+b_2(x)y^2$ 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 $η\ge 1$ (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 $η\ge 1$ we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is $2η$ (resp. $3$) when $η\ge 2$ (resp…
In this paper, we explain a procedure based on a classical result of Sturm that can be used to determine rigorously whether a given trigonometric polynomial is nonnegative in a certain interval or not. Many examples are given. This technique has been employed by the author in several recent works. The procedure often involves tedious computations that are time-consuming and error-prone. Fortunately, symbolic computation software is available to automate the procedure. In this paper, we give the details of its implementation in MAPLE 13. Some who are strongly attached to a more traditional theo…
We prove, informally put, that it is not a coincidence that $\cos{(n θ)} + 1 \geq 0$ and that the roots of $z^n + 1 =0$ are uniformly distributed in angle -- a version of the statement holds for all trigonometric polynomials with `few' real roots. The Erdős-Turán theorem states that if $p(z) =\sum_{k=0}^{n}{a_k z^k}$ is suitably normalized and not too large for $|z|=1$, then its roots are clustered around $|z| = 1$ and equidistribute in angle at scale $\sim n^{-1/2}$. We establish a connection between the rate of equidistribution of roots in angle and the number of sign changes of the correspo…
We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that $\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.
It is well known that the expected number of real zeros of a random cosine polynomial $ V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2π) $, with the $ a_j $ being standard Gaussian i.i.d. random variables is asymptotically $ 2n / \sqrt{3} $. On the other hand, some of the previous works on the random cosine polynomials with dependent coefficients show that such polynomials have at least $ 2n / \sqrt{3} $ expected real zeros lying in one period. In this paper we investigate two classes of random cosine polynomials with pairwise equal blocks of coefficients. First, we prove that a rando…
The average value of the number of intersection points of two plane curves given by trigonometrical polynomial maps of degree $N$ and bounded $L_2$- or $W_2^r$- norms is calculated
Let $\mathbb R_t[θ]$ be the ring generated over $\mathbb R$ by $\cosθ$ and $\sinθ$, and $\mathbb R_t(θ)$ be its quotient field. In this paper we study the ways in which an element p of $\mathbb R_t[θ]$ can be decomposed into a composition of functions of the form $p=R(q),$ where $\mathbb R\in \mathbb R(x)$ and $q\in \mathbb R_t(θ)$. In particular, we describe all possible solutions of the functional equation $R_1(q_1)=R_2(q_2)$, where $R_1, R_2 \in \mathbb R[x]$ and $q_1,q_2\in \mathbb R_t[θ].$
