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.
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.
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…
A Method of Proving a Class of Inequalities of Mixed Trigonometric Polynomial Functions
本文考虑一类不等式（1）的一种证明方法。该方法是基于精确的近似的正弦和余弦函数的麦克劳林多项式给定的顺序。利用这种方法，我们给出了C.-C. P. Chen，W.- S. Cheung [J. Inequal. Appl.2012：72（2012）]和Z. J. Sun，L. Zhu [ISRN Math. Anal.（2011）]。
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]}{|…
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.
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 $q$.
Four Jacobi settings are considered in the context of Hardy's inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy's inequality is proved for the type parameters $α,β\in(-1,\infty)^d$, whereas in the function systems for $α,β\in[-1/2,\infty)^d$. The ranges of these parameters are the widest in which the corresponding orthonormal bases are composed of bounded functions. Moreover, the sharp $L^1$-analogues of Hardy's inequality are obtained with the same restrictions on the parameters $α$ and $β$.
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.
We say that a function $f\in C[a,b]$ is $q$-monotone, $q\ge3$, if $f\in C^{q-2}(a,b)$ and $f^{(q-2)}$ is convex in $(a,b)$. Let $f$ be continuous and $2π$-periodic, and change its $q$-monotonicity finitely many times in $[-π,π]$. We are interested in estimating the degree of approximation of $f$ by trigonometric polynomials which are co-$q$-monotone with it, namely, trigonometric polynomials that change their $q$-monotonicity exactly at the points where $f$ does. Such Jackson type estimates are valid for piecewise monotone ($q=1$) and piecewise convex (q=2) approximations. However, we prove, t…
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.
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…
In the paper we obtain sufficient conditions for a trigonometric polynomial to be a refinement mask corresponding to a tight wavelet frame. The condition is formulated in terms of the roots of a mask. In particular, it is proved that any trigonometric polynomial can serve as a mask if its associated algebraic polynomial has only negative roots (at least one of them, of course, equals $-1$).
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…
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 $q_1,q_2\in \mathbb R_t[θ].$
