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.
Various miscellaneous functional inequalities are deduced for the so-called generalized inverse trigonometric and hyperbolic functions. For instance, functional inequalities for sums, difference and quotient of generalized inverse trigonometric and hyperbolic functions are given, as well as some Grünbaum inequalities with the aid of the classical Bernoulli inequality. Moreover, by means of certain already derived bounds, bilateral bounding inequalities are obtained for the generalized hypergeometric ${}_3F_2$ Clausen function.
In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications to these polynomials are presented, including a unified approach to the asymptotic expansion of the error term in many numerical quadrature formulae, and many new and sharp inequalities, that bound some trigonometric sums.
In this sequel to our recent note it is shown, in a unified manner, by making use of some basic properties of certain special functions, such as the Hurwitz zeta function, Lerch zeta function and Legendre chi function, that the values of all derivatives of four trigonometric functions at rational multiples of $π$ can be expressed in closed form as simple finite sums involving the Bernoulli and Euler polynomials. In addition, some particular cases are considered.
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 $β$.
Let $\left( p,q\right) \mapsto β\left( p,q\right) $ be a function defined on $\mathbb{R}^{2}$. We determine the best or better $p,q$ such that the inequality% \begin{equation*} \left( \frac{\sin x}{x}\right) ^{p}<\left( >\right) 1-β\left( p,q\right) +β\left( p,q\right) \cos ^{q}x \end{equation*}% holds for $x\in \left( 0,π/2\right) $, and obtain a lot of new and sharp Cusa type inequalities for trigonometric functions. As applications, some new Shafer-Fink type and Carlson type inequalities for arc sine and arc cosine functions, and new inequalities for trigonometric means are established.
We prove that for $p\in (0,1]$, the double inequality% \begin{equation*} \tfrac{1}{3p^{2}}\cos px+1-\tfrac{1}{3p^{2}}<\frac{\sin x}{x}<\tfrac{1}{% 3q^{2}}\cos qx+1-\tfrac{1}{3q^{2}} \end{equation*}% holds for $x\in (0,π/2)$ if and only if $0<p\leq p_{0}\approx 0.77086$ and $\sqrt{15}/5=p_{1}\leq q\leq 1$. While its hyperbolic version holds for $% x>0$ if and only if $0<p\leq p_{1}=\sqrt{15}/5$ and $q\geq 1$. As applications, some more accurate estimates for certain mathematical constants are derived, and some new and sharp inequalities for Schwab-Borchardt mean\ and logarithmic means are estab…
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]}{|…
Let $I_ν(x)$ and $K_ν(x)$ be the first and second kind modified Bessel functions. It is shown that the nullclines of the Riccati equation satisfied by $x^α Φ_{i,ν}(x)$, $i=1,2$, with $Φ_{1,ν}=I_{ν-1}(x)/I_ν(x)$ and $Φ_{2,ν}(x)=-K_{ν-1}(x)/K_ν(x)$, are bounds for $x^α Φ_{i,ν}(x)$, which are solutions with unique monotonicity properties; these bounds hold at least for $\pm α\notin (0,1)$ and $ν\ge 1/2$. Properties for the product $P_ν(x)=I_ν(x)K_ν(x)$ can be obtained as a consequence; for instance, it is shown that $P_ν(x)$ is decreasing if $ν\ge -1$ (extending the known range of this result) an…
The purpose of this paper is to define generalized Apostol--Bernoulli polynomials with including a new cosine and sine parametric type of generating function using the quasi-monomiality properties and trigonometric functions. In this study, the Apostol-Bernoulli polynomials with three variable are defined with two new generating functions cosine and sine parameters. Then, we investigate multiplicative and derivative operators, diffrential equations, some summation formulas and partial differential equations for these polynomials. Moreover, we introduce Gould--Hopper--Apostol--Bernoulli type po…
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$-sp…
This article is the collection of the six research papers, recently written by the authors. In these papers authors refine the inequalities of trigonometric and hyperbolic functions such as Adamovic-Mitrinovic inequality, Cusa-Huygens inequality, Lazarevic inequality, Huygens inequality, Jordan's inequality, Carlson's inequality, Wilker's inequality, Redheffer's inequality, Wilker-Anglesio inequality, Becker-Stark inequality, Kober's inequality, Shafer's inequality and Shafer-Fink's inequality. The relation between trigonometric and hyperbolic functions has been built too. In the last paper, t…
We prove limit relations between the sharp constants in the multivariate Bernstein-Nikolskii type inequalities for trigonometric polynomials and entire functions of exponential type with the spectrum in a centrally symmetric convex body.
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…
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…
Let an algebraic polynomial $P_n(ζ)$ of degree $n$ be such that $|P_n(ζ)|\le 1$ for $ζ\in E\subset\mathbb{T}$ and $|E|\ge 2π-s$. We prove the sharp Remez inequality $$ \sup_{ζ\in\mathbb{T}}|P_n(ζ)|\le \mathfrak{T}_{n}\left(\sec \frac{s} 4\right),$$ where $\mathfrak{T}_{n}$ is the Chebyshev polynomial of degree $n$. The equality holds if and only if $$ P_n(e^{iz})=e^{i(nz/2+c_1)}\mathfrak{T}_n\left(\sec\frac s 4\cos \frac {z-c_0} 2\right), \quad c_0,c_1\in\mathbb{R}. $$ This gives the solution of the long-standing problem on the sharp constant in the Remez inequality for trigonometric polynomia…
Parabolic Cylinder functions (PCFs) are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and bounds for these functions. We obtain very sharp bounds for the ratio $Φ_n(x)=U(n-1,x)/U(n,x)$ and the double ratio $Φ_n(x)/Φ_{n+1}(x)$ in terms of elementary functions (algebraic or trigonometric) and prove the monotonicity of these ratios; bounds for $U(n,z)/U(n,y)$ are also made available. The bounds are very sharp as $x\rightarrow \pm \infty$ and $n\rightarrow +\infty$, and this simultaneou…
In this paper we consider anisotropic Lorentz-Karamata space $2π$ of periodic functions of $m$ variables and Nikol'skii--Besov's class . In this paper, we establish order-sharp estimates of the best approximation by trigonometric polynomials with harmonic numbers from the step hyperbolic cross of functions from the Nikol'skii - Besov class in the norm of the anisotropic Lorentz-Karamata space.
Let $f \in C^2(\mathbb{T}^2)$ have mean value 0 and consider $$ \sup_{γ~{\tiny \mbox{closed geodesic}}}{~~~ \frac{1}{|γ|} \left| \int_γ{ f ~~d\mathcal{H}^1}\right| },$$ where $γ$ ranges over all closed geodesics $γ:\mathbb{S}^1 \rightarrow \mathbb{T}^2$ and $|γ|$ denotes their length. We prove that this supremum is always attained. Moreover, we can bound the length of the geodesic $γ$ attaining the supremum in terms of \textit{smoothness} of the function: for all $s \geq 2$, $$ |γ|^{s} \lesssim_s \left( \max_{|α| = s}{ \| \partial_α f \|_{L^{1}(\mathbb{T}^2)}} \right) \| \nabla f \|_{L^2}^{} \…
On exact estimates of the order of approximation of functions of several variables in the anisotropic Lorentz-Zygmund space
Lomonosov Moscow University, Kazakhstan Branch（莫斯科国立大学哈萨克斯坦分校）；Institute of mathematics and mathematical modeling（数学与数学建模研究所）；Ural Federal University, Institute of Natural Sciences and Mathematics（乌拉尔联邦大学自然科学与数学学院）
