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A kind of sharp Wirtinger inequality
Journal of Inequalities and Applications volume 2019, Article number: 166 (2019)
Abstract
In this study, we give a kind of sharp Wirtinger inequality
where f is defined on \([0,1]\) and satisfies \(f^{(k_{1})}(0)=f^{(k _{2})}(0)=\cdots =f^{(k_{s})}(0)=f^{(m_{s+1})}(1)=\cdots =f^{(m_{r})}(1)=0\) with \(0\le k_{1}< k_{2}<\cdots <k_{s}\le r-1\) and \(0\le m_{s+1}< m_{s+2}< \cdots <m_{r}\le r-1\). First, based on the Birkhoff interpolation, we refer the computation of \(C_{r,p,q}\) to the norm of an integral-type operator. Second, we refer the values of \(C_{r,1,1}\) and \(C_{r,\infty ,\infty }\) to explicit integral expressions and the value of \(C_{r,2,2}\) to the computation of the maximal eigenvalue of a Hilbert–Schmidt operator. Finally, we give three examples to show our method.
1 Introduction
Let \(\mathbb{N}\) and \(\mathbb{R}\) be the sets of positive integers and real numbers, respectively. For \(1 \le p\le \infty \), let \(L_{p}[a,b]\) be the space of pth-power Lebesgue-integrable functions \(f :[a,b] \to \mathbb{R}\) with the corresponding \(L_{p}\)-norms \(\Vert \cdot \Vert _{p}\). Denote by \(W_{p}^{r}\), \(r\in \mathbb{N}\), the class of all continuous real-valued functions f defined on the interval \([a,b]\) such that \(f^{(r-1)}\) (with \(f^{(0)} :=f\)) is absolutely continuous and \(\Vert f^{(r)}\Vert _{p}<\infty \).
The relationships among the norms of a function and its derivatives play an important role in the study of harmonic analysis and function approximation theory. There are many well-known inequalities in this area, for example, the Landau–Kolmogorov, Gorny, Wirtinger, Schmidt, Sobolev, Bernstein, and Markov inequalities. Wirtinger-type inequalities is a kind of the most important inequalities in this aspect. The first result appeared in [1, p. 105]: for every locally absolutely continuous and 2π-periodic function f with the first-order derivative \(f'\in L_{2}[0,2\pi ]\) and \(\int _{0}^{2\pi }f(x)\,dx=0\), we have
where the equality is valid if and only if \(f\in \operatorname{span}\{\cos , \sin \}\).
Since then, many results on Wirtinger-type inequalities appeared. For example, in the case \(r=1\), Schmidt [2] proved the following two results.
Let \(0< p \le \infty \) and \(1\le q \le \infty \). Then for an arbitrary function \(f\in W_{q}^{1}\) [a,b] satisfying \(f(a)=0\) (or equivalently \(f(b)=0\)), we have the sharp inequality
where \(q'\) is the conjugate exponent of q, and \(1/\infty \) is to be interpreted (in the usual way) as 0. At the same time, for an arbitrary function \(f\in W_{q}^{1} [a,b]\) satisfying \(f(a)=f(b)=0\), we have the sharp inequality
Further generalizations and applications of (1.1) and (1.2) can be found in [3,4,5,6,7,8,9,10,11,12].
In the case \(r>1\) the most important result is that if \(f\in W_{q} ^{r} [a,b]\) with zero a of multiplicity k and zero b of multiplicity \(r-k\), \(0\le k \le r\), and \(1\le p,q \le \infty \), then we have the inequality
But as far as we know, the best constants \(C(r,k,p,q)\) are known only for \(q=\infty \) and in some particular cases for \(r=2\). At the same time, some papers closely related to the Wirtinger inequality, such as Shadrin [13] and Waldron [14], considered the problem of estimating the best constant \(C(r,j,p,q)\) in the inequality
where \(H_{\varTheta }f\) is the Hermite interpolation to f at some multiset of r points in \([a,b]\) and \(0\le j< r\). However, the best constant was determined also only for \(p=q=\infty \). Xu and Zhang [15] considered the corresponding estimate of (1.4) for the cubic Hermite interpolation (for which the number of points in the multiset is greater than r), but the best constant was determined only in the cases \(p=q=\infty \) and \(p=q=1\). Liu, Wu, and Xu [16] obtained the constant in (1.3) for the particular case of \(k=r\) and \(p=q=2\). Recently, Xu, Liu, and Xiong [17] obtained the best constant in (1.3) with zeros of multiplicity m at both a and b for \(m+1 \le r\le 2m+2\) and \(p=q=2\).
In this paper, we give an extension of (1.3). Let \(1\le s \le r-1\) be an integer, \(0\le k_{1}< k_{2}<\cdots <k_{s}\le r-1\), and \(0\le m_{s+1}< m _{s+2}<\cdots <m_{r}\le r-1\). Let \(D_{g}\) denote the number of \(k_{i}\) and \(m_{j}\) for which \(k_{i}=g\) or \(m_{j}=g\). We assume that
For \(f\in W_{p}^{r}[a,b]\) with \(f^{(k_{i})}(a)=0\) for \(1\le i \le s\) and \(f^{(m_{j})}(b)=0\) for \(s+1\le j \le r\), we will prove that (1.3) also holds and give the corresponding best constant.
The paper is organized as follows. Section 2 contains our main theorem and its proof. In Sect. 3, we give three examples to show our method.
2 Basic concepts and our main theorem
First, we introduce some known facts about Birkhoff interpolation related to our problems (see [18]). Let \(x_{0}, x_{1}, \ldots ,x_{n}\) be points of \([a,b]\), not necessarily distinct. Also, let \(k_{0},k _{1},\ldots ,k_{n}\) be integers such that \(0\le k_{i}\le n-1\), \(i=0,1,\ldots ,n\). The \(n+1\) pairs of numbers \((x_{i},k_{i})_{i=0} ^{n}\) are supposed to be distinct. Furthermore, let \(D_{g}\) denote the number of pairs of the system \((x_{i},k_{i})_{i=0}^{n}\) for which \(k_{i}=g\). We assume that
Under assumption (2.1), for every \(f\in W_{p}^{n} [a,b]\), from [18] we know that there exists a unique polynomial (Birkhoff interpolation) \(L_{n}(f,t)\) of degree ≤n that satisfies
Now we introduce a remainder theorem about the Birkhoff interpolation (see [18]). Denote \(l_{i}=n-k_{i}-1\) for \(0\le i\le n\), and let
where \(\frac{1}{s!}=0\) for \(s<0\). Besides, we define the discontinuous function \(\chi _{t}(s)\) as follows:
Then from (8) in [18] we obtain
In particular, if \(f^{(k_{i})}(x_{i})=0\) for \(1\le i \le n\) and \(\Delta _{0}\ne 0\), \(x_{0}=x\), \(k_{0}=0\), then (2.4) turns into
where
Combining (2.3) with the last equality, we obtain
In particular, for r pairs of numbers \((0,k_{1}), (0,k_{2}), \ldots , (0,k_{s}),(1,m_{s+1}), \ldots , (1,m_{r})\), we obtain the problem presented in the introduction (here \(a=0\) and \(b=1\)). For simplicity, we represent these r pairs of numbers as \(\{k_{1},\ldots ,k_{s}\} \cup \{m_{s+1},\ldots ,m_{r}\}\).
Now we introduce some information about the norms of the integral operators. Let \(K(x,t)\) be a piecewise continuous function on \([0,1]\times [0,1]\). We define
It is known that S is a linear continuous operator from \(L_{q}[0,1]\) to \(L_{p}[0,1]\) for all \(1\le p,q\le \infty \). Let \(\|S\|_{p,q}\) be the operator norm of S from \(L_{q}[0,1]\) to \(L_{p}[0,1]\). It is known that
Besides, let \(S^{*}\) be the dual operator of S, and let
where
Then W is a Hilbert–Schmidt operator. Let \(\{(\lambda _{j},e_{j})\} _{j\in \mathbb{N}}\) be the sequence of eigenpairs of W with nonincreasing eigenvalues, that is, \(\lambda _{1}\ge \lambda _{2}\ge \cdots \) and \(W(e_{j})=\lambda _{j}e_{j}\). Then (see [16])
In this paper, we obtain the following results.
Theorem 2.1
Let \(\{k_{1},\ldots ,k_{s}\}\cup \{m_{s+1}, \ldots ,m_{r}\}\) satisfy (1.5). Then for an arbitrary \(f\in W_{q}^{r}[0,1]\) with \(f^{(k_{i})}(0)=0\) for \(1\le i\le s\) and \(f^{(m_{j})}(1)=0\) for \(s+1\le j\le r\), we have the sharp inequality
where \(C(p,q)\) is the norm of the operator (with \(B(x,t)\) given by (2.6))
from \(L_{q}[0,1]\) to \(L_{p}[0,1]\), that is, \(C(p,q)=\|T\|_{p,q}\) depends on \(\{k_{1},\ldots ,k_{s}\}\cup \{m_{s+1},\ldots ,m _{r}\}\). Furthermore, the following relations hold:
where \(\lambda _{1}\) is the maximal eigenvalue of the operator
with kernel
Proof
If \(f^{(k_{i})}(0)=0\) for \(1\le i\le s\) and \(f^{(m_{j})}(1)=0\) for \(s+1\le j\le r\), then it follows from (2.5) with \(n=r\) that
From (2.19) and (2.13) we conclude that
On the other hand, for any \(g\in L_{q}[0,1]\), let
and let
where \(L_{r}\) is the Birkhoff interpolation based on \(\{k_{1},\ldots ,k_{s}\}\cup \{m_{s+1},\ldots ,m_{r}\}\). It is known that \(L_{r}( \bar{f},x)\) is an algebraic polynomial of degree at most \(n=r-1\). Then we easily check that \(f^{(r)}=g\) and \(f^{(k_{i})}(0)=0\) for \(1\le i \le s\) and \(f^{(m_{j})}(1)=0\) for \(s+1\le j\le r\). Hence (2.19) becomes
From (2.21) we obtain
From (2.20) and (2.22) we obtain (2.12). Besides, from (2.13) and (2.8), (2.9), (2.11) we obtain (2.14)–(2.16), respectively. This completes the proof of Theorem 2.1. □
For \(f\in W_{q}^{r}[a,b]\) with \(f^{(k_{i})}(a)=0\) for \(1\le i\le s\) and \(f^{(m_{j})}(b)=0\) for \(s+1\le j\le r\), letting \(g(t)=f(a+(b-a)t)\), we obtain the following result.
Corollary 2.2
Let \(\{k_{1},\ldots ,k_{s}\}\cup \{m_{s+1}, \ldots ,m_{r}\}\) satisfy (1.5). Then for an arbitrary \(f\in W_{q}^{r}[a,b]\) with \(f^{(k_{i})}(a)=0\) for \(1\le i\le s\) and \(f^{(m_{j})}(b)=0\) for \(s+1\le j\le r\), we have the sharp inequality
Remark 2.3
It is obvious that if \(k_{i}=i-1\) for \(1\le i \le s\) and \(m_{j}=j-s-1\) for \(s+1\le j \le r\), then Corollary 2.2 is the case (1.3) for \(k=s\). In this case, we give the best constant \(C(r,k,p,q)\) in (1.3) by the corresponding \(C(p,q)\).
3 Some examples
In this section, we give three examples showing how to compute the values of \(C(1,1)\), \(C(\infty ,\infty )\), and \(C(2,2)\). It is obvious that these three examples are not in the case of (1.3).
Example 1
For \(\{0\}\cup \{1\}\), we have \(C(1,1)= \frac{1}{2}\), \(C(\infty ,\infty )=\frac{1}{2}\), and \(C(2,2)=\frac{4}{ \pi ^{2}}\).
Proof
Since \(x_{1}=0\), \(x_{2}=1\), \(k_{1}=0\), and \(k_{2}=1\), it follows from (2.6) that (2.13) holds with
We first consider \(C(1,1)\). From (2.14) and (3.1) by a direct computation we obtain
Now we consider \(C(\infty ,\infty )\). From (2.15) and (3.1) it follows that
Finally, we consider \(C(2,2)\). Since \(\min\{x,t\}\) is a reproducing kernel, it follows from the computation of [19, p. 55] that
This completes the proof of Example 1. □
Example 2
For \(\{0,2\}\cup \{1\}\), we have \(C(1,1)= \frac{1}{3}\), \(C(\infty ,\infty )=\frac{1}{3}\), and \(C(2,2)=\frac{8}{ \pi ^{3}}\).
Proof
Since \(x_{1}=x_{2}=0\), \(x_{3}=1\), \(k_{1}=0\), \(k_{2}=2\), and \(k_{3}=1\), it follows from (2.6) that (2.13) holds with
We first consider \(C(1,1)\). From (2.14) and (3.4) by a direct computation it follows that
Now we consider \(C(\infty ,\infty )\). From (2.15) and (3.4) by a direct computation it follows that
Finally, we consider \(C(2,2)\). We will use (2.16). From (3.4) and (2.18) by a direct computation we obtain
Let g be an eigenvector corresponding to a positive eigenvalue λ of the operator W given by (2.17), that is, \(\lambda g(x) =W(g,x)\). Then from (3.7) and the relations \(\max \{a,b\}= \frac{a+b+|a-b|}{2}\) and \(\min \{a,b\}=\frac{a+b-|a-b|}{2}\) by a direct computation it follows that
Taking \(x=1\) in (3.8), we obtain
Differentiating both sides of (3.8), we obtain
Let \(x=0\) in (3.10). Then we obtain
Differentiating both sides of (3.10), we obtain
Let \(x=1\) in (3.12). Then we obtain
Differentiating both sides of (3.12), we obtain
Let \(x=0\) in (3.14). Then we obtain
Differentiating both sides of (3.14), we obtain
Let \(x=1\) in (3.16). Then we obtain
Differentiating both sides of (3.16), we obtain
Let \(x=0\) in (3.18). Then we obtain
Differentiating both sides of (3.18), we obtain
Let \(\mu =\frac{1}{\sqrt[6]{\lambda }} \). Then the general solution of equation (3.20) is
From (3.21) it follows that
Substituting (3.9), (3.11), (3.13), (3.15), (3.17), and (3.19) into (3.21)–(3.26), respectively, by a simplification we obtain the following linear equations in the six unknown numbers \(C_{j}\), \(1\le j\le 6\):
Subtracting two sides of equation (3.31) from two sides of equation (3.29), by a simple simplification we obtain
Subtracting two sides of equation (3.27) from two sides of equation (3.29) and adding \(\frac{\sqrt{3}}{2}\) times two sides of equation (3.33), we obtain
Equations (3.27) and (3.34) imply that
Thus by (3.28), (3.30), (3.32), (3.33), (3.34), and (3.35), if λ is a positive eigenvalue of the operator W and g is a nonzero eigenfunction corresponding to λ, then \(\mu =\lambda ^{-\frac{1}{6}} \) is a positive zero of the function F defined by
Due to \((e^{\frac{\sqrt{3}}{2}t}-e^{-\frac{\sqrt{3}}{2}t} ) ^{2}\sin ^{2} \frac{t}{2}+ (e^{\frac{\sqrt{3}}{2}t}+e^{-\frac{ \sqrt{3}}{2}t} )^{2}\cos ^{2} \frac{t}{2}>0\), by (3.36) we see that the set of all positive eigenvalues of W is a subset of the set
which is the set of all positive roots of the equation \(\cos t=0\). Furthermore, it is easy to verify that \(g_{k}(x)=\cos (k-\frac{1}{2}) \pi x\) is an eigenvector corresponding to the eigenvalue \(((k- \frac{1}{2})\pi )^{-6}\) for all \(k\in \mathbb{N}\). Therefore the set of all positive eigenvalues of W is given by (3.37), and hence the maximal eigenvalue of W is \(\lambda _{1}=(\frac{\pi }{2})^{-6}\). Combining this fact with (2.16), we obtain that \(C(2,2)=\frac{8}{\pi ^{3}} \). □
Example 3
For \(\{0,1\}\cup \{1\}\), we have \(C(1,1)=\frac{ \sqrt{3}}{27}\), \(C(\infty ,\infty )=\frac{1}{12}\), and
where \(\mu _{1}\) is the minimal positive root of the equation
Proof
Since \(x_{1}=x_{2}=0\), \(x_{3}=1\), \(k_{1}=0\), and \(k_{2}=k_{3}=1\), it follows from (2.6) that (2.13) holds with
We first consider \(C(1,1)\). From (2.14) and (3.39) by a direct computation it follows that
Now we consider \(C(\infty ,\infty )\). From (2.15) and (3.39) by a direct computation it follows that
Finally, we consider \(C(2,2)\). We will use (2.16). From (3.39) and (2.18) by a direct computation we obtain
Let g be an eigenvector corresponding to a positive eigenvalue λ of the operator W given by (2.17). Then from (3.42) and the relations \(\max \{a,b\}=\frac{a+b+|a-b|}{2}\) and \(\min \{a,b \}=\frac{a+b-|a-b|}{2}\) by a direct computation it follows that
Taking \(x=0,1\) in (3.43), we obtain
Differentiating twice both sides of (3.43), we obtain
Let \(x=1\) in (3.45). Then we obtain
Differentiating both sides of (3.45), we obtain
Let \(x=0\) in (3.47). Then we obtain
Differentiating both sides of (3.47), we obtain
Let \(x=0,1\) in (3.49). Then we obtain
Differentiating twice both sides of (3.49), we obtain (3.21). By (3.21) we obtain (3.22)–(3.26). Substituting (3.44), (3.46), (3.48), and (3.50) into (3.21), (3.23), (3.24), and (3.25), respectively, by a simple simplification we obtain the following linear equations in the six unknown numbers \(C_{j}\), \(1\le j\le 6\):
Substituting (3.51) into (3.55), by a simple simplification we obtain
Subtracting both sides of equation (3.56) from both sides of equation (3.53), by a simple simplification we obtain
Subtracting both sides of equation (3.52) from both sides of equation (3.53) and adding \(\frac{\sqrt{3}}{2}\) times both sides of equation (3.58), we obtain
Equations (3.51), (3.52), and (3.59) imply
Thus by (3.51), (3.54), (3.57), (3.60), (3.58), and (3.59), if λ is a positive eigenvalue of the operator W and if g is a nonzero eigenfunction corresponding to λ, then \(\mu =\lambda ^{-1/6}\) is a positive zero of the function \(F_{1}\) defined by
On the other hand, if \(F_{1}(\mu )=0\), then we can find \(\overline{C} _{3}\), \(\overline{C}_{4}\), \(\overline{C}_{5}\), \(\overline{C}_{6}\) such that (3.57)–(3.60) hold, and let \(\overline{C}_{1}=-\overline{C}_{3}- \overline{C}_{5}\), \(\overline{C}_{2}=\overline{C}_{4}+\overline{C}_{6}\). Then, instead of \(C_{1}\), \(C_{2}\), \({C}_{3}\), \({C}_{4}\), \({C}_{5}\), \({C}_{6}\) in (3.21), take \(\overline{C}_{1}\), \(\overline{C}_{2}\), \(\overline{C} _{3}\), \(\overline{C}_{4}\), \(\overline{C}_{5}\), \(\overline{C}_{6}\) such that the corresponding \(g(x)\) satisfies (3.44), (3.46), (3.48), and (3.50). It is easy to verify that this \(g(x)\) is an eigenfunction of W corresponding to the eigenvalue \(\lambda =\mu ^{-6}\). Therefore the set of all positive eigenvalues of W is
where \(\{\mu _{k}\}\) is the sequence of all positive zeros of the function \(F_{1}\), and \(\mu _{1}\le \mu _{2} \le \cdots \) . This shows that \(\lambda _{1}=\mu_{1}^{-6}\), and hence (2.16) implies (3.38). □
References
Blaschke, W.: Kreis und Kugel, Leipzig, Leipzig (1916)
Schmidt, E.: Über die Ungleichung, welche die Integrale über eine Potenz einer Funktion und über eine andere Potenz ihrer Ableitung verbindet. Math. Ann. 117, 301–326 (1940)
Agarwal, R.P., Pang, P.Y.H.: Remarks on the generalizations of Opial’s inequality. J. Math. Anal. Appl. 190(2), 559–577 (1995)
Chen, R., Zhang, D.F.: A generalized Wirtinger’s inequality with applications to a class of ordinary differential equations. J. Inequal. Appl. 2009, 710475 (2009). https://doi.org/10.1155/2009/710475
Giova, R., Ricciardi, T.: A sharp weighted Wirtinger inequality and some related functional spaces. Bull. Belg. Math. Soc. Simon Stevin 17(2), 209–218 (2010)
Jaros, J.: On an integral inequality of the Wirtinger type. Appl. Math. Lett. 24(8), 1389–1392 (2011)
Lee, C.F., Chin, C., Hong, C.H., Agarwal, R.P.: Lyapunov and Wirtinger inequalities. Appl. Math. Lett. 17(7), 847–853 (2004)
Lee, C.F.: Applications of Wirtinger inequalities on the distribution of zeros of the Riemann Zeta-function. J. Inequal. Appl. 2010, 215416 (2010). https://doi.org/10.1155/2010/215416
Liu, K., Fridman, E.: Wirtinger’s inequality and Lyapunov-based sampled-data stabilization. Automatica 48(1), 102–108 (2012)
Seuret, A., Gouaisbaut, F.: Wirtinger-based integral inequality: application to time-delay systems. Automatica 49(9), 2860–2866 (2013)
Swanson, C.A.: Wirtinger’s inequality. SIAM J. Math. Anal. 9(3), 484–491 (1978)
Waldron, S.: Schmidt’s inequality. East J. Approx. 3(2), 117–135 (1997)
Shadrin, A.: Error bounds for Lagrange interpolation. J. Approx. Theory 80, 25–49 (1995)
Waldron, S.: \(L_{p}\) error bounds for Hermite interpolation and the associated Wirtinger inequalities. Constr. Approx. 13, 461–479 (1997)
Xu, G.Q., Zhang, Z.: Simultaneous approximation of Sobolev classes by piecewise cubic Hermite interpolation. Numer. Math., Theory Methods Appl. 7(3), 317–333 (2014)
Liu, Y.P., Wu, W.Y., Xu, G.Q.: The best constants in the Wirtinger inequality. Int. J. Wavelets Multiresolut. Inf. Process. 14(6), 1650048-1-13 (2016)
Xu, G.Q., Liu, Y.P., Xiong, L.Y.: Exact constants for simultaneous approximation of Sobolev classes by piecewise Hermite interpolation. Anal. Math. (2019, in press). https://doi.org/10.1007/s10476-019-0985-y
Birkhoff, G.D.: General mean value and remainder theorems with applications to mechanical differentiation and quadrature. Trans. Am. Math. Soc. 7, 107–136 (1906)
Ritter, K.: Average-Case Analysis of Numerical Problems. Springer, Berlin (2000)
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Xu, G., Liu, Z. & Lu, W. A kind of sharp Wirtinger inequality. J Inequal Appl 2019, 166 (2019). https://doi.org/10.1186/s13660-019-2121-8
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DOI: https://doi.org/10.1186/s13660-019-2121-8