Abstract
We consider a special 2k-order modulus of continuity W 2k (f,h) of 2π-periodic continuous functions and prove an analog of the Bernstein–Nikolsky–Stechkin inequality for trigonometric polynomials in terms of W 2k . We simplify the main construction from the paper by Foucart et al. (Constr. Approx. 29(2), 157–179, 2009) and give new upper estimates of the Jackson–Stechkin constants. The inequality W 2k (f,h)≤3∥f∥∞ and the Bernstein–Nikolsky–Stechkin type estimate imply the Jackson–Stechkin theorem with nearly optimal constant for approximation by periodic splines.
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1 Introduction
The paper is devoted to the question of equivalence of two types of direct theorems in approximation theory:
-
(a)
the case of smooth functions (Favard estimates);
-
(b)
the case of arbitrary continuous functions (Jackson–Stechkin estimates).
We show that the Jackson–Stechkin inequality with the optimal constants follows from the Favard inequality (see the proof of Theorem 2 and (5.3)). The main tool in the proof of this statement is the function W 2k , measuring the smoothness of an integrable periodic function. Modulus W 2k is the special case of the generalized modulus of smoothness introduced by H. Shapiro [2, 15]. This characteristic is more delicate than the standard modulus of continuity of order 2k. The function W 2k allows us to obtain asymptotically sharp results for the approximation by Favard-type operators. For example, we obtain the Jackson–Stechkin inequality for the periodic splines with constants close to optimal.
The following two facts play a key role here:
-
1.
Uniform (in k) boundedness of operators W 2k :
$$W_{2k}(f,h)\le3\|f\|, \quad f\in C(\mathbb{T}),\ h>0. $$ -
2.
The Bernstein–Nikolsky–Stechkin inequality in terms of W 2k .
The paper is organized as follows. In the second section, we introduce notation. In the third section, we consider the smooth characteristic W 2k and prove the uniform boundedness of W 2k (Lemma 1). The technical details of the proof can be found in the Appendix. Section 4 is devoted to the analog of the classical Bernstein–Nikolsky–Stechkin estimate in terms of W 2k (Theorem 1). The next important result in the paper is Theorem 2, which gives a simple and general proof of the Jackson–Stechkin theorem. We improve and simplify the main constructions from [9]. In the fifth section, we introduce Favard-type operators and show that Favard-type operators give Jackson–Stechkin theorems with almost optimal constants. That result is a consequence of the sharp inequality for the trigonometric approximation. We will show that to prove Jackson–Stechkin theorems with almost optimal constants, it is sufficient to obtain a Favard-type inequality (Theorem 3). Theorem 4 is devoted to approximation by periodic splines. Finally, we give in Theorems 5 and 6 the classical almost sharp variants of Theorems 1 and 3.
2 Notation
Let \(\mathbb{I}\) denote either a one-dimensional torus \(\mathbb{T}=[-\pi,\pi)=\mathbb{R}/(2\pi\mathbb{Z})\) or the real line \(\mathbb{R}=(-\infty,\infty)\), and let \(L(\mathbb{I})\) be the space of integrable functions \(f:\mathbb{I}\rightarrow\mathbb{R}\) with the norm \(\|f\|_{L(\mathbb{I})}=\int_{\mathbb{I}}|f(t)|\,dt\). The space of continuous 2π-periodic functions with the norm
is denoted by \(C(\mathbb{T})\). In this paper, we are interested in the approximation of a real continuous function \(f \in C (\mathbb{T})\) by trigonometric polynomials τ∈T n of degree n:
By ∗ we denote the convolution operation in \(L(\mathbb{R})\) (see [16, Chap. 1, Sect. 1]):
and by ⊛ the periodic convolution operation in \(L(\mathbb{T})\) (see [10, Chap. 1, Sect. 1.5.4], [4, Chap. 3, Sect. 3.1]):
Let χ h (x), h>0, be the characteristic function of the interval (−h/2,h/2) normalized in \(L(\mathbb{R})\):
We will use the well-known periodization method (see [16, Chap. 7, Sect. 2, (2.1)]), which for a given \(f\in{L(\mathbb{R})}\) provides the 2π-periodic function \(\widetilde{f}\) from \({L(\mathbb{T})}\) by the formula
and (see the proof of Theorem 2.4 from [16, Chap. 7, Sect. 2])
For a nonnegative \(f\in{L(\mathbb{R})}\), the inequality (2.2) changes to the equality
For example, the 2π-periodization of χ h is given by the formula
and (2.1), (2.3) imply that \(\|\widetilde{\chi}_{h}\|_{L(\mathbb{T})}=1\) for each h>0. The Fourier series for \(\widetilde{\chi}_{h}\) is
where
Following [14, Chap. 3, Sect. 2, (III, 2; 96)], we write
and consider the convolution squares
To prove equalities (2.6), it is sufficient to apply (2.4) and properties of periodic convolution (see [4, Part 3, Sect. 3.1]). Notably, the convolution f ⊛ g of the functions f,g from \(L(\mathbb{T})\) belongs to \(L(\mathbb{T})\), and for the Fourier coefficients of the convolution we have the following identity:
where we use the standard notation for Fourier coefficients of \(f\in{L(\mathbb{T})}\):
The last equality in (2.6) follows from the general fact about periodization (see [16, Part 7, Sect. 2, Theorem 2.4, Corollary 2.6]).
Note that (2.3), (2.5), (2.6) imply that
for each h>0.
3 Special Modulus of Continuity
For fixed h>0 and \(k\in\mathbb{N}\), consider the following operator from \(C(\mathbb{T})\) to \(C(\mathbb{T})\):
where
is a central difference of order 2k with the step t. The following representation is valid (cf. [9, Sect. 3]):
Here
For a 2π-periodic function \(f\in{C(\mathbb{T})}\), this gives the following representation:
where
Note that for the positive numbers a j =a j (k), j=1,…,k, we have
The formulas (2.6), (3.6) and the first equality in (3.7) imply that
We will show that
For h>0 and \(f\in{C}(\mathbb{T})\), let
The definitions (3.8), (3.9), and (3.1) imply that
Lemma 1
If h>0, \(k\in\mathbb{N}\) and \(f \in {C}(\mathbb{T})\), then
Proof
For every h>0 and \(f\in{C}(\mathbb{T})\), we have (see (3.2)–(3.6))
where
First, consider the case k=1. In this case, we have (see (2.5), (2.6))
From this and from (2.7), we obtain
Hence, Lemma 1 in the case k=1 is proved.
Let \(k\in\mathbb{N}\), k≥2. It is sufficient to consider the case h=1. In this case, Λ k :=Λ k,1 is an even, piecewise linear function (see Fig. 1) with vertices at the points (i,b i,k ), i=−k,…,k, where
and
The inequalities (see Appendix, Lemma A)
imply (see (3.8), (3.10), and (2.2)) that
Now Lemma 1 is completely proved. □
Remark 1
It is clear that the exact constant in Lemma 1 is equal to \(1 + \|{\varLambda}_{k,1}\|_{L(\mathbb{R})}\). We have the following estimates of \(n_{k}:=\|{\varLambda}_{k,1}\|_{L(\mathbb{R})}\) for small k≥2:
4 Bernstein–Nikolsky–Stechkin Inequality
The Bernstein–Nikolsky–Stechkin inequality [17] (see also [10, Theorem 3.5.3], [3, p. 131, (12.1)]) is the generalization of the classical Bernstein inequality for trigonometric polynomials
The Bernstein–Nikolsky–Stechkin inequality is given by
where
Note that in the case r=1, h=π/n, the inequality (4.2) was proved by M. Riesz [12, §4] in 1914.
Let e j (x):=exp(ijx), c j (x):=cos(jx). It is easily seen that
where
Lemma 2
For α∈(1,2], we have the following inequality:
For α=2, we have in (4.3) equality.
Proof
Set s:=πα/2. We have
□
The main result in this section is the following analog of the Bernstein–Nikolsky–Stechkin inequality.
Theorem 1
If τ∈T n , \(k,n\in\mathbb{N}\), then
In particular,
It is clear that Theorem 1 is sharp. We have the equality in (4.4) for τ=c n .
Theorem 1 implies Bernstein’s inequality (4.1) for even derivatives. This follows from Lemma 3.
Lemma 3
If τ∈T n , \(k,n\in\mathbb{N}\), then
Proof of Theorem 1
Theorem 1 follows from Lemmas 4, 5 below. Specifically, if
then
and
Now Lemmas 4 and 5 imply that
□
Lemma 4
(cf. [19, p. 361])
Suppose that q(t) is a nonnegative, even, convex on [−n,n] function. Then for real \(\tau(t)=\sum_{j=-n}^{n}\widehat{\tau}_{j}e_{j}(t)\), the following inequality is valid:
Lemma 5
Let \(n,k\in\mathbb{N}\), h∈(0,2π/n]. Then the function \(q(t):=t^{2k}\lambda^{-1}_{h,k}(t)\) satisfies the conditions of Lemma 4.
Proof
We have the following formula:
where (see (2.5))
The function t 2k/sin2k(tu/2) satisfies the conditions of Lemma 4. In other words,
We need to prove the inequality
Let
Then the function
is convex on (0,n]. We are going to prove that for 0<h≤2π/n, the function
is convex on (0,n].
The properties
imply that
The condition of positivity for the second derivative takes the following form:
Consider the function
where
Let
After the change of variable, we may assume that u,s∈(0,h], h∈(0,π].
Consider the case k≥2. First, reduce the value of F 2 by omitting the positive quantity t 2 u 2−4sin2(tu/2). Then, let
To get (4.5) for k≥2, it is sufficient to show that
It is sufficient to prove (4.6) for the symmetric function \(F_{2}^{*}\) defined by
where
To prove (4.6), it is sufficient to prove positivity of the following one-dimensional integral:
The function φ(u)/sin(u) is increasing, and the function φ(a−u)/sin(a−u) is decreasing on the interval (max(0,a−π),a/2). Consequently, the function
is increasing on (max(0,a−π),a/2) from 0 to 1, and the function
has exactly one zero on the interval (max(0,a−π),a/2). The functions \(\sin(u) \sin(a-u)\), ϕ h (u)ϕ h (a−u) are increasing and positive on (max(0,a−π),a/2). These facts and the inequality \(F_{2}^{*}(a/2,a/2)>0\) imply that it is sufficient to consider only the case k=2 and to prove that the following integral is positive:
Furthermore, it is sufficient to prove that I(a):=I π (a)>0 (0<a<2π). The proof of this inequality can be found in the Appendix (see Sect. A.2), where a special simple case k=1 is also considered. □
In the proof of Lemma 3, we will use the following Lemma 6.
Lemma 6
Let \(n,k\in\mathbb{N}\), h∈(0,π/(2n)). Then the function q(t):=λ h,k (t) satisfies the conditions of Lemma 4.
Proof of Lemma 6
This follows from the formula
□
Proof of Lemma 3
Lemma 3 follows from Lemmas 4 and 6. For
we have
and Lemmas 4 and 6 imply that
□
5 Jackson–Stechkin Theorem
5.1 Jackson–Stechkin Inequality for Polynomial Approximation
In 1936, Jean Favard [6, 7] proved that the following Euler–MacLaurin formula for smooth 2π-periodic functions g with \(\widehat{g}_{0}=0\):
gives a simple proof of the Bohr–Favard inequality (generalization of H. Bohr result [1], r=1). He used in [8] equality (5.1) to obtain a famous sharp inequality:
In the present paper, the following “telescoping identity” by C. Neumann [11] (see also [13, p. 146], [14, (III,2;96)]) will be used. For every \(f\in L(\mathbb{T})\) and m=2,3,… we have
This equality gives a simple proof with new almost optimal constants of the following Jackson–Stechkin type theorem (see [9]).
Theorem 2
Let \(f\in C(\mathbb{T})\) and α>1, \(k,n\in\mathbb{N}\). Then
Proof
For arbitrary \(f\in{C(\mathbb{T})}\), the representation (5.2) is valid. Using a subadditive property for E n−1(f), we obtain
where
So
Therefore, it is sufficient to estimate the best L-approximation of the smooth functions \(\widetilde{\varLambda}_{k,h}^{j\,\circledast}\). This was done in [9]. Specifically, Lemma 4.2 from [9] contains the result, which reads (in our notation) as follows:
where \(T^{\perp}_{n-1}\) is the subspace of all functions from \(C(\mathbb {T})\) that are orthogonal to T n−1,
and the numbers a l =a l (k), l=1,…,k, are defined by (3.4).
The inequality (5.3) implies that
Since α>1 and \({\mathcal{K}}_{2j}\le{4/\pi}\), for \(f\in {C(\mathbb{T})}\) we have the following:
and
Here, we use the well-known expansion for the secant function (cf. [5, pp. 561–562, (6), (8)]). □
5.2 Favard-Type Operators
Consider a family \(F:=\{F_{n,k}:\; n,k\in\mathbb{N}\}\) of operators
with the properties
where the constant 0<C F <∞ does not depend on g, k, n. We will call F n,k ∈F a Favard-type operator.
Theorem 3
Let \(f\in{C(\mathbb{T})}\). If F n,k is a Favard-type operator and τ ∗∈T n−1 is the best uniform approximation of f, then
with
Proof
Suppose that
and let
Theorem 1 and Lemma 1 imply that
Thus,
□
5.3 Approximation by Periodic Splines
We say that \(s\in\mathcal{S}\equiv\mathcal{S}_{2n,2k-1}\) if \(s^{(2k-2)} \in C(\mathbb{T})\) and s (2k−2)(x)=s j =const for x∈Δ j :=[2πj/(2n),2π(j+1)/(2n)), j=0,…,2n−1.
The space \(\mathcal{S}\) is the space of smooth periodic splines of degree 2k−1 with minimal defects (=1) on the uniform partition of \(\mathbb{T}=\bigcup_{j=0}^{2n-1}\varDelta _{j}\). Define the operator of interpolation at the endpoints of Δ j :
V.M. Tihomirov [18] (see also [10, Theorem 5.2.6, p. 223]) proved that I n,k is an operator of Favard type:
Therefore, we obtain the following Jackson–Stechkin theorem for periodic splines.
Theorem 4
For \(f\in{C(\mathbb{T})}\), α>1, \(n,k\in\mathbb{N}\),
where
5.4 Two Results for the Classical Modulus of Continuity ω r
First, we improve the main result from [9] (see [9, Theorem 2.1]).
Theorem 5
For \(f\in{C(\mathbb{T})}\), \(n, r\in \mathbb{N}\), α>1,
with
In particular,
where
Proof
The proof follows from Theorem 1, keeping in mind the inequalities
and the lower estimate from [9, Sect. 8, Theorem 8.2]. □
Now, let us rewrite Theorem 4 in standard form:
Theorem 6
For \(f\in {C(\mathbb{T})}\), \(n,k\in\mathbb{N}\), r∈{2k−1,2k}, α>1,
Note that Lemma 8.1 and Theorem 8.2 from [9] provide the lower estimate for the constants in (5.4) equal to \(c_{r}\gamma_{r}^{*}\). Therefore, the estimate (5.4) is asymptotically sharp. For example, for α=2,
References
Bohr, H.: Ein allgemeiner Satz über die Integration eines trigonometrischen Polynoms. Pr. Mat.-Fiz. 43, 273–288 (1935). (Collected Mathematical Works II, C 36)
Boman, J., Shapiro, H.: Comparison theorems for a generalized modulus of continuity. Ark. Mat. 9, 91–116 (1971)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften, vol. 303. Springer, Berlin (1993). 449 pp.
Edwards, R.E.: Fourier Series. A Modern Introduction, Vol. 1, 2nd edn. Graduate Texts in Mathematics, vol. 64. Springer, New York (1979). 224 pp.
Elkies, N.D.: On the sums \(\sum^{\infty}_{k=-\infty}(4k+1)^{-n}\). Am. Math. Mon. 110(7), 561–573 (2003)
Favard, J.: Sur l’pproximation des functions périodiques par des polynomes trigonométriques. C. R. Acad. Sci. (Paris) 203, 1122–1124 (1936)
Favard, J.: Application de la formule summatoire d’Euler à la démostration de quelques propriétés extrémales des intégrales des fonctions périodiques et presque–périodiques. Mat. Tidskr. Københaven B. H. 4, 81–94 (1936)
Favard, J.: Sur les meilleurs procedes d’approximation de certaies clasess de fonctions par des polynomes trigonometriques. Bull. Sci. Math. 61, 209–224 (1937), 243–256
Foucart, S., Kryakin, Yu., Shadrin, A.: On the exact constant in Jackson–Stechkin inequality for the uniform metric. Constr. Approx. 29(2), 157–179 (2009)
Korneichuk, N.P.: Exact Constants in Approximation Theory. Encyclopedia of Mathematics and Its Applications, vol. 38. Cambridge University Press, Cambridge (1991). 452 pp.
Neumann, C.: Untersuchungen über das Logarithmische und Newton’sche potential. Teubner, Leipzig (1877)
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Acknowledgements
A.G. Babenko is supported by the Russian Foundation for Basic Research (project no. 11-01-00462) and by the Integration Project for Fundamental Research of the Ural and Siberian Divisions of the Russian Academy of Sciences (project no. 12-C-1-1018).
The authors are very grateful to A.Yu. Shadrin for a number of valuable remarks and suggestions.
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Communicated by Kamen Ivanov.
Dedicated to the Memory of Vladimir A. Yudin.
Appendix
Appendix
1.1 A.1 Proof of Some Inequalities for Lemma 1
The estimates look simple enough, and one can check them by symbolic computations. The actual proof of these estimates is not as simple as it may seem at first glance.
Recall the notation:
Define
Lemma A
Let \(k\in\mathbb{N}\), k≥2. Then
Proof
Let 0≤δ≤1/2. Introduce an auxiliary notation
It is clear that
We first show that for arbitrary \(k\in\mathbb{N}\), we have inequalities
By (A.4), these inequalities turn into (A.1) if δ=0. Further, we will show that (A.5) implies (A.2) if δ=1/2. In order to prove (A.5), let
Observe that
Thus,
The following equality:
is a key ingredient. It implies
Let the expression in the square brackets be denoted by A i,k (δ). After simplification, it becomes
where
By change of variables ν=j+1, j=ν−1 in S 2(δ), we obtain
Therefore,
Now
A nice formula for A i,k (0) (see Lemma B below) yields
It follows from (A.6) that b i,k+1(δ)−b i,k (δ) is positive for δ≤1/2.
Let 0≤δ≤1/2, k=2r, and r=n(i+1), n≥2. The estimates
imply
The inequalities (A.5) are proved. It was mentioned above that (A.5) contains (A.1) as a particular case because
and
Thus, (A.1) is proved. It is a principal inequality, which implies
and
Now
and (A.3) follows. In order to derive (A.2), we note that the first inequality in (A.1) implies
From this and (A.5), we get (A.2). Lemma A is proved. □
Lemma B
Proof
We need to prove that
An equivalent form is
For \(k\in\mathbb{N}\), i≤k−1, we have the equality
One can prove (A.7) by induction on j=k−i−1 for fixed k:
□
1.2 A.2 Proof of Lemma 5. Computations
First, we consider the case k=1 in Lemma 5.
A.2.1
To prove inequality
it is sufficient to prove the convexity of the function
on the interval (0,π). The condition
is equivalent to the inequality
We have φ 1(0)=φ 1(π)=0, and the inequality follows from the power expansions of the functions φ 1(x), φ 1(π−x) at 0:
The proof of the inequality
is similar. We have
and
A.2.2
Below we will prove the following inequality:
where
We will show that I(a)>0 for 0<a<2π (see Fig. 2) by direct calculations. The proof will be divided into several steps.
-
1.
For 0<a≤π, we obtain
$$I(a)= \varphi_3(a) + \varphi_4(a), $$where
$$\begin{aligned} \varphi_3(a) & := \frac{87}{256} a- \frac{1}{32}a^3+ \biggl(\frac{179}{576} a - \frac{1}{48} a^3 \biggr) \cos(2a)+ \biggl(-\frac{1145}{3456} + \frac{1}{8} a^2 \biggr)\sin(2a), \\ \varphi_4(a) & :=- \frac{19}{6144} a \cos(4a)+ \biggl( \frac{835}{221184}-\frac{1}{1536} a^2 \biggr)\sin(4a). \end{aligned}$$Consider the following cases.
-
1.1.
Suppose that \(0<a \le\sqrt{15}/2 \). The Taylor expansion of I(a) gives
$$\begin{aligned} I(a) & = {\frac{1}{405405}} {a}^{13}-{\frac{4}{6081075}} {a}^{15}+{\frac{101}{1206079875}} {a}^{17} \\ &\quad {}-{\frac{248}{36395233875}}{a}^{19}+ \cdots. \end{aligned} $$It is easily seen that
$$I(a)>\frac{a^{13}}{405405} \biggl( 1 -{\frac{4}{15}} {a}^2 \biggr) \ge 0, \quad a \in(0, \sqrt{15}/2]. $$ -
1.2.
In the case \(\sqrt{15}/2 <a \le\pi\), the inequality I(a)>0 follows from the estimates:
$$\begin{aligned} \varphi_3(a)& >0,\quad a \in[\sqrt{15}/2,3\pi/4); \qquad \varphi_3(a)>0.01, \quad a \in[3\pi/4, \pi); \\ \varphi_4(a)& >0, \quad a \in[\sqrt{15}/2,3\pi/4); \qquad \varphi_4(a)>-0.01, \quad a \in[3\pi/4,\pi). \end{aligned} $$
-
1.1.
-
2.
Now we turn to the case π<a<2π. Let us compute I 1(b):=I(2π−b):
$$\begin{aligned} I_1(b)& =\frac{1}{32} \bigl(3\pi+\bigl(87/8-3\pi^{2}\bigr)b-3\pi b^{2}+b^{3} \bigr) \\ &\quad {}+ \biggl(- \frac{\pi}{18} - \frac{\pi^{2}b}{8} + \frac{179b}{576} + \frac{\pi b^{2}}{8}- \frac{b^{3}}{48} \biggr) \cos(2b) \\ &\quad {}+ \biggl(\frac{\pi^{2}}{16} - \frac{1145}{3456} - \frac{11b\pi}{48} + \frac{b^{2}}{8} \biggr) \sin(2b) \\ &\quad {}+ \biggl( - \frac{11\pi}{288} - \frac{\pi^{2}b}{64} - \frac{19b}{6144} + \frac{\pi b^{2}}{64} \biggr)\cos(4b) \\ &\quad {} + \biggl(\frac{7\pi^{2}}{256} + \frac{835}{221184} - \frac{37b\pi}{768} - \frac{b^{2} }{1536} \biggr)\sin(4b). \end{aligned}$$We will show that I 1(b)>0 for 0<b<π. Consider the following cases.
-
2.1.
Assume that 0<b<π/2. The following is a series expansion of I 1(b):
$$\begin{aligned} I_1(b) & =\frac{1}{105} \pi^2 b^7 - \frac{2}{315} \pi^2 b^9 + \frac{2}{945} \pi b^{10}+ \frac{29}{17325}\pi^2 b^{11} \\ &\quad {}- \frac{127}{155925} \pi b^{12}+ \biggl(\frac{1}{405405} - \frac{514}{2027025} \pi^2 \biggr) b^{13} + \cdots. \end{aligned}$$Under the above conditions on b, the omitted terms are positive. Therefore, it is sufficient to prove that polynomial p 6(b):=I 1(b)/b 7 is positive on (0,π/2). It is easy to check that derivative \(p'_{6}(b)<0\), b∈(0,π/2) and that p 6(π/2)>0.
-
2.2.
For π/2<b<3π/4, let I 1,1(c):=I 1(c+π/2):
$$\begin{aligned} I_{1,1}(c)&=\frac{135\,\pi}{512} - \frac{7 \pi^3 }{256} + \frac{87 c}{256} - \frac{3 \pi^2 c }{128} + \frac{3 \pi c^2}{64} - \frac{c^3}{32} \\ &\quad {}+ \biggl( - \frac{115 \pi}{1152} + \frac{13 \pi^3}{384} - \frac{179 c}{576} + \frac{\pi^2 c}{64} - \frac{3\pi c^2}{32} + \frac{c^3}{48} \biggr) \cos(2c) \\ &\quad {}+ \biggl( \frac{1145}{3456}+\frac{\pi^2}{48} + \frac{5 \pi c}{48} - \frac{c^2}{8} \biggr) \sin(2c) \\ &\quad {}+ \biggl(- \frac{1465 \pi}{36864} - \frac{\pi^3}{256} - \frac{19 c}{6144} + \frac{\pi c^2}{64} \biggr)\cos(4c) \\ &\quad {}+ \biggl(\frac{835}{221184}+\frac{19\pi^2}{6144}- \frac{25 \pi c}{512} - \frac{c^2}{1536} \biggr) \sin(4c). \end{aligned}$$To prove the estimate I 1,1(c)>0, we can take five terms in the power expansion of I 1,1:
$$\begin{aligned} I_{1,1}(c) & > \frac{\pi^3}{384} + \frac{1525 \pi}{12288} + \biggl( \frac{71\pi^2}{1536} + \frac{19}{27} \biggr) c + \biggl(\frac{767 \pi}{1536} - \frac{7 \pi^3}{192} \biggr) c^2 \\ &\quad {}+ \biggl(-\frac{53 \pi^2}{576} - \frac{8}{81} \biggr) c^3+ \biggl(- \frac{53 \pi}{1152} - \frac{11 \pi^3}{576} \biggr) c^4 > 0, \\ &\qquad\ c \in (0,\pi/4). \end{aligned} $$ -
2.3.
In the case 3π/4≤a<π, we let I 1,2(d):=I 1(π−d), and in the same way we obtain the inequality
$$\begin{aligned} I_{1,2}(d) & > \biggl(\frac{11935\,\pi}{18432}-\frac{5\pi^3}{96} \biggr)+ \frac {35\pi^2 d}{384}+ \biggl(\frac{\pi^3}{24}-\frac{595\pi}{2304} \biggr)d^2 \\ &\quad {}-\frac{5\pi^2}{144} d^3 + \biggl( -\frac{35\pi}{1728} - \frac{\pi^3}{72} \biggr) d^4 -\frac{\pi^2 d^5}{72} > 0, \quad d \in(0,\pi/4]. \end{aligned} $$
-
2.1.
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Babenko, A.G., Kryakin, Y.V. & Staszak, P.T. Special Moduli of Continuity and the Constant in the Jackson–Stechkin Theorem. Constr Approx 38, 339–364 (2013). https://doi.org/10.1007/s00365-013-9210-6
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DOI: https://doi.org/10.1007/s00365-013-9210-6