On modulus of continuity of differentiation operator on weighted Sobolev classes
- Vladislav F Babenko^{1} and
- Oleg V Kovalenko^{1}Email author
https://doi.org/10.1186/s13660-015-0822-1
© Babenko and Kovalenko 2015
Received: 19 December 2014
Accepted: 16 September 2015
Published: 23 September 2015
Abstract
In this paper we investigate the modulus of continuity of kth order differential operator, \(k\in\mathbb{N}\), on the classes of functions defined on half-line that have positive non-increasing continuous majorants of functions and their higher derivatives.
Keywords
1 Introduction
The following theorem was proved in [1] (1912) by Hardy and Littlewood (see also [2], Theorem 1.1.2).
Theorem A
- 1.If \(x(t)=O(f(t))\), \(x''(t)=O(g(t))\), then$$x'(t)=O \bigl(\sqrt{f(t)g(t)} \bigr). $$
- 2.If \(x(t)=o(f(t))\), \(x''(t)=O(g(t))\), then$$x'(t)=o \bigl(\sqrt{f(t)g(t)} \bigr). $$
- 3.If \(x(t)=O(f(t))\), \(x''(t)=o(g(t))\), then$$x'(t)=o \bigl(\sqrt{f(t)g(t)} \bigr). $$
This theorem had a great impact on formation of the whole field of problems connected with inequalities between derivatives. To confirm this, it is sufficient to note that fundamental possibility of inequalities for upper bounds of derivatives of functions defined on the whole real line or half-line can be easily derived from this theorem (see [2], p.18).
In 1928 Mordell [3] (see also [2], Theorem 1.4.1) proved the following refinement of Theorem A for non-increasing functions.
Theorem B
Later inequalities of type (2) for functions defined on \({\mathbb{R}}\) and \({\mathbb{R}}_{+}\) were generalized in many directions by many mathematicians. One of the brightest and the most important results in the whole field is Kolmogorov’s inequality [5–7] for functions defined on the real line \({\mathbb{R}}\). After this result inequalities of type (2) are called Kolmogorov type inequalities. In articles [8–13] and monographs [2, 14], one can find a detailed overview of classical results about sharp inequalities for derivatives and further references. Articles [15] and [16] are devoted to inequalities between derivatives on classes with non-constant restrictions on the higher derivatives; in [17] and [18] inequalities for derivatives on classes of functions defined on a finite interval are considered; in [19] discrete analogues of inequalities are considered; in [20] and [21] one can find results connected to inequalities for fractional derivatives and further references.
We discuss some of the results for functions defined on the half-line in a more detailed way.
Let \(T_{r}(t):=\cos r \arccos t\), \(t\in[-1,1]\), be Chebyshev polynomials of the first kind. Matorin in 1955 proved the following theorem (see [22]).
Theorem C
For \(r>3\), inequality (3) is not sharp. Sharp inequality that estimates \(\|x^{(k)}\|_{C({\mathbb{R}}_{+})}\) using \(\|x\| _{C({\mathbb{R}}_{+})}\) and \(\|x^{(r)}\|_{L_{\infty}({\mathbb{R}}_{+})}\) for functions \(x\in L_{\infty,\infty}^{r}({\mathbb{R}}_{+})\) was received by Schoenberg and Cavaretta (see [23, 24]) in 1970 (see also [2], Section 3.3).
Note that in the case \(r = 2\) Theorem B gives an estimate for the modulus of continuity \(\omega(D^{1},\delta)\leq2 \delta^{\frac{1}{2}}\), \(\delta> 0\).
The result by Schoenberg and Cavaretta gives a sharp Kolmogorov type inequality for arbitrary orders of derivatives \(k< r\). The exact constant \(C(k,r)\) in this inequality is given implicitly in terms of a limit of a perfect splines sequence. In the case when \(f=g\equiv1\), the sharp Kolmogorov type inequality is equivalent to the equality \(\omega(D^{k},\delta) = C(k,r)\delta^{1-\frac{k}{r}}\), \(\delta>0\). So we can think that in the case of constant f and g this result gives the value of \(\omega(D^{k},\delta)\) for all \(\delta\geq0\) (although rather implicitly). Theorem C gives \(\omega(D^{k},\delta)\), \(\delta\geq 0\) for \(r\leq3\) (with explicit constant).
Information about the connection between modulus of continuity of differentiation operator and Kolmogorov type inequalities and further references can be found in [2], Section 7 and Chapter 7.
The aim of this article is to study the function \(\omega(D^{k},\delta)\) for arbitrary \(k,r\in{\mathbb{N}}\), \(k< r\) and non-increasing continuous positive functions f and g.
The article is organized in the following way. In Section 2 some auxiliary statements and in Section 3 main statements are given. Sections 4 and 5 are devoted to proofs.
2 Auxiliary results
- (a)
the derivative \(G^{(r)}\) exists for all \(t\in(t_{i},t_{i+1})\), \(i=0,1,\ldots,n\), where \(t_{0}:=a\) and \(t_{n+1}:=b\);
- (b)
there exists \(\epsilon\in\{1,-1\}\) such that \(\frac {G^{(r)}(t)}{g(t)}\equiv\epsilon\cdot(-1)^{i}\) for \(t\in(t_{i},t_{i+1})\), \(i=0,1,\ldots,n\).
Denote by \(\Gamma^{r}_{n,g}[0,a]\) the set of all perfect g-splines G defined on \([0,a]\) of order r with not more than n knots.
Below f and g will denote continuous positive non-increasing on \([0,\infty)\) functions.
The next theorem proves existence and some properties of the perfect g-spline \(G_{r,n,f,a}\in\Gamma_{n,g}^{r}[0,a]\) that least deviates from zero in \(\|\cdot\|_{C[0,a],f}\) norm.
Theorem 1
For \(a>0\), set \(\varphi_{r,n,f}(a):= \|G_{r,n,f,a}\|_{C[0,a],f}\). Then \(\varphi_{r,n,f}(a)\) is a continuous and non-decreasing function of \(a\in(0,\infty)\).
Remark
We do not prove the uniqueness of the spline \(G_{r,n,f,a}\in\Gamma ^{r}_{n,g}[0,a]\) satisfying (5). However, from arguments similar to the ones used in the proof of Theorem 2, it follows that if two splines \(G_{1},G_{2}\in\Gamma^{r}_{n,g}[0,a]\) satisfy (5), then \(G_{1}^{(k)}(0)=G_{2}^{(k)}(0)\) for all \(k=1,\ldots, r-1\).
The role of perfect g-splines becomes clearer due to the following theorem.
Theorem 2
3 Main results
If \(f(t)\equiv1\) and \(g(t)\equiv1\), then \(\varphi_{r,n,f}(\infty ):=\lim_{a\to+\infty} \varphi_{r,n,f}(a)=\infty\) for all \(r\in{\mathbb{N}}\), \(n\in{\mathbb{Z}}_{+}\). In the case when f, g are arbitrary positive non-increasing continuous functions, this is not always true.
Set \(g_{k}(t):=\int_{0}^{t}g_{k-1}(s)\,ds\), \(k=1,2,\ldots,{r}\), where \(g_{0}:=g\). The following theorem holds.
Theorem 3
Remark
From Theorem 3 it follows that for all \(r\in {\mathbb{N}}\), \(n\in{\mathbb{Z}}_{+}\), \(\varphi_{r,n,f}(\infty)<\infty\) if and only if \(\varphi_{r,0,f}(\infty)<\infty\).
If \(\varphi_{r,0,f}(\infty)=\infty\), then, in virtue of Theorems 1 and 3, for all \(r\in{\mathbb{N}}\), \(n\in{\mathbb{Z}}_{+}\) and \(\delta>0\), there exists a number \(\delta_{r,n} = \delta_{r,n}(\delta) >0\) such that \(\|G_{r,n,f,\delta_{r,n}}\| _{C[0,\delta_{r,n}],f}=\delta\) (if such number \(\delta _{r,n}\) is not unique, we can take the minimal value). In this case, for all \(\delta>0\), the function \(\omega(D^{k},\delta)\) is characterized by the following theorem.
Theorem 4
Information about the function \(\omega(D^{k},\delta)\) in the case when \(\varphi_{r,0,f}(\infty)<\infty\) is given by the following theorem.
Theorem 5
In the case when \(\varphi_{r,0,f}(\infty)<\infty\), information about asymptotic behavior of the function \(\varphi_{r,n,f}(\infty)\) as \(n\to \infty\) and fixed r is given by the following theorem.
Theorem 6
Let \(r\in{\mathbb{N}}\) and \(\varphi_{r,0,f}(\infty)<\infty\). \(\lim_{n\to\infty} \varphi_{r,n,f}(\infty)>0\) if and only if \(\varliminf _{t\to\infty}\frac{f(t)}{\vert P_{r}(t)\vert }<\infty\), where the function \(P_{r}(t)\) is defined in (8).
4 Proofs of the auxiliary results
4.1 Proof of Theorem 1
Proof of existence and uniqueness of the perfect g-spline \(G_{r,n,f,a}\) uses ideas that were used to prove Theorem 3.3.1 in monograph [2].
Then we have \((G_{\xi}^{a})^{(r)}=g_{\xi}^{a}\) and hence \(G_{\xi}^{a}\) is a g-spline with knots at the points of partition. Let \(Q^{\xi ,a}_{n+r-1}(t)=\sum_{i=0}^{n+r-1}{a_{i}(\xi)t^{i}}\) be the polynomial on which \(\inf_{Q_{n+r-1}}\|G_{\xi}^{a}-Q_{n+r-1}\|_{C[0,a],f}\) over all polynomials of degree less than or equal to \(n+r-1\) is attained. Consider the mapping \(\phi:S^{n}\to {\mathbb{R}}^{n}\), \(\phi(\xi):=(a_{r}(\xi),\ldots,a_{n+r-1}(\xi))\). From the definition of ϕ and properties of polynomials of the best approximation, it follows that ϕ is continuous and odd. Hence from Borsuk’s theorem it follows that there exists \(\xi_{0}\in S^{n}\) such that \(\phi(\xi_{0})=0\). This means that the polynomial \(Q^{\xi_{0},a}_{n+r-1}\) has order less than or equal to \(r-1\). Therefore, for the function \(G_{r,n,f,a}:=G_{\xi_{0}}-Q^{\xi_{0},a}_{n+r-1}\), we have \(G_{r,n,f,a}^{(r)}=g_{\xi_{0}}\). Due to the generalization of Chebyshev’s theorem about oscillation (see, for example, [25, 26], Chapter 9, Section 5) \(G_{r,n,f,a}\) has \(n+r+1\) oscillation points \(0\leq t_{1}< t_{2}<\cdots <t_{n+r+1}\leq a\) and hence at least \(n+r\) sign changes. Thus, in view of Rolle’s theorem, \(G_{r,n,f,a}^{(r)}\) has at least n sign changes (and hence exactly n sign changes due to construction). This means that \(G_{r,n,f,a}\) is a perfect g-spline with exactly n nodes, in particular, \(G_{r,n,f,a}\in\Gamma_{n,g}^{r}[0,a]\).
Let us prove that \(t_{n+r+1}=a\). Assume the converse. Since f is non-increasing, we get that \(G_{r,n,f,a}'\) has \(n+r\) sign changes and hence \(G_{r,n,f,a}^{(r)}\) has \(n+1\) sign changes. However, this is impossible. Multiplying, if needed, the function \(G_{r,n,f,a}\) by −1, we get a perfect g-spline for which equalities (5) hold.
The fact that for fixed \(r\in{\mathbb{N}}\) and \(n\in {\mathbb{Z}}_{+}\) the function \(\varphi_{r,n,f}\) is non-decreasing follows from its definition. The continuity of the function \(\varphi_{r,n,f}\) follows from the continuity of functions f and g. The theorem is proved.
4.2 Proof of Theorem 2
We need the following lemma.
Lemma 1
Remark
Notation \(\operatorname{sgn}x^{(r)}(0) = \pm1\) means that there exists \(\varepsilon> 0\) such that \(\operatorname{sgn}x^{(r)}(t)=\pm1\) almost everywhere in the interval \((0,\varepsilon)\).
From conditions of the lemma it follows that the function \(x^{(s)}\) has exactly \(n+r-s\) sign changes, \(s = 0,1,\ldots,r\). Hence the function \(x^{(s)}\) changes sign on each of its monotonicity intervals, \(s = 0,1,\ldots,r-1\). This implies that \(x^{(s)}(0)\neq0\), \(s = 0,1,\ldots,r-1\), and that equalities (9) hold. The lemma is proved.
Due to (12), \((-1)^{k} G_{r,n,f,a}(t_{1})>0\), where \(t_{1}\) is the first oscillation point of \(G_{r,n,f,a}\). This means that \((-1)^{k}\Delta(t_{1}) < 0\). Since all sign changes of the function Δ are located inside the interval \((t_{1},a)\), we get \((-1)^{k}\Delta(0) < 0\), and hence, in virtue of Lemma 1, we get \(\Delta^{(k)}(0) < 0\). But this contradicts (11). The theorem is proved.
5 Proofs of the main results
5.1 Proof of Theorem 3
We prove first that the statement of the theorem is true in the case \(n=0\). In the case \(n=0\), we write \(\varphi_{r,f}\) instead of \(\varphi _{r,0,f}\) and \(G_{r,f,M}\) instead of \(G_{r,0,f,M}\). To prove the theorem, we need the following lemma.
Lemma 2
We will proceed using induction on m in order to prove the first inequality.
Let \(m=1\). \(P_{1}(t)=-A_{0}+g_{1}(t)\). Since conditions (6) hold and \(P_{1}(0)=-A_{0}\), we have \(P_{1}(\infty)=0\). Since the function \(h_{1}(t)=g_{1}(t)+C\) has one zero, \(C\in(-A_{0},0]\). This means that \(|h_{1}(t)|<|P_{1}(t)|\) for all \(t\in[0,\alpha_{1}]\).
Let the statement of the lemma hold in the case \(m=k\leq r-1\). We will show that it holds in the case \(m=k+1\) too.
From Lemma 1 it follows that \(\operatorname{sgn} h_{m}(0) = \operatorname{sgn}P_{m}(0)\). From the proved part of the lemma it now follows that the function \(P_{m}(t)-P_{m}(0)+h_{m}(0)\) has exactly one zero. The second inequality in the statement of the lemma can be proved using arguments similar to the ones used in the proof of the first inequality. The lemma is proved.
Let us return to the proof of the theorem in the case when \(n=0\).
Let conditions (6) and (7) hold. Set \(K_{r}:=\sup_{t\in [0,\infty)}\frac{|P_{r}(t)|}{f(t)}\). Assume the converse, let \({\varphi _{r,f}(\infty)=\infty}\). This means that there exists \(M>0\) such that \(\varphi_{r,f}(M)>K_{r}\). Due to Lemma 2 the inequality \(|G_{r,f,M}(t)|<|P_{r}(t)|\) holds on the interval \([0,\alpha_{r}^{M}]\). Moreover, \(|P_{r}(t)|\leq K_{r} f(t)<\varphi_{r,f}(M) f(t)\). Thus, on the interval \([0,\alpha_{r}^{M}]\), the inequality \(|G_{r,f,M}(t)|<\varphi _{r,f}(M) f(t)\) holds. However, in this case \(G_{r,f,M}(t)\) has not more than r oscillating points. Contradiction. Sufficiency is proved.
Since functions \(f(t)\) and \(g(t)\) are bounded, we get that all functions \(Q_{r}^{(k)}(t)\), \(k=1,\ldots,r-1\), are also bounded on \([0,\infty)\). Note that the only bounded on \([0,\infty)\) primitives of the function \(g(t)\) of order \(k\in{\mathbb{N}}\) are functions \(P_{k}(t) + C_{k}\), where \(C_{k}\in{\mathbb{R}}\), and only in the case when corresponding conditions (6) hold. This means that conditions (6) hold. Necessity of conditions (6) are proved.
Note that from arguments above it follows that the following lemma holds.
Lemma 3
Let \(r\in{\mathbb{N}}\), \(r\geq2\) and \(\varphi_{r,f}(\infty)<\infty\). Then \(\vert G_{r,f,M}^{(r-k-1)}(0)\vert \to A_{k}\) and \(\alpha_{k+1}^{M}\to \infty\) when \(M\to\infty\), where \(\alpha_{k+1}^{M}\) is the first zero of the function \(G_{r,f,M}^{(r-k-1)}\), \(k=0,1,\ldots,r-2\).
Let n be an arbitrary natural number now.
We will prove that for all \(r,n\in{\mathbb{N}}\), \(\varphi_{r,n,f}(\infty )<\infty\) if and only if \(\varphi_{r,f}(\infty)<\infty\).
It is clear that \(\varphi_{r,f}(M)\geq\varphi_{r,n,f}(M)\) for all \(M>0\), and hence \(\varphi_{r,f}(\infty)<\infty\) implies \(\varphi _{r,n,f}(\infty)<\infty\).
Assume that \(\varphi_{r,n,f}(\infty)<\infty\). Denote by \(t_{n,k}^{M}\) the kth knot of the g-spline \(G_{r,n,f,M}(t)\), \(k=1,2,\ldots,n\). Set \(t_{n,0}^{M}:=0\), \(t_{n,n+1}^{M}:=M\). Let \(1\leq k\leq n+1\) be the smallest number of the knots of the g-spline \(G_{r,n,f,M}(t)\) for which the set \(\{t_{n,k}^{M}: M>0 \}\) is unbounded. We can choose an increasing sequence \(\{M_{l} \}_{l=1}^{\infty}\), \(M_{l}\to\infty\) as \(l\to\infty\) such that \(t_{n,s}^{M_{l}}\to t_{n,s}<\infty\), \(s\leq k-1\) and \(t_{n,k}^{M_{l}}\to\infty\) as \(l\to \infty\).
Remark
5.2 Proof of Theorem 4
- 1.
\(\|G_{r,\delta}\|_{C[0,\infty),f}= \delta\) and either \(G_{r,\delta }^{(r)}\equiv g\) or \(G_{r,\delta}^{(r)}\equiv-g\) on the intervals \((y_{k},y_{k+1})\) (\(k=0,1,2,\ldots\)).
- 2.
For all \(c>0\), the sequences \(\{ G_{r,n,f,\delta _{r,n}}^{(k)} \}_{n=0}^{\infty}\) (\(k=0,1,\ldots,r-1\)) (whose elements are defined on \([0,c]\) for big enough n) converge to \(G_{r,\delta}^{(k)}\) uniformly on \([0,c]\).
\(\{\delta_{r,n} \}_{n=0}^{\infty}\) is a non-decreasing sequence. Moreover, this sequence is unbounded because otherwise we would get a perfect g-spline G with arbitrarily close oscillating points; this is impossible because the functions G and \(G^{(r)}\) (and hence \(G'\)) are bounded.
Denote by \(t_{n,k}\) (\(k=1,\ldots,n\), \(n=1,2,\ldots\)) the knots of the g-spline \(G_{r,n,f,\delta_{r,n}}\). We can choose a sequence \(n_{s}\) (\(n_{s}\to\infty\) as \(s\to\infty\)) such that every sequence \(\{t_{n_{s},k} \}_{n_{s}\geq k}^{\infty}\) (\(k=1,2,\ldots\)) has a (finite or infinite) limit.
Let \(0\leq y_{1}< y_{2}<\cdots\) be all distinct finite limits of these sequences, ordered in an ascending way. The number of the nodes \(y_{k}\) is infinite since from the statement of the theorem, we have \(\varphi_{r,n,f}(\infty)=\infty\) for all \(n\in{\mathbb{N}}\).
For all \(i\in{\mathbb{N}}\) and for all small enough \(\varepsilon>0\), there exists \(N=N(i,\varepsilon)\) such that for every \(n>N(i,\varepsilon )\), \(G_{r,n,f,\delta_{r,n}}^{(r)}\equiv g\) or \(G_{r,n,f,\delta _{r,n}}^{(r)}\equiv-g\) on \(I_{i}(\varepsilon):=(y_{i-1}+\varepsilon ,y_{i}-\varepsilon)\). In other words, for each \(i\in {\mathbb{N}}\) starting with some \(n = N(i,\varepsilon)\), the restriction of the g-spline \(G_{r,n,f,\delta_{r,n}}\) to the interval \(I_{i}(\varepsilon)\) is a primitive of order r of g or −g. Since \(\varepsilon>0\) is arbitrary, on each interval \((y_{i-1},y_{i})\) we get existence of point-wise limit \(\lim_{n\to \infty}G_{r,n,f,\delta_{r,n}}=:G_{r,\delta}\); moreover, on the intervals \((y_{i},y_{i+1})\), \(G_{r,\delta}^{(r)}\equiv g\) or \(G_{r,\delta }^{(r)}\equiv-g\) (\(i=1,2,\ldots\)). It is clear that \(\|G_{r,\delta}\| _{C[0,\infty),f}= \delta\). Using arguments similar to the ones used to prove that \(\lim_{n\to\infty}{\delta_{r,n}}=+\infty\), we can prove that \(y_{k}\to\infty\) (\(k\to\infty\)).
Let us fix some \(c>0\). Starting with some n, all g-splines \(G_{r,n,f,\delta_{r,n}}\) are defined on \([0,c]\). From \(\|G_{r,n,f,\delta _{r,n}}\|_{C[0,\delta_{r,n}],f}=\delta\) and the fact that f is non-increasing (and hence is bounded) it follows that the sequence \(\{G_{r,n,f,\delta_{r,n}} \} _{n=0}^{\infty}\) is uniformly bounded on \([0,c]\); from \(\vert G_{r,n,f,\delta_{r,n}}^{(r)}(t)\vert \leq g(t)\) almost everywhere on \([0,\infty)\) and the fact that g is non-increasing (and hence is bounded) it follows that sequences \(\{G_{r,n,f,\delta_{r,n}}^{(k)} \}_{n=0}^{\infty}\), \(k=0,\ldots ,r-1\), are uniformly bounded on \([0,c]\) and equicontinuous. The later implies uniform convergence on \([0,c]\) of the sequence \(G_{r,n,f,\delta_{r,n}}\) to \(G_{r,f}\). The theorem is proved.
5.3 Proof of Theorem 5
Let \(n\geq0\). We can choose an increasing sequence \(\{M_{k} \} _{k=1}^{\infty}\), \(M_{k}\to\infty\) as \(k\to\infty\) in such a way that all sequences \(t_{n,s}^{M_{k}}\), \(1\leq s\leq n\) (as above, \(t_{n,s}^{M_{k}}\) is the sth knot of the g-spline \(G_{r,n,f,M_{k}}\)) have limits (finite or infinite). Let \(t_{n,1}<\cdots<t_{n,m}\) be all distinct finite limits of these sequences in ascending order. Analogously to the proof of Theorem 4, we get uniform on each segment \([0,c]\), \(c>0\) convergence of the sequence \(G_{r,n,f,M_{k}}\) to the g-spline \(P_{r,n,f,\{M_{k}\}}\) with m knots (defined on the whole half-line) together with all derivatives up to the order \(r-1\) inclusively. For brevity we will write \(P_{r,n,f}\) instead of \(P_{r,n,f,\{M_{k}\}}\).
In view of (22) and (23) \((-1)^{s}\Delta _{k}(0) < 0\), and hence due to Lemma 1 we get \(\Delta_{k}^{(s)}(0) < 0\). However, this contradicts (21).
In virtue of property (19) proved above, the limit \(\lim_{M_{k}\to\infty} \vert G^{(k)}_{r,n,f,M_{k}}(0) \vert \) does not depend on the choice of the sequence \(\{M_{k}\}_{k=1}^{\infty}\). This finishes the proof of the theorem.
5.4 Proof of Theorem 6
We will need the following lemmas.
Lemma 4
Lemma 5
Let \(\varphi_{r,n,f}(\infty) < \infty\) and \(\lim_{t\to\infty}\frac {f(t)}{P_{r}(t)} = \infty\). Then the number of oscillation points of the g-spline \(P_{r,n,f}\) tends to infinity as \(n\to\infty\).
Let some \(n\in{\mathbb{N}}\) be fixed. Suppose that \(M>0\) is such that for all \(t>M\), \(\frac{f(t)}{P_{r}(t)} > \frac{2}{\varphi_{r,n,f}(\infty )}\). Let the g-spline \(P_{r,n,f}\) have k oscillating points \(0\leq a_{1}< a_{2}<\cdots<a_{k}\). Denote by \(0\leq b_{1}< b_{2}<\cdots<b_{n+r+1}\) all oscillation points of the g-spline \(G_{r,n,f,K}\), where K is chosen so big that \(\operatorname{sgn} G_{r,n,f,K}(b_{s})=\operatorname{sgn} P_{r,n,f}(b_{s})\), \(s=1,2,\ldots,k\), \(b_{k+1}>\max\{a_{k}, M\}\) and \(\varphi_{r,n,f}(K) > \frac{1}{2} \varphi_{r,n,f}(\infty)\). Choose \(\varepsilon>0\) so small that \((1-\varepsilon )\varphi_{r,n,f}(K) > \frac{1}{2} \varphi_{r,n,f}(\infty)\).
Let \(t_{n,s}^{K}\), \(s = 1,\ldots, n\), be the knots of the g-spline \(G_{r,n,f,K}\). Note that for all \(s = 1,2,\ldots, n\), the g-spline \(G_{r,n,f,K}\) has at least s oscillating points on the interval \([0,t_{n,s}^{k}]\). Really, assume the converse, suppose for some \(1\leq s\leq n\) that the g-spline \(G_{r,n,f,K}\) has less than s oscillation points on the interval \([0,t_{n,s}^{K}]\). Then the g-spline \(G_{r,n,f,K}\) has more than \(n+r+1-s\) oscillation points on the interval \((t_{n,s}^{K}, K]\), and hence more than \(n+r-s\) sign changes. This means that the function \(G^{(r)}_{r,n,f,K}\) has more than \(n-s\) sign changes on the interval \((t_{n,s}^{K}, K]\). However, this is impossible.
This means that the limiting g-spline \(P_{r,n,f}\) has at least \(n-1\) oscillation points. The lemma is proved.
Lemma 6
We will prove the statement of the lemma using induction. In the case \(s=r\), inequality (25) holds with equality sign. Let inequality (25) hold with \(s=k\geq2\). We will prove that it is true for \(s=k-1\).
Let us return to the proof of the theorem.
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable remarks and suggestions.
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Authors’ Affiliations
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