Abstract
A quantitative estimate for the Trotter’s approximation theorem for the limiting semigroup of operators generated by the multidimensional Bernstein operators on a simplex is obtained. For this, an essential step consists in an explicit representation of the derivatives of higher order of multidimensional Bernstein operators.
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1 Introduction
Let X be a Banach space, endowed with norm \(\Vert \cdot \Vert\). Denote by L(X) the space of bounded linear operators \(T:X\rightarrow X\), endowed with norm \(\Vert L\Vert =\sup \{\Vert Lx\Vert ,\; x\in X, \Vert x\Vert =1\}.\) A \(C_0\) semigroup of operators on the space X is a family of operators \(\{T(t)\}_{t\ge 0}\), \(T(t)\in L(X)\), with the properties
-
a)
\(T(t+s)=T(t)T(s)\), for \(t,s\ge 0\);
-
b)
\(\lim _{t \rightarrow 0+}T(t)x=x\), for any \(x\in X\), in the sense of norm of X.
As a general bibliography of the subject we mention [1,2,3, 5, 10, 17, 18]. A basic result concerning \(C_0\) semigroups of operators is given by Trotter’s approximation theorem.
Theorem A
[21] Let \((L_n)_{n\in {\mathbb {N}}}\) be a sequence of bounded linear operators on a Banach space X and let \((\rho _n)_{n\in {\mathbb {N}}}\) be a decreasing sequence of positive real numbers tending to 0. Suppose that there exist \(M \ge 0\) and \(\omega \in {\mathbb {R}}\) such that
Moreover, assume that D is a dense subspace of X and for every \(f\in D\) the following Voronovskaja-type formula holds
If \((\lambda I-A)(D)\) is dense in X for some \(\lambda >\omega\), then there exists a \(C_0\)-semigroup \((T(t))_{t\ge 0}\) such that for every \(f \in X\) and every sequence \((k(n))_n\in {\mathbb {N}}\) of positive integers satisfying \(\lim _{n\rightarrow +\infty } k(n)\cdot {\rho _n} = t\), we have
A version of Trotter’s approximation theorem is the following
Theorem B
([3], a part of Corollary 2.2.11) Let \((L_n)_{n\ge 1}\) be a sequence of linear operators on the Banach space E, with \(\Vert L_n\Vert \le 1\) and let \((\rho _n)_{n\ge 1}\) be a sequence of positive real numbers such that \(\lim _{n\rightarrow \infty } \rho (n)=0\). Let \(A_0:D_0\rightarrow E\) be a linear operator defined on a subspace \(D_0\) of E and assume that (i) there is a family \((E_i)_{i\in I}\) of finite dimensional subspaces of \(D_0\) which are invariant under each \(L_n\) and whose union \(\bigcup _{i\in I} E_i\) is dense in E; (ii) \(\lim \limits _{n\rightarrow \infty }\frac{L_n(u)-u}{\rho (n)}=A(u)\) for every \(u\in D_0\).
Then \(A_0\) is closable and its closure \(A:D(A)\rightarrow E\) is the generator of a contraction \(C_0\)-semigroup \((T(t))_{t\ge 0}\) on E satisfying the following condition: if \((k(n))_{n\ge 1}\) is a sequence of positive integers with \(\lim \limits _{n\rightarrow \infty }k(n)/\rho (n)=t\), then, for every \(f\in E\),
The iterates and the limiting semigroup generated by Bernstein operators were studied in [6,7,8,9, 12, 14, 16] among others. The semigroup generated by multidimensional Bernstein operators was considered in [6, 7, 15]. For the limiting groups generated by other positive linear operators we cite [4, 11, 13, 15, 19, 20].
2 Additional results for multidimensional Bernstein operators
We fix the following notation. Let \({\mathbb {N}}=\{1,2,\ldots \}\) and \({\mathbb {N}}_0={\mathbb {N}}\cup \{0\}\). Let \(d\in {\mathbb {N}}\) be fixed. For a multi-index \({\overline{k}}\in {\mathbb {N}}_0^d\), \({\overline{k}}=(k_1,\ldots ,k_d)\), denote \(|{\overline{k}}|=k_1+\ldots +k_d\) and \({\overline{k}}!=k_1!\ldots k_d!\). For \(n\in {\mathbb {N}}\), if \({\overline{k}}\in {\mathbb {N}}_0^d\), \(|{\overline{k}}|\le n\), define \(\left( {\begin{array}{c}n\\ {\overline{k}}\end{array}}\right) =\frac{n!}{{\overline{k}}!(n-|{\overline{k}}|)!}\).
Define the d-simplex
Vectors \({\overline{e}}_i=(0,\ldots ,0,1,0,\ldots ,0)\), \(1\le i\le d\), form the standard base of the space \({\mathbb {R}}^d\). If \({\overline{x}}=(x_1,\ldots ,x_d)\in \varDelta _d\) put \(|{\overline{x}}|=x_1+\ldots +x_d\). Hence \(|{\overline{x}}|\le 1\). If, in addition we take \({\overline{k}}=(k_1,\ldots ,k_d)\in \varLambda _d^n\), then define \({\overline{x}}^{{\overline{k}}}=x_1^{k_1}\ldots x_d^{k_d}\). With this notation we define now
We extend the definition of \(p_{n,{\overline{k}}}({\overline{x}})\), for \({\overline{k}}\in {\mathbb {Z}}^d\), putting
In the case \(d=1\) and \({\overline{k}}=k\), \({\overline{x}}=x\) we write simply \(p_{n,k}(x)\) instead of \(p_{n,{\overline{k}}}({\overline{x}})\).
With these preparations we can define the Bernstein operator on the simplex \(\varDelta _d\):
where \(\frac{{\overline{k}}}{n}=\Big (\frac{k_1}{n},\ldots \frac{k_d}{n}\Big )\), \(n\in {\mathbb {N}}\), \(f:\varDelta _d\rightarrow {\mathbb {R}}\), \({\overline{x}}\in \varDelta _d\).
Let \(\alpha =(\alpha _1,\ldots ,\alpha _d)\in {\mathbb {N}}_0^d\). Suppose \(|\alpha |\ge 1\), where \(|\alpha |=\alpha _1+\ldots +\alpha _d\). If \(f\in C^{|\alpha |}(\varDelta _d)\) define
If \(|\alpha |=0\), define \(\frac{\partial ^{\alpha }f}{\partial {\overline{x}}^{\alpha }}:=f\).
For \(\alpha \in {\mathbb {N}}_0^d\) denote by \(C^{\alpha }(\varDelta _d)\) the space of functions \(f:\varDelta _d\rightarrow {\mathbb {R}}\) which admits the partial derivative \(\frac{\partial ^{\alpha }f}{\partial {\overline{x}}^{\alpha }}\) continuous on \(\varDelta _d\). For \(1\le i\le d\) consider functions \(\pi _i:\varDelta _d\rightarrow {\mathbb {R}}\), \(\pi _i({\overline{x}})=x_i\).
The next lemma is easy to obtain and in great part well known, see for instance [2, Section 6.2].
Lemma 1
For \({\overline{x}}=(x_1,\ldots ,x_d)\in \varDelta _d\) we have
-
i)
\(B_n(\pi _i-x_i,{\overline{x}})=0\), \((1\le i\le d)\);
-
ii)
\(B_n((\pi _i-x_i)(\pi _j-x_j),{\overline{x}})=-\frac{x_i x_j}{n}\), \((1\le i,j\le d,\;i\not =j)\);
-
iii)
\(B_n((\pi _i-x_i)^2,{\overline{x}})=\frac{x_i(1-x_i)}{n}\), \((1\le i \le d)\);
-
iv)
\(B_n((\pi _i-x_i)^3,{\overline{x}})=\frac{x_i(1-x_i)(1-2x_i)}{n^2},\;(1\le i\le d)\);
-
v)
\(B_n((\pi _i-x_i)^2(\pi _j-x_j),{\overline{x}})=\frac{x_ix_j(2x_i-1)}{n^2};\;(1\le i,j\le d,\;i\not =j)\);
-
vi)
\(B_n((\pi _i-x_i)(\pi _j-x_j)(\pi _m-x_m),{\overline{x}})=\frac{2x_i x_jx_m}{n^2},\;(1\le i<j<m\le d)\);
-
vii)
\(B_n((\pi _i-x_i)^4,{\overline{x}})=\frac{1}{n^2}\left( 3-\frac{6}{n}\right) x_i^2(1-x_i)^2+\frac{x_i(1-x_i)}{n^3},\) \((1\le i\le d)\);
-
viii)
\(B_n((\pi _i-x_i)^2(\pi _j-x_j)^2,{\overline{x}})=\frac{1}{n^2}\left( 3-\frac{6}{n}\right) x_i^2x_j^3+\left( -\frac{1}{n^2}+\frac{2}{n^3}\right) (x_i^2x_j+x_ix_j^2)+\frac{n-1}{n^3}x_ix_j,\) \((1\le i,j\le d,\;i\not =j)\).
Theorem 1
Let \(\alpha \in {\mathbb {N}}_0^d\), \(|\alpha |\ge 1\). Then for any \(f\in C^{|\alpha |}(\varDelta _d)\), \(n\in {\mathbb {N}}\), \(n\ge |\alpha |\) and \({\overline{x}}\in \varDelta _d\) we have
where \(I_{\alpha }=\{i\in \{1,\ldots ,d\}\mid \alpha _i\ge 1\}\) and
In the case \(|\alpha |=0\), the term \(\int \!\!\!\int \ldots \int _{\left[ 0,\frac{1}{n}\right] ^{|\alpha |}}\frac{\partial ^{\alpha }}{\partial {\overline{t}}^{\alpha }}f\Big (\frac{{\overline{k}}}{n}+\sum _{i\in I_{\alpha }}\Big (\sum _{j=1}^{\alpha _i}t_{i,j}\Big ){\overline{e}}_i\Big )d{\overline{t}}_{\alpha }\) is reduced to \(f\left( \frac{{\overline{k}}}{n}\right) .\)
Proof
We consider only the case \(d\ge 2\), since the proof the case \(d=1\) can be easily deduced from the case \(d\ge 2\).
The following formula is well-known.
We induct on \(r:=|\alpha |\). For \(r=0\) relation (7) is obvious. Suppose that relation (7) holds for any \(d\ge 1\) and any \(\alpha\) with \(|\alpha |=r\) and let show that it is a true for a multi-index \(\beta =(\beta _1,\ldots ,\beta _d)\) with \(|\beta |=r+1\). Then there are a multi-index \(\alpha =(\alpha _1,\ldots ,\alpha _d)\) with \(|\alpha |=r\) and an index \(1\le i\le d\) such that \(\beta _i=\alpha _i+1\) and \(\beta _j=\alpha _j\), for \(1\le j\le d\), \(j\not =i\). To simplify the notation, we can suppose that \(i=d\). In other cases we make a renumbering of the variables.
Let \({\overline{x}}\in \varDelta _d\), \({\overline{x}}=(x_1,\ldots ,x_d)\). Denote \(|{\overline{x}}|=x_1+\ldots + x_d\). Suppose \(x_d>0\). Define \({\overline{z}}=(x_1,\ldots ,x_{d-1})\) and \(|{\overline{z}}|=x_1+\ldots x_{d-1}\). Then \(|{\overline{z}}|<1\). Denote also \(y:=\frac{x_d}{1-|{\overline{z}}|}\in [0,1]\) and \(m:=n-|\alpha |=n-r\).
Let \({\overline{k}}\in {\mathbb {N}}_0^d\), with \(|{\overline{k}}|=m\). Denote \({\overline{\ell }}:=(k_1,\ldots k_{d-1})\). Then \(|{\overline{k}}|=|{\overline{\ell }}|+k_d\). We can write
For \({\overline{k}}\in {\mathbb {N}}_0^d\), \(|{\overline{k}}|\le m\), denote
By the hypothesis of induction we have
Using relation (9) and the decomposition of the sum
we can write
By relation (11) it follows
where the first \(d-1\) components of \({\overline{k}}\) are fixed and form vector \({\overline{\ell }}\). Then
Now, using similar relations to (9) and (10), but with \(m-1\) instead of m we obtain
Finally, we have
Because \(\beta _d=\alpha _d+1\ge 1\) it follows that \(d\in I_{\beta }\). Then we can denote s by \(t_{d,\beta _d}\). Let us use the notation \(d{\overline{t}}_{\beta } =\prod _{i\in I_{\beta }}\prod _{j=1}^{\beta _i}dt_{i,j}\). Then
Also, \(\left[ 0,\frac{1}{n}\right] ^{|\alpha |}\times \left[ 0,\frac{1}{n}\right] =\left[ 0,\frac{1}{n}\right] ^{|\beta |}\), \(dt_{d,\beta _d}d{\overline{t}}_{\alpha }=d{\overline{t}}_{\beta }\) and \(\frac{\partial }{\partial x_d}\frac{\partial ^{\alpha }}{\partial {\overline{t}}^{\alpha }}=\frac{\partial ^{\beta }}{\partial {\overline{t}}^{\beta }}\). Thus,
From relations (12) and (13) and since \(m-1=n-|\beta |\) one obtains
The relation above can be also extended by continuity at a point \({\overline{x}}\) with \(x_d=0\). The induction step is proved. \(\square\)
Let \(\alpha \in {\mathbb {N}}_0^d\). Denote
The following corollaries are immediate.
Corollary 1
For any \(n\in {\mathbb {N}}\) we have
Let \(\alpha \in {\mathbb {N}}_0^d\). If \(f\in C^{\alpha }(\varDelta _d)\) denote \(\left\| \frac{\partial ^{\alpha }f}{\partial {\overline{x}}^{\alpha }}\right\| =\max _{{\overline{x}}\in \varDelta _d}\left| \frac{\partial ^{\alpha }f}{\partial {\overline{x}}^{\alpha }}({\overline{x}})\right|\).
Corollary 2
For any \(n\in {\mathbb {N}}\), any \(\alpha \in {\mathbb {N}}_0^d\) and any \(f\in C^{\alpha }(\varDelta _d)\) we have
By induction one obtains
Corollary 3
For any \(n\in {\mathbb {N}}\), any \(\alpha \in {\mathbb {N}}_0^d\), any \(j\in {\mathbb {N}}_0\) and any \(f\in C^{\alpha }(\varDelta _d)\) we have
Remark 1
For \(|\alpha |\ge 2\) it follows
For \(k\in {\mathbb {N}}\), \(f\in C^k(\varDelta _d)\) define
Corollary 4
For any \(n\in {\mathbb {N}}\), any \(j\in {\mathbb {N}}_0\), any \(k\in {\mathbb {N}}\), \(k\ge 2\), and any \(f\in C^{k}(\varDelta )\) we have
Proof
Let \(j\ge 0\). There exists \(\alpha _0\in {\mathbb {N}}_0^d\) with \(|\alpha _0|=k\) such that \(\mu _k((B_n)^{j+1}(f))\) \(=\Big \Vert \frac{\partial ^{\alpha _0}}{\partial {\overline{x}}^{\alpha _0}}(B_n)^{j+1}(f)\Big \Vert\). Then using relation (17) and Remark 1 we obtain
So that we can apply the induction. \(\square\)
Corollary 5
We have \(B_n(\varPi _m)\subset \varPi _m\), \(m\ge 0\), where \(\varPi _m\) is the set of polynomials with d variables with total degree at most m.
Proof
Take a monomial function \(f({\overline{x}})={\overline{x}}^{\gamma }\), \(\gamma =(\gamma _1,\ldots ,\gamma _d)\), with \(|\gamma |\le m\). Then \(\frac{\partial ^{\gamma _j+1}}{\partial x_j^{\gamma _j+1}}f=0\) on \({\mathbb {R}}^d\), for \(1\le j\le d\). From Theorem 1 we deduce that \(\frac{\partial ^{\gamma _j+1}}{\partial x_j^{\gamma _j+1}}B_n(f)=0\) on \(\varDelta _d\), for every \(1\le j\le d\). It is easy to see that \(B_n(f)\) is a polynomial of the form \(\sum _{s\in I}a_s{\overline{x}}^{\beta _s}\), where I is finite, \(\beta _s=(\beta _{s,1},\ldots ,\beta _{s,d})\in {\mathbb {N}}^d\), \(\beta _{s,j}\le \gamma _j\), for \(1\le j\le d\), \(s\in I\) and \(a_s\in {\mathbb {R}}\), for \(s\in I\). Therefore \(B_n(f)\in \varPi _m\). It follows \(B_n(\varPi _m)\subset \varPi _m\). \(\square\)
3 A quantitative estimate for Trotter’s theorem
Consider operator
In the following lemma we give a Voronovskaja type theorem for operators \(B_n\).
Lemma 2
Remark 2
There exists a semigroup of bounded linear operators \(\{T(t)\}_{t\ge 0}\), \(T(t):C(\varDelta _d)\rightarrow C(\varDelta _d)\), such that
for any sequences of integers \((m_n)_n\) such that \(\frac{m_n}{n}=t\). This fact follows, for instance, from Theorem B, with the choices: \(L_n=B_n\), \(E=C(\varDelta _d)\), \(D_0=C^2(\varDelta _d)\), \(A_0=A\) and \(E_i=\varPi _i\), \(i\ge 0\), where \(\varPi _i\) is the space of polynomials with total degree i, see Corollary 5.
Lemma 3
For \(g\in C^4(\varDelta _d)\) we have
where
and \(\mu _3(g)\) is defined in (19).
Proof
For \(g\in C^4(\varDelta _d)\), \({\overline{x}},{\overline{t}}\in \varDelta _d\) we get
where \(\xi\) belongs to the interval \([{\overline{x}},{\overline{t}}]\subset \varDelta _d\). Then, using relation (21) and Lemma 1 we obtain \(B_n(g,{\overline{x}})=g({\overline{x}})+\frac{1}{n}A(g,{\overline{x}})+R_3({\overline{x}})\), and
\(\square\)
Lemma 4
For any \(g\in C^4(\varDelta _d)\) and \(t\ge 0\) we have
where
and \(\mu _k(g)\), \(k=2,3,4\) are defined in (19).
Proof
First we use the known inequality
In the sequel we use abbreviated notations for sums of the form \(\sum _i a_i\), \(\sum _{i,j} a_{i,j}\), \(\sum _{i,j,k} a_{i,j,k}\), \(\sum _{i,j,k,\ell } a_{i,j,k,\ell }\). We suppose that all the indices are in the set \(\{1,2,\ldots , d\}\) and are different from each other in the case of these sums. The terms are unique taken as indicated in the generic form described by the sum. For instance, \(\sum _{i,j}x_ix_j\) is the abbreviation of \(\sum _{1\le i<j\le d}x_ix_j\) and \(\sum _{i,j}x_i^2x_j\) is the abbreviation of \(\sum _{1\le i,j\le d,\;i\not =j}x_i^2x_j\). We also use the convention that if the number of indices is strictly greater than d, then the corresponding sum is null.
From (21) one obtains, after certain calculations, for \(g\in C^4(\varDelta _d)\) and \({\overline{x}}\in \varDelta _d\):
Therefore
\(\square\)
The main result is the following.
Theorem 2
For \(f\in C^4(\varDelta _d)\), \(m\in {\mathbb {N}}\), \(n\in {\mathbb {N}}\), \(t\ge 0\) we have
where \(C^k_d\), \(k=1,2,3,4\) are given in (23) and (24).
Proof
The method of proof is a modification of the method used in [12] and consists in a modification of a telescopic sum argument.
Since \(\Vert (B_n)^m\Vert =1\), for \(m\ge 1\) it follows \(\Vert T(t)\Vert =1\), \(t>0\).
Consider the decomposition
From relation (21) we deduce \(\Vert Af\Vert \le \Big (\frac{1}{2}d+\frac{d(d-1)}{2}\Big )\mu _2(f)=\frac{d^2}{2}\mu _2(f)\). We obtain successively:
For the second term one can use a telescopic sum:
We can write
From Lemmas 3 and 4 it results for any j
But using Corollary 4 we have \(\mu _k((B_n)^jf)\le \Big (\frac{n-1}{n}\Big )^j\mu _k(f)\), for \(j\ge 0\), \(k=2,3,4\). Then by using (28), (29), (30) and (31) one obtains
From (27) and (32) it results (25). \(\square\)
Finally, we compare our result with others, obtained previously.
Remark 3
A quantitative version of Trotter’s theorem for the semigroup generated by Bernstein operators defined on the simplex \(\varDelta _d\) was obtained by Campiti and Tacelli [6, 7] for functions belonging to the space \(C^{2,\alpha }(\varDelta _d)\), with \(0<\alpha <1\). The space \(C^{2,\alpha }(\varDelta _d)\) consists of real functions f defined on \(\varDelta _d\), which admit second derivatives on \(\varDelta _d\) and for which the following condition
is satisfied. In [6, Theorem 2.3], completed in [7], the following estimate is obtained:
for every \(t\ge 0\), \(f\in C^{2,\alpha }(\varDelta _d)\) and sequence \((k(n))_{n\ge 1}\) of positive integers, where \(\psi (f)\) depends only on f. On other hand, relation (25) with m replaced by k(n) is of the form
The first remark is that in the hypothesis \(k(n)/n\rightarrow t\), \((n\rightarrow \infty )\), relation (33) is generally stronger, because it is valid for the greater space \(C^{2,\alpha }(\varDelta _d)\), instead of space \(C^4(\varDelta _d)\).
In the case when \(f\in C^4(\varDelta _d)\), in order to make an asymptotic comparison, let fix f and t and denote \(\beta =\frac{\alpha }{4(\alpha +1)}\in (0,1/8)\). We can make this comparison in two cases.
If \(\liminf _{n\rightarrow \infty }\left| \frac{k(n)}{n}-t\right| n^{\beta }\in (0,\infty )\cup \{\infty \}\), then the two estimates have the same order of convergence to 0, namely \(\mathrm{O}\left( \left| \frac{k(n)}{n}-t\right| \right)\),
In the case when \(\left| \frac{k(n)}{n}-t\right| =\mathrm{o}\left( n^{-\beta }\right) \;(n\rightarrow \infty )\) relation (34), i.e., (25) is stronger than relation (33).
Remark 4
Another estimate for approximation of the semigroup generated by the Bernstein operators on a simplex was given by Mangino and Raşa [15] in the form:
where \(C_n=\frac{1}{n}+\frac{1}{6}d^3\sqrt{n}\sup _{{\overline{x}}\in \varDelta _d}\sqrt{B_n(\Vert {\overline{x}}-\bullet \Vert ^4,{\overline{x}})}\), \(f\in C^3(\varDelta _d)\), \(\Vert f\Vert _3=\sum \limits _{|\alpha |\le 3}\Vert D^{\alpha }f\Vert\). This estimate has also a larger domain of applicability: \(C^3(\varDelta )\). It remains to compare (35) with (25) for \(f\in C^4(\varDelta )\). Fix d and \(t>0\). Consider a sequence \((k(n))_n\), such that \(\lim _{n\rightarrow \infty }\frac{k(n)}{n}=t\). We can make the comparison in two cases.
If \(\liminf _{n\rightarrow \infty }\left| \frac{k(n)}{n}-t\right| \sqrt{n}\in (0,\infty )\cup \{\infty \}\), then, by taking into account that \(C_n=\mathrm{O}\left( \frac{1}{\sqrt{n}}\right)\), it follows that the two estimates have the same order, namely \(\mathrm{O}\left( \left| \frac{k(n)}{n}-t\right| \right)\).
In the case when \(\left| \frac{k_n}{n}-t\right| =\mathrm{o}\left( n^{-\frac{1}{2}}\right)\), estimate (25) is stronger than relation (35).
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Communicated by Markus Haase.
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Păltănea, R., Smuc, M. Quantitative results for the limiting semigroup generated by the multidimensional Bernstein operators. Semigroup Forum 102, 235–249 (2021). https://doi.org/10.1007/s00233-020-10146-x
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DOI: https://doi.org/10.1007/s00233-020-10146-x
Keywords
- Multidimensional Bernstein operators on a simplex
- Trotter’s approximation theorem
- Limiting semigroup of operators