1 Introduction
In this note, we work with two fixed real parameters
α and
β satisfying
\(\alpha \geq \beta\geq-1/2\). We use the following notations:
$$ \varrho^{ \alpha,\beta}(x) = (1 - x)^{\alpha}(1 + x)^{\beta}, \quad x \in(- 1, 1), $$
(1)
and, for
\(1 \leq p < \infty\),
$$L^{p}_{(\alpha,\beta)} = \biggl\{ f:[-1,1]\to\mathbb{R}: \Vert f \Vert _{p} = \biggl( \int_{-1}^{1}\bigl| f(x)\bigr|^{p} \varrho^{ \alpha,\beta}(x)\,dx \biggr)^{1/p} < \infty \biggr\} . $$
Moreover, for each
\(n\in\mathbb{N}_{0}\),
\(\mathbb{P}_{n}\) is the family of all algebraic polynomials of degree not greater than
n,
$$ w^{\alpha,\beta}_{n} = \frac{(2n + \alpha +\beta + 1)\Gamma(n + \alpha+\beta+ 1)\Gamma(n + \alpha + 1)}{\Gamma(n +\beta + 1) \Gamma(n + 1)(\Gamma(\alpha+ 1))^{2}} $$
(2)
(Γ stands for the gamma function) and
$$ \lambda_{n}=n(n+\alpha+\beta+1). $$
(3)
Since α and β are fixed, we set X for one of the spaces \(C[-1, 1]\) or \(L^{p}_{(\alpha, \beta)}\).
For
\(n \in\mathbb{N}\), the Jacobi polynomial
\(R^{(\alpha,\beta)}_{n}\) is the unique polynomial of degree
n which satisfies
$$R^{(\alpha,\beta)}_{n} (1) = 1 \quad\text{and}\quad \int_{-1}^{1} Q_{n- 1}(x)R^{(\alpha,\beta)}_{n} (x) \varrho^{ \alpha,\beta }(x)\,dx= 0 $$
for all
\(Q_{n-1} \in\mathbb{P}_{n- 1}\). We also take
\(R^{(\alpha,\beta)}_{0} (x) = 1\).
For
\(f \in X \), the Fourier–Jacobi coefficients are defined by
$$\bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle = \int_{-1}^{1} f(x)R^{(\alpha,\beta)}_{n} (x) \varrho^{ \alpha,\beta}(x)\,dx,\quad n \in\mathbb{N}_{0}, $$
and the associated expansion is
$$ f(x)\sim\sum_{n=0}^{\infty}\bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle w^{(\alpha,\beta)}_{n} R^{(\alpha,\beta)}_{n} (x). $$
(4)
It is known that each
\(f \in L^{1}_{ (\alpha,\beta)}\) is completely determined a.e. by its Fourier–Jacobi coefficients.
Of course the kernel \(W_{t, \gamma}\) and the operator \(C_{t, \gamma}\) also depend on α and β but, for simplicity, we omit these indexes. The (classical) Jacobi–Weierstrass operators correspond to \(\gamma= 1\).
The generalized Jacobi–Weierstrass operators have been studied in different papers, but only for parameters satisfying
\(0 < \gamma\leq 1\). This restriction was considered because in such a case the kernels
\(W_{t, \gamma}\) are positive and the family
\(\{C_{t, \gamma}\}\) can be considered as formed by positive operators (see [
2,
3], [
7], pp. 96–97) and/or as a semigroup of contractions (see [
2], pp. 49–52, and [
18]). For
\(\gamma > 1\), one cannot expect the positivity of
\(W_{t, \gamma}\). For instance, it is known that the analogous generalized Weierstrass kernels for trigonometric expansion are not positive when
\(\gamma > 1\) (see [
6], p. 263).
In this paper we will prove that the operators \(C_{t, \gamma}\) can be used as a realization of some K-functionals which usually appear in some approximation problems related to Jacobi expansions.
For fixed real
\(\gamma> 0\), let
\(\Phi^{\gamma}(X )\) denote the family of all
\(f \in X \) for which there exists
\(\Psi^{\gamma}(f) \in X \) satisfying
$$\Psi^{\gamma}(f) (x) \sim\sum_{n=0}^{\infty}\lambda_{n}^{\gamma}\bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle w^{(\alpha,\beta)}_{n} R^{(\alpha,\beta)}_{n} (x). $$
The associated
K-functional is defined by
$$ K_{\gamma}(f, t)=K_{\gamma}(f, t)_{\alpha,\beta} = \inf_{ g \in \Psi^{\gamma}(X )} \bigl\{ \Vert f - g \Vert _{X} + t \bigl\Vert \Psi^{\gamma}(g) \bigr\Vert _{X } \bigr\} $$
(7)
for
\(f \in X \) and
\(t > 0\). For different realizations of these
K-functionals, see [
8], Theorem 7.1, and [
10], Lemma 2.3. We will not use the characterization of these
K-functionals in terms of moduli of smoothness. We will show that, for any
\(\gamma > 0\),
$$\sup_{ 0< s\leq t} \bigl\Vert (I - C_{s,\gamma}) (f) \bigr\Vert _{X} \approx K_{\gamma}(f, t). $$
The notation
\(A(f, t) \approx B(f, t)\) means that there exists a positive constant
C such that
\(C^{-1}A(f, t) \leq B(f, t) \leq CA(f, t)\) with
C independent of
f and
t.
Following [
19], for
\(\gamma > 0\), define
$$ (I - C_{t, 1})^{\gamma}= \sum _{j=0}^{\infty}(- 1)^{j} \binom{\gamma}{j} C_{jt, 1}, $$
(8)
where
$$\binom{\gamma}{0} = 1 \quad\text{and} \quad\binom{\gamma}{j} = \prod _{k=1}^{j} \frac{\gamma- k + 1}{ k} \quad\text{for } j \in \mathbb{N}. $$
For these operators, we will show the relations
$$K_{\gamma}\bigl(f, t^{\gamma}\bigr) \approx\sup _{ 0< s\leq t} \bigl\Vert (I - C_{s,1})^{\gamma}(f) \bigr\Vert _{X } \approx\sup_{ 0< s\leq t^{\gamma}} \bigl\Vert (I - C_{s,\gamma}) (f) \bigr\Vert _{X }. $$
It is known that, if
\(Q_{n}\) is a trigonometric polynomial of degree not greater than
n and
\(r \in\mathbb{N}\), then
$$\bigl\Vert Q^{(r)}_{n} \bigr\Vert _{p} \leq \biggl(\frac{ n}{ 2 \sin(nh)} \biggr)^{r} \bigl\Vert (1 - T_{h})^{r}(Q_{n}) \bigr\Vert _{p},\quad h \in(0, \pi/n), $$
where
\(\Vert\cdot\Vert_{p}\) denotes the
\(L^{p}\)-norm of 2
π-periodic functions and
\(T_{h}\) is the translation operator. That is,
\(T_{h}Q(x) = Q(x + h)\). These inequalities are due to Nikolskii [
11] and Stechkin [
13]. For similar inequalities for algebraic polynomials, see [
4] and the references given there. Here we will verify an analogous inequality by considering the operators
\(\Psi^{r}\) and the linear combination of the Jacobi–Weierstrass operators
\(C_{t,1}\).
In Sect.
2 we collect some definitions and results which will be needed later. The main results are given in Sect.
3, where the result concerning simultaneous approximation is also included. Finally, in Sect.
4 we present a Nikolskii–Stechkin type inequality.
2 Auxiliary results
We need a convolution structure due to Askey and Wainger (see [
1]).
For
\(j > \alpha + 1/2\) and
\(f \in X \), let
$$S^{j}_{m} (f) = \sum_{k=0}^{m} \frac{A^{j}_{ m-k}}{ A^{j}_{m}} \bigl\langle f,R^{\alpha,\beta)}_{k} \bigr\rangle w^{\alpha,\beta}_{k} R^{(\alpha,\beta)}_{k} (x),\quad A^{j}_{m} = \binom{m+j}{m}, $$
be the
mth Cesàro means of order
j. It is known that there exists a constant
C such that
$$ \bigl\Vert S^{j}_{m} \bigr\Vert \leq C, $$
(10)
and, for each
\(f \in X \), one has ([
2], Corollary 3.3.3, or [
7], Theorem A)
$$ \lim_{ m\to\infty} \bigl\Vert f - S^{j}_{m} (f) \bigr\Vert _{X } = 0. $$
(11)
We need some classical results related to Banach spaces.
Let
Y,
\(B(Y )\) and
\(\{T(t): t > 0\}\) be an equi-bounded semigroup as in Definition
2.2. Let
\(D(Q)\) be the family of all
\(g \in Y\), for which there exists
\(Q(g) \in Y\) such that
$$ Q(g) = \lim_{ t\to0+} \frac{1}{t} \bigl[T(t) - I \bigr]g $$
(13)
(the limit is considered in the norm of
Y). The operator
\(Q: D(Q) \to Y\) is called the infinitesimal generator of the semigroup
\(\{T(t): t \geq0\}\). It is known that
Q is a closed linear operator and
\(D(Q) \) is dense in
Y. For properties of semigroups of operators, see [
5].
For
\(r \in\mathbb{N}\), set
$$D \bigl(Q^{r+1} \bigr) = \bigl\{ f \in Y: f \in D \bigl(Q^{r} \bigr) \text{ and } Q^{r}(f) \in D(Q) \bigr\} $$
and, for
\(f \in D(Q^{r+1})\),
$$ Q^{r+1}(f) = Q \bigl(Q^{r}(f) \bigr). $$
(14)
A family of operators
\(S = \{S_{t},: t > 0\}\),
\(S_{t} \in B(Y )\) for each
\(t > 0\) is called a (commutative) strong approximation process for
Y if, for all
\(f \in Y\) and
\(s, t > 0\),
$$S_{s} \bigl(S_{t}(f) \bigr) = S_{t} \bigl(S_{s}(f) \bigr),\qquad \bigl\Vert S_{t}(f) \bigr\Vert _{Y} \leq \Lambda \Vert f \Vert _{Y} \quad\text{and} \quad\lim _{ t\to0+} \bigl\Vert f -S_{t}(f) \bigr\Vert _{Y} = 0, $$
where Λ is a constant. In such a case, we set
$$\theta_{S}(f, t) =\sup_{ 0< s\leq t} \bigl\Vert f - S_{s}(f) \bigr\Vert _{Y}. $$
Let
\(\phi: [0,1)\to \mathbb{R}^{+}\) be a positive increasing function,
\(\phi(t)\to0\) as
\(t \to0\), and
\(Y_{0}\) be a subspace of
Y. We say that
S is saturated with order
ϕ and with trivial subspace
\(Y_{0}\) if every
\(f \in Y\) satisfying
$$\lim_{ t\to0+}\frac{\theta_{S} (f, t)}{\phi(t)} = 0 $$
belongs to
\(Y_{0}\) and there exists
\(f \in Y\setminus Y_{0}\) satisfying
\(\theta_{S}(f, t) \leq C(f)\phi(t)\). The following assertion is known (for instance, see [
2], Theorem 2.4.2).
3 The operators \(C_{t, \gamma}\) as a semigroup
In fact, it is known that, for
\(x \in(-1,1)\),
\(| R^{(\alpha,\beta)}_{n} (x) |< 1\), [
14], pp. 163–164, and there exists a constant
C such that, for each
\(n\in\mathbb{N}_{0}\),
$$ w^{(\alpha,\beta)}_{n} \leq Cn^{2\alpha+1}. $$
(16)
These relations can be used to prove that the series in (
5) converges absolutely and uniformly in
\([-1, 1]\). Thus
\(W_{t, \gamma} \in L^{1}_{(\alpha,\beta)}\) and, for each
\(f \in L^{1}_{ (\alpha,\beta)}\), the series
\(C_{t, \gamma} (f)\) converges absolutely and uniformly in
\([-1, 1]\). Moreover,
$$C_{t, \gamma} (f, x) = (W_{t, \gamma}* f) (x)=\sum _{n=0}^{\infty}e^{{ -t\lambda_{n}^{\gamma}}} \bigl\langle f,R_{n}^{(\alpha,\beta)} \bigr\rangle w_{n}^{(\alpha,\beta)} R_{n}^{(\alpha,\beta)}(x). $$
For these assertions, see [
2], p. 30.
Our first result seems to be known. For convenience of the reader, we include a proof.
Proof. It follows from Theorem 3.9 of [
15] that the family of operators
\(\{ C_{t, \gamma}: t > 0\}\) is uniformly bounded.
Condition (
12) is derived from the properties of the convolution. In fact, it follows from (
9) that, for each
\(f \in X\) and
\(k \in\mathbb{N}_{0}\),
$$\begin{aligned} \bigl\langle C_{s+t} (f),R^{(\alpha,\beta)}_{k} \bigr\rangle &= e^{{ -(s + t)\lambda _{n}^{\gamma}}} \bigl\langle f,R^{(\alpha,\beta)}_{k} \bigr\rangle = e^{{ -s\lambda_{n}^{\gamma}}} \bigl\langle C_{t, \gamma} (f),R^{(\alpha,\beta)}_{k} \bigr\rangle \\ &= \bigl\langle C_{s,\gamma} \bigl(C_{t, \gamma} (f) \bigr),R^{(\alpha,\beta)}_{k} \bigr\rangle \end{aligned}$$
and this implies
\(C_{s+t}(f) = (C_{s,\gamma} \circ C_{t, \gamma}) (f)\).
Finally, for each
\(k \in\mathbb{N}_{0}\),
$$ C_{t, \gamma} \bigl(R^{(\alpha,\beta)}_{k} \bigr) (x) = e^{{ -t\lambda_{n}^{\gamma}}} R^{(\alpha,\beta)}_{k} (x). $$
(17)
Hence
$$\lim_{t\to0+} \bigl\Vert R^{(\alpha,\beta)}_{k} - C_{t, \gamma} \bigl(R^{(\alpha,\beta)}_{k} \bigr) \bigr\Vert _{X} = 0. $$
Since the operators
\(C_{t, \gamma}\) are linear and uniformly bounded and the polynomials are dense in
X, the last equation holds for every
\(f \in X\).
Taking into account Theorem
3.1, we denote by
\(A_{\gamma}\) the infinitesimal generator of
\(C_{t, \gamma}\) and by
\(D(A_{\gamma})=D(A_{\gamma}(\alpha, \beta))\) the domain of
\(A_{\gamma}\). In the next result we give a description of the infinitesimal generator.
The arguments used in the proof of Theorem
3.2 can be used to derive similar relations concerning the fractional powers of the Jacobi–Weierstrass operators
\(\{C_{t, 1}\}\).
Recall that
\(A_{1}: D(A_{1}) \to X\) is the infinitesimal generator of
\(\{ C_{t, 1}, t > 0\}\). For
\(\gamma> 0\), let
\(D((-A_{1})^{\gamma},X )\) be the family of all
\(f \in X \), for which there exists an element
\((-A_{1})^{\gamma}(f) \in X \) satisfying
$$ \lim_{ t\to0+}\biggl\Vert (-A_{1})^{\gamma}(f) -\frac{1}{t^{\gamma}} (I - C_{t, 1})^{\gamma}(f)\biggr\Vert _{X } = 0, $$
(20)
where
\((I - C_{t, 1})^{\gamma}(f)\) is defined by (
8). This induces a map
$$\bigl(- A^{1} \bigr)^{\gamma}: D \bigl( \bigl(-A^{1} \bigr)^{\gamma},X \bigr) \to X $$
which is called the fractional power of order
γ of
\(- A_{1}\).
Some result concerning simultaneous approximation can be derived from the ones given above.
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