# Generalized Jacobi–Weierstrass operators and Jacobi expansions

## Abstract

We present a realization for some K-functionals associated with Jacobi expansions in terms of generalized Jacobi–Weierstrass operators. Fractional powers of the operators as well as results concerning simultaneous approximation and Nikolskii–Stechkin type inequalities are also considered.

## 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.

### Definition 1.1

For fixed $$\gamma> 0$$ and $$t > 0$$, the generalized Jacobi–Weierstrass kernel is defined by

$$W_{t,\gamma} (x) = \sum_{n=0}^{\infty}e^{{ - t\lambda_{n}^{\gamma}}} w^{(\alpha,\beta)}_{n} R^{(\alpha,\beta)}_{n} (x),\quad x \in[- 1, 1].$$
(5)

For $$f \in X$$, the generalized Jacobi–Weierstrass (or Abel–Cartwright) operator is defined by

$$C_{t,\gamma} (f, x) = \int_{-1}^{1} \tau_{y}(f, x)W_{t,\gamma} (y)\varrho^{ \alpha,\beta}(y)\,dy,\quad x \in[- 1, 1],$$
(6)

where $$\tau_{y}(f, x)$$ is the translation given in Theorem 2.1 below.

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 2π-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]).

### Theorem 2.1

For each $$h \in[-1, 1)$$, there exists a function $$\tau_{h}: X \to X$$ with the following properties:

1. (i)

For each $$f \in X$$, one has

$$\Vert \tau_{h} f \Vert _{X } \leq \Vert f \Vert _{X}, \qquad \lim_{ h\to1-} \bigl\Vert \tau_{h}(f) - f \bigr\Vert _{X } = 0$$

and

$$\bigl\langle \tau_{h}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle = R^{(\alpha,\beta)}_{n}(h) \bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle ,\quad n \in\mathbb{N}_{0}.$$
2. (ii)

For $$f\in X$$ and $$g \in L^{1}_{\alpha,\beta}$$, the integral

$$(f * g) (x):= \int_{-1}^{1} \tau_{y}(f, x) g(y) \varrho^{ \alpha,\beta}(y)\,dy$$

exists a.e. in $$[-1.1]$$,

$$f * g = g * f,\qquad f * g \in X,\qquad \Vert f * g \Vert _{p}\leq \Vert g \Vert _{1} \Vert f \Vert _{X}$$

and

$$\bigl\langle f * g,R^{(\alpha,\beta)}_{n} \bigr\rangle = \bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle \bigl\langle g,R^{(\alpha,\beta)}_{n} \bigr\rangle ,\quad n \in\mathbb{N}_{0}.$$
(9)

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.

### Definition 2.2

Let Y be a real Banach space and $$B(Y )$$ be the Banach algebra of all bounded linear operators $$B: Y \to Y$$. A uniformly bounded family of operators $$\{T(t): t \geq0\}$$ in $$B(Y )$$ is called an equi-bounded semigroup of class $$(C_{0})$$ if

$$T(s)T(t) = T(s + t)\quad \text{for }s, t \geq 0,\qquad T(0) = I,$$
(12)

and $$\lim_{ t\to0+} \Vert f - T(t)f\Vert _{Y} = 0$$ for each $$f \in Y$$.

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).

### Theorem 2.3

Assume that Y is a Banach space, $$D(B)$$ is a dense subspace of Y, and $$B: D(B) \to Y$$ is a closed linear operator. Let $$S=\{ S_{t}: t > 0\}$$ be a strong approximation process in Y satisfying $$S_{t}(f) \in D(B)$$ for any $$f \in Y$$ and each $$t > 0$$. If there exists a constant $$\gamma_{0}$$ such that, for all $$g \in D(B)$$,

$$\lim_{ t\to0+}\biggl\Vert \frac{S_{t}(g) - g}{ t^{\gamma_{0}}} - B(g)\biggr\Vert _{Y} = 0,$$
(15)

then the strong approximation process S is saturated with order $$t^{\gamma_{0}}$$ and the trivial space is the kernel of B.

## 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.

### Theorem 3.1

For each $$\gamma > 0$$, the family of operators $$\{ C_{t, \gamma}: t > 0\}$$ is an equi-bounded semigroup of operators in X.

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.

### Theorem 3.2

If $$\gamma, t > 0$$ and $$A_{\gamma}: D(A_{\gamma}) \to X$$ is the infinitesimal generator of $$C_{t, \gamma}$$, then

$$D(A_{\gamma}) = \Psi^{\gamma}(X)\quad \textit{and} \quad{-} A_{\gamma}(f) = \Psi^{\gamma}(f)$$

for each $$f \in\Psi^{\gamma}(X)$$.

Moreover, for each $$r \in\mathbb{N}$$ and $$f \in D(A^{r}_{\gamma})$$,

$$D \bigl(A^{r}_{\gamma}\bigr) = \Psi^{r\gamma} (X)\quad \textit{and}\quad (-1)^{r}A^{r}_{\gamma}(f) = \Psi^{r\gamma} (f),$$
(18)

where $$A^{r}_{\gamma}$$ is defined as in (14).

### Proof

Since $$A_{\gamma}$$ is the infinitesimal generator of the semi-group (see (13)), $$A_{\gamma}: D(A_{\gamma}) \to X$$ is a closed operator.

If $$f \in D(A_{\gamma})$$, then

$$\bigl\langle A_{\gamma}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle = \lim_{ t\to0+}\frac{ 1}{t} \bigl( e^{{ -t\lambda_{n}^{\gamma}}} - 1 \bigr) \bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle = -\lambda_{n}^{\gamma}\bigl\langle f, R^{(\alpha,\beta)}_{n} \bigr\rangle .$$
(19)

Thus $$f\in\Psi^{\gamma}(X)$$ and

$$\Psi^{\gamma}(f) = -A_{\gamma}(f).$$

In particular, for each polynomial P, one has $$P \in D(A_{\gamma})$$ and $$\Psi^{\gamma}(P) = -A_{\gamma}(P)$$.

On the other hand, fix an integer $$j > \alpha+ 1/2$$. For $$f \in\Psi ^{\gamma}(X)$$, let $$S^{j}_{m} (f)$$ and $$S^{j}_{m}( \Psi^{\gamma}(f))$$ be the mth Cesàro means of order j of f and $$\Psi^{\gamma}(f)$$, respectively. We know that (see (11))

$$S^{j}_{m} (f) \to f, \quad m \to \infty$$

and

$$-A_{\gamma}\bigl(S^{j}_{m} (f) \bigr) = \Psi^{\gamma}\bigl(S^{j}_{m} (f) \bigr) = S^{j}_{m} \bigl(\Psi^{\gamma}(f) \bigr) \to \Psi^{\gamma}(f).$$

Since $$-A_{\gamma}$$ is a closed operator, $$f \in D(A_{\gamma})$$ and $$-A_{\gamma}(f) = \Psi^{\gamma}(f)$$.

Equations (18) can be proved by recurrence. For instance, (19) can be written as

$$\bigl\langle A^{2}_{\gamma}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle = \bigl\langle A_{\gamma}\bigl(A_{\gamma}(f) \bigr),R^{(\alpha,\beta)}_{n} \bigr\rangle =-\lambda_{n}^{\gamma}\bigl\langle A_{\gamma}(f), R^{(\alpha,\beta)}_{n} \bigr\rangle = \lambda_{n}^{2\gamma} \bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle .$$

□

### Theorem 3.3

(i) If for $$\gamma, t > 0$$, and $$f \in X$$

$$\theta_{\gamma}(f, t)= \theta_{\gamma}(f, t)_{\alpha,\beta} = \sup _{ 0< s\leq t} \bigl\Vert (I - C_{s,\gamma} ) (f) \bigr\Vert ,$$

and $$K_{\gamma}(f,t)$$ is defined by (7), then

$$\theta_{\gamma}(f, t) \approx K_{\gamma}(f, t).$$

(ii) The strong approximation process $$\{ C_{t, \gamma}; t > 0\}$$ is saturated with order t and the trivial class consists of the constant functions.

### Proof

(i) From Theorem 3.2 we know that $$-\Psi^{\gamma}$$ is the infinitesimal generator of $$\{C_{t, \gamma}\}$$ and $$D(A_{\gamma}) =\Psi^{\gamma}(X)$$. Thus, the result is a simple consequence of [17], Theorem 1.1, or [5], p. 192.

(ii) We will derive the result from Theorem 2.3, with $$B =\Psi^{\gamma}$$ and $$D(B) = D(A_{\gamma})$$. We should verify that $$C_{t, \gamma} (f) \in D(A_{\gamma})$$ for any $$f \in X$$ and each $$t > 0$$.

For any $$f\in X$$, the Fourier–Jacobi coefficients of f are bounded by $$\Vert f\Vert_{L^{1}_{(\alpha,\beta)}}$$. Taking into account (16), for every $$x\in[-1,1]$$,

\begin{aligned} &\Biggl\vert \sum_{n=1}^{\infty}\lambda_{n}^{\gamma}\exp\{ -t\lambda_{n}\} \bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle w^{(\alpha,\beta)}_{n} R^{(\alpha,\beta)}_{n} (x) \Biggr\vert \\ &\quad\leq \Vert f \Vert _{L^{1}_{(\alpha,\beta)}} \sum_{n=1}^{\infty}\lambda_{n}^{\gamma}\exp \bigl\{ -t\lambda_{n}^{\gamma}\bigr\} w^{(\alpha,\beta)}_{n} \\ &\quad\leq C \Vert f \Vert _{L^{1}_{(\alpha,\beta)}} \sum_{n=1}^{\infty}\lambda_{n}^{\gamma}\exp \bigl\{ -t\lambda_{n}^{\gamma}\bigr\} n^{2\alpha+1} < \infty. \end{aligned}

Since the series converges absolutely and uniformly, it defines a function $$g_{t}\in X$$ satisfying

$$\bigl\langle g_{t},R^{(\alpha,\beta)}_{n} \bigr\rangle = \lambda_{n}^{\gamma}\exp \bigl\{ -t\lambda_{n}^{\gamma}\bigr\} \bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle = \lambda_{n}^{\gamma}\bigl\langle C_{t,\gamma}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle ,\quad n\in\mathbb{N}.$$

By definition of the operator $$\Psi^{\gamma}$$, $$C_{t, \gamma} (f) \in \Psi^{\gamma}(X )$$ (Theorem 3.2) and

$$\Psi^{\gamma}\bigl(C_{t, \gamma} (f) \bigr) = g_{t}.$$

We have proved that $$C_{t, \gamma} (X) \in D(A_{\gamma})$$.

If $$g\in\Psi^{\gamma}(X) =D(A_{\gamma})$$, by definition of the infinitesimal generator,

$$\lim_{ t\to0+}\biggl\Vert \frac{C_{t,\gamma}(g) - g}{ t} - A_{\gamma}(g)\biggr\Vert _{Y} = 0.$$

If $$f \in\Psi^{\gamma}(X )$$ and $$A_{\gamma}(f)=-\Psi^{\gamma}(f) = 0$$, then $$\langle f,R^{(\alpha,\beta)}_{n} \rangle = 0$$ for all $$n \in \mathbb{N}$$. Therefore f is a constant.

From part (i), if $$g \in\Psi^{\gamma}(X )$$, then

$$\theta_{\gamma}(g, t) \leq C K _{\gamma}(g, t) \leq C t \bigl\Vert \Psi^{\gamma}(g) \bigr\Vert _{X }.$$

Hence, the family

$$\bigl\{ f \in X: \exists C(f) \text{ such that } \theta_{\gamma}(f, t) \leq C(f) t \bigr\}$$

contains nonconstant functions.

Now, from Theorem 2.3, we know that the strong approximation process $$\{ C_{t, \gamma}: t > 0\}$$ is saturated with order t. □

### Remark 3.4

Some characterizations of the saturation class of the strong approximation process $$\{C_{t, \gamma}: t > 0\}$$ can be given as in [2], Theorems 5.1.1 and 7.4.1, where the case $$\gamma=1$$ was considered. When $$\gamma > 0$$ is not an integer, fractional derivatives should be considered. This task would lead us far from our main topic.

### Remark 3.5

A relation similar to (i) in Theorem 3.3 is asserted in [16], p. 2885, for the discrete case and Gauss–Weierstrass type means

$$\widetilde{W}_{\Omega(n),\gamma} (f,x) =\sum_{n=0}^{\infty}e^{{ -(\Omega(k)/\Omega(n))^{\gamma}}} \bigl\langle f,R_{n}^{(\alpha,\beta )} \bigr\rangle w^{(\alpha,\beta)}_{n} R^{(\alpha,\beta)}_{n} (x),$$

with Ω varying in a specified class of functions. The proof suggested there is different from the one given here (it does not use the semi-group structure). The main argument in [16] is that some abstract Riesz means are equivalent (as approximation processes) to some Gauss–Weierstrass type means. This kind of equivalence can also be derived by using Corollary 5.4 of [9]. Anyway, the arguments of [16] and the proof given here are related because both use [15], Theorem 3.9, to obtain a uniformly bounded family of multipliers. Apart from this, other topics considered here are not connected with [16].

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 $$- A_{1}$$.

### Proposition 3.6

If $$\gamma> 0$$ and $$(- A_{1})^{\gamma}$$ is the fractional power of order γ of $$- A_{1}$$, then

$$D \bigl((- A_{1})^{\gamma},X \bigr) = \Psi^{\gamma}(X )$$

and, for each $$f \in\Psi^{\gamma}(X )$$,

$$\Psi^{\gamma}(f) = \lim_{ t\to0+} \frac{1}{t^{\gamma}} (I - C_{t, 1})^{\gamma}(f) = \lim _{ t\to0+} \frac{1}{t} \bigl( f-C_{t, \gamma} (f) \bigr).$$
(21)

### Proof

If γ is a positive integer or $$| a |< 1$$, the Taylor expansion gives

$$(1 - a)^{\gamma}= \sum_{j=0}^{\infty}(-1)^{j} \binom{\gamma}{j} a^{j}.$$

Notice that

\begin{aligned} \bigl\langle (I - C_{t, 1})^{\gamma}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle &= \sum_{k=0}^{\infty}(-1)^{k} \binom{\gamma}{k} \bigl\langle C_{kt, 1}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle \\ &= \sum_{k=0}^{\infty}(-1)^{k} \binom{\gamma}{k} \bigl\langle W_{kt},R^{(\alpha,\beta)}_{n} \bigr\rangle \bigl\langle f,R^{(\alpha,\beta )}_{n} \bigr\rangle \\ &= \bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle \sum _{k=0}^{\infty}(-1)^{k} \binom {\gamma}{k} \exp ( - kt\lambda_{n}) ) \\ &= \bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle \bigl( 1-\exp(-t\lambda_{n}) \bigr)^{\gamma}. \end{aligned}
(22)

Therefore, if $$f \in D((- A_{1})^{\gamma},X )$$, then

$$\bigl\langle (-A_{1})^{\gamma}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle = (\lambda_{n})^{\gamma}\bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle .$$

Hence $$f \in\Psi^{\gamma}(X )$$ and $$(- A_{1})^{\gamma}(f) =\Psi^{\gamma}(f)$$.

It is clear that, for each polynomial P, one has $$P \in D((- A_{1})^{\gamma},X)$$ and

$$(- A_{1})^{\gamma}(P) = \Psi^{\gamma}(P).$$

On the other hand, fix an integer $$j > \alpha + 1/2$$. For $$f \in\Psi ^{\gamma}(X )$$, let $$S^{j}_{m} (f)$$ and $$S^{j}_{m} (\Psi^{\gamma}(f))$$ be the mth Cesàro means of order j of f and $$\Psi^{\gamma}(f)$$, respectively. From (11), as in the proof of Theorem 3.2, one has $$\lim_{ m\to\infty} \Vert S^{j}_{m}(f) - f\Vert_{X } = 0$$ and

\begin{aligned} \lim_{ m\to\infty} \bigl\Vert \bigl(- A^{1} \bigr)^{\gamma}\bigl(S^{j}_{m} (f) \bigr) - \Psi^{\gamma}(f) \bigr\Vert _{X }& = \lim_{ m\to\infty} \bigl\Vert \Psi^{\gamma}\bigl(S^{j}_{m} (f) \bigr) - \Psi^{\gamma}(f) \bigr\Vert _{X } \\ &= \lim_{ m\to\infty} \bigl\Vert S^{j}_{m} \bigl( \Psi^{\gamma}(f) \bigr) - \Psi^{\gamma}(f) \bigr\Vert _{X} = 0. \end{aligned}

It was proved in [19], Theorem 4, that $$D((- A_{1})^{\gamma},X)$$ is dense in X and $$(- A_{1})^{\gamma}$$ is a closed operator. Hence $$f \in D((- A_{1})^{\gamma},X)$$ and $$(- A_{1})^{\gamma}(f) = \Psi^{\gamma}(f)$$.

The last equality in (21) was proved in Theorem 3.2, because $$\Psi^{\gamma}$$ is the infinitesimal generator of $$\{ C_{t, \gamma}, t > 0\}$$. □

### Theorem 3.7

For fixed $$\gamma > 0$$, one has

$$K_{\gamma}\bigl(f, t^{\gamma}\bigr) \approx \sup _{ 0< s\leq t} \bigl\Vert (I - C_{s,1})^{\gamma}(f) \bigr\Vert _{X } \approx \theta_{\gamma}\bigl(f, t^{\gamma}\bigr)$$

for each $$f \in X$$ and $$t > 0$$.

### Proof

From Theorems 3.1 and 3.2 we know that the family $$\{C_{t, 1}, t \geq0\}$$ is a semi-group of operators of class $$(C_{0})$$ with the infinitesimal generator $$A_{1} =-\Psi^{1}$$. From Theorem 1.1 of [17], we know that, for all $$f \in X$$ and $$t > 0$$,

$$\inf_{ g \in D((- A_{1})^{\gamma},X )} \bigl( \Vert f - g \Vert _{X } + t^{\gamma}\bigl\Vert (- A_{1})^{\gamma}(g) \bigr\Vert _{X } \bigr) \approx\sup_{ 0< s\leq t} \bigl\Vert (I - C_{s,1})^{\gamma}(f) \bigr\Vert _{X },$$

where $$(- A_{1})^{\gamma}$$ is given as in (20). But it was verified in Proposition 3.6 that $$\Psi^{\gamma}(X ) = D((- A_{1})^{\gamma},X )$$ and $$(- A_{1})^{\gamma}(g) = \Psi^{\gamma}(g)$$ for each $$g \in\Psi^{\gamma}(X )$$.

The equivalence with $$\theta_{\gamma}(f, t^{\gamma})$$ follows from Theorem 3.3. □

### Remark 3.8

When γ is an integer, Theorem 3.7 is similar to the Main Theorem in [18], p. 390, but the authors assumed that the operators are positive (plus other conditions).

### Remark 3.9

The results of Theorem 3.7 allow us to obtain equivalent relations between fractional powers $$(I - C_{s,1})^{\gamma}$$ and some Riesz means as in Theorem 5.1 of [9].

Some result concerning simultaneous approximation can be derived from the ones given above.

### Theorem 3.10

If $$\gamma,\sigma$$, and t are positive real numbers and $$f \in \Psi^{\sigma}(X )$$, then

\begin{aligned} &C_{t, \gamma} (f), (I - C_{t, 1})^{\gamma}(f) \in \Psi^{\sigma}(X ), \\ &\bigl\Vert \Psi^{\sigma}(f) - \Psi^{\sigma}\bigl(C_{t, \gamma} (f) \bigr) \bigr\Vert _{X } \leq C \theta_{\gamma}\bigl( \Psi^{\sigma}(f), t \bigr) \end{aligned}

and

$$\bigl\Vert \Psi^{\sigma}\bigl((I - C_{t, 1})^{\gamma}(f) \bigr) \bigr\Vert _{X } \leq C \theta_{\gamma}\bigl( \Psi^{\sigma}(f), t^{\gamma}\bigr),$$

where the constant C is independent of f and t.

### Proof

If $$f \in\Psi^{\sigma}(X )$$ and $$n \in\mathbb{N}_{0}$$, from (17) we obtain

\begin{aligned} \bigl\langle C_{t, \gamma} \bigl( \Psi^{\sigma}(f) \bigr),R^{(\alpha,\beta)}_{n} \bigr\rangle &= \exp \bigl(- t \lambda_{n}^{\gamma}\bigr) \bigl\langle \Psi^{\sigma}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle \\ &= \lambda_{n}^{\sigma}\exp \bigl(- t\lambda_{n}^{\gamma}\bigr) \bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle = \lambda_{n}^{\sigma}\bigl\langle C_{t,\gamma}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle \end{aligned}

and from (22) one has

\begin{aligned} \bigl\langle (I - C_{t, 1})^{\gamma}\bigl( \Psi^{\sigma}(f) \bigr),R^{(\alpha,\beta)}_{n} \bigr\rangle &= \bigl( 1- \exp(- t \lambda_{n}) \bigr)^{\gamma}\bigl\langle \Psi^{\sigma}(f);R^{(\alpha,\beta)}_{n} \bigr\rangle \\ &= \lambda_{n}^{\sigma}\bigl( 1 - \exp(- t\lambda_{n}) \bigr)^{\gamma}\bigl\langle f,R^{(\alpha,\beta)}_{n} \bigr\rangle = \lambda_{n}^{\sigma}\bigl\langle (I - C_{t, 1})^{\gamma}(f),R^{(\alpha,\beta)}_{n} \bigr\rangle . \end{aligned}

Therefore $$C_{t, \gamma} (f), (I - C_{t, 1})^{\gamma}(f) \in \Psi^{\sigma}(X)$$,

$$\Psi^{\sigma}\bigl(C_{t, \gamma} (f) \bigr) = C_{t,\gamma} \bigl( \Psi^{\sigma}(f) \bigr) \quad\text{and} \quad\Psi^{\sigma}\bigl((I - C_{t, 1})^{\gamma}(f) \bigr) = (I - C_{t, 1})^{\gamma}\bigl( \Psi^{\sigma}(f) \bigr).$$

Now, from Theorem 3.3 one has

$$\bigl\Vert \Psi^{\sigma}(f) - \Psi^{\sigma}(C_{t, \gamma}) \bigr\Vert _{X} = \bigl\Vert (I - C_{t, \gamma} ) \bigl( \Psi^{\sigma}(f) \bigr) \bigr\Vert _{X } \leq C \theta_{\gamma}\bigl( \Psi^{\sigma}(f), t \bigr),$$

and using Theorem 3.7 we obtain

$$\bigl\Vert \Psi^{\sigma}\bigl((I - C_{t, 1})^{\gamma}(f) \bigr) \bigr\Vert _{X } = \bigl\Vert (I - C_{t, 1})^{\gamma}\bigl( \Psi^{\sigma}(f) \bigr) \bigr\Vert _{X } \leq C \theta_{\gamma} \bigl( \Psi^{\sigma}(f), t^{\gamma}\bigr).$$

□

## 4 A Nikolskii–Stechkin type inequality

### Theorem 4.1

For each $$r \in\mathbb{N}$$, there exists a constant C, depending upon r, such that, for every $$\lambda\geq1$$ and for each polynomial $$P \in\mathbb{P}_{\xi(\lambda)}$$,

$$\bigl\Vert \Psi^{r}(P) \bigr\Vert _{X} \leq C \lambda^{r} \sup_{ 0< h\leq1/\lambda} \bigl\Vert (I - C_{h,1})^{r}(P) \bigr\Vert _{X },$$

where

$$\xi(\lambda) = \max \bigl\{ k \in\mathbb{N}_{0}: k(k + \alpha+\beta + 1) < \lambda \bigr\} .$$

### Proof

In this proof the infinitesimal generator of $$\{C_{t, 1}: t > 0\}$$ is denoted by A.

From the proof of Lemma 1 in [12] we know that, given $$r \in\mathbb{N}$$, there exists a constant $$C_{1} = C(r)$$ such that, for each $$f \in X$$ and $$t > 0$$, there is $$g_{t} \in D(A^{r+1})$$ satisfying

\begin{aligned} & \Vert f - g_{t} \Vert _{X } \leq\sup _{ 0< h\leq t} \bigl\Vert (I - C_{h,1})^{r}f \bigr\Vert _{X }, \end{aligned}
(23)
\begin{aligned} & \bigl\Vert A^{r+1}(g_{t}) \bigr\Vert _{X } \leq C_{1} \frac{1}{t^{r+1}} \sup_{0< h\leq t} \bigl\Vert (I - C_{h,1})^{r}f \bigr\Vert _{X } \end{aligned}
(24)

and

$$\bigl\Vert (- A)^{r}(g_{t}) \bigr\Vert _{X } \leq C_{1} \frac{1}{ t^{r}} \sup_{ 0< h\leq t} \bigl\Vert (I - C_{h,1})^{r}f \bigr\Vert _{X }.$$
(25)

As in [9], for $$\lambda> 0$$ and $$f \in X$$, consider the best approximation

$$E_{\lambda}(f) = \inf \bigl\{ \Vert f - P \Vert _{X }: P \in \mathbb{P}_{\xi(\lambda)} \bigr\} .$$

It was proved there (Theorem 6.1) that there exists a constant $$C_{2} = C(r, \alpha, \beta)$$ such that, for $$\lambda> 0$$ and $$f \in X$$,

$$E_{\lambda}(f) \leq C_{2} K_{r+1} \bigl(f, \lambda^{- r- 1} \bigr),$$
(26)

and (Theorem 3.2) for each $$Q \in\mathbb{P}_{\xi(\lambda)}$$,

$$\bigl\Vert \Psi^{r}(Q) \bigr\Vert _{ X } \leq C_{2} \lambda^{r} \Vert Q \Vert _{ X }.$$
(27)

Now, fix $$\lambda > 0$$ and $$P \in\mathbb{P}_{\xi(\lambda)}$$. Let $$g_{t} \in D(A^{r+1}) = \Psi^{r+1}(X)$$ (see (18)) be given as (23)–(25) with $$t = 1/\lambda$$ and $$f = P$$.

For $$\varepsilon > 0$$ and $$k \in\mathbb{N}_{0}$$, choose

$$q(g_{t}, k) \in\mathbb{P}_{\xi(2^{k}\lambda)}$$
(28)

such that

$$\bigl\Vert g_{t} - q(g_{t}, k)) \bigr\Vert _{X} \leq (1 + \varepsilon)E_{2^{k}\lambda} (g_{t}).$$
(29)

From (26), (18), and (24) we know that

\begin{aligned} \bigl\Vert g_{t} - q(g_{t}, k))\bigr\Vert _{X } &\leq C_{2}(1 + \varepsilon)K_{r+1} \bigl(g_{t}, \bigl(2^{k}\lambda \bigr)^{- r- 1} \bigr) \\ &\leq\frac{C_{2}(1 + \varepsilon)}{ (2^{k}\lambda)^{r+1}} \bigl\Vert \Psi ^{r+1}(g_{t}) \bigr\Vert _{X } \\ &= \frac{C_{2}(1 + \varepsilon)}{ (2^{k}\lambda)^{r+1}} \bigl\Vert A^{r+1}(g_{t}) \bigr\Vert _{X} \\ &\leq\frac{C_{1}C_{2}(1 + \varepsilon)}{ (2^{k}\lambda)^{r+1}}\frac{1}{t^{r+1}} \sup_{ 0< h\leq t} \bigl\Vert (I - C_{h,1})^{r}P \bigr\Vert _{X } \\ &= \frac{C_{1}C_{2}(1 + \varepsilon)}{(2^{k})^{r+1}}\sup_{ 0< h\leq1/\lambda} \bigl\Vert (I - C_{h,1})^{r}P \bigr\Vert _{X }. \end{aligned}

On the other hand, from the identity

$$q(g_{t}, 0) - g_{t} = \sum_{k=0}^{\infty}\bigl(q(g_{t}, k) - q(g_{t}, k + 1) \bigr),$$

(28), (27), (29), and (26), one has

\begin{aligned} \bigl\Vert \Psi^{r} \bigl(q(g_{t}, 0) - g_{t} \bigr) \bigr\Vert _{X } &\leq\sum_{k=0}^{\infty}\bigl\Vert \Psi^{r} \bigl(q(g_{t}, k) - q(g_{t}, k +1) \bigr) \bigr\Vert _{X } \\ &\leq C_{2}\sum_{k=0}^{\infty}\bigl(2^{k+1}\lambda \bigr)^{r} \bigl\Vert q(g_{t}, k) - q(g_{t}, k + 1) \bigr\Vert _{X } \\ &\leq C_{2} \sum_{k=0}^{\infty}\bigl(2^{k+1}\lambda \bigr)^{r} \bigl( \bigl\Vert q(g_{t}, k) - g_{t} \bigr\Vert _{X } + \bigl\Vert g_{t} - q(g_{t}, k + 1) \bigr\Vert _{X } \bigr) \\ &\leq2C_{1}C_{2}^{2} (1 + \varepsilon) \sup _{ 0< h\leq1\lambda} \bigl\Vert (I - C_{h,1})^{r}P \bigr\Vert _{X } \sum_{k=0}^{\infty}\bigl(2^{k+1}\lambda \bigr)^{r} \frac{1}{(2^{k})^{r+1}} \\ &= 2^{r+1}C_{1}C_{2}^{2} (1 + \varepsilon) \lambda^{r} \sup_{ 0< h\leq1\lambda} \bigl\Vert (I - C_{h,1})^{r}P \bigr\Vert _{X } \sum _{k=0}^{\infty}\frac{1}{2^{k}} \\ &= C_{3}(1 + \varepsilon)\lambda^{r} \sup _{ 0< h\leq1/\lambda} \bigl\Vert (I - C_{h,1})^{r}P \bigr\Vert _{X }. \end{aligned}

We also need the inequality (see (18) and (25))

\begin{aligned} \bigl\Vert \Psi^{r}(g_{t}) \bigr\Vert _{X } &= \bigl\Vert A^{r}(g_{t}) \bigr\Vert _{X } \leq C_{1} \frac{1}{ t^{r}}\sup_{ 0< h\leq t} \bigl\Vert (I - C_{h,1})^{r}P \bigr\Vert _{X } \\ &= C_{1}\lambda^{r} \sup_{ 0< h\leq1/\lambda} \bigl\Vert (I - C_{h,1})^{r}P \bigr\Vert _{X }. \end{aligned}

From the inequalities given above, for $$P \in\mathbb{P}_{\xi(\lambda )}$$, we obtain

\begin{aligned} \bigl\Vert \Psi^{r}(P) \bigr\Vert _{X } &\leq \bigl\Vert \Psi^{r} \bigl(P - q(g_{t}, 0) \bigr) \bigr\Vert _{X} + \bigl\Vert \Psi^{r} \bigl(q(g_{t}, 0) \bigr) \bigr\Vert _{X } \\ &\leq C_{2}\lambda^{r} \bigl\Vert P - q(g_{t}, 0) \bigr\Vert _{X } + \bigl\Vert \Psi ^{r}(g_{t}) \bigr\Vert _{ X } + \bigl\Vert \Psi^{r} \bigl(g_{t} - q(g_{t}, 0) \bigr) \bigr\Vert _{X } \\ &\leq C_{1}\lambda^{r} ( \Vert P - g_{t} \Vert _{X } + \bigl\Vert g_{t} - q(g_{t}, 0) \bigr\Vert _{X_{\alpha,\beta}} + C_{4}\lambda^{r} \sup _{ 0< h\leq1/\lambda} \bigl\Vert (I - C_{h,1})^{r}P \bigr\Vert _{X } \\ &\leq C_{5} \lambda^{r} \sup_{ 0< h\leq1/\lambda} \bigl\Vert (I - C_{h,1})^{r}P \bigr\Vert _{X }. \end{aligned}

□

### Remark 4.2

The problem of obtaining a Nikolskii–Stechkin inequality for fractional derivatives is open.

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## Acknowledgements

The authors would like to thank the referees for their careful reading of the manuscript and for their helpful comments.

## Funding

The coauthors Adell, J.A. and Quesada, J.M. are partially supported by Research Project MTM2015-67006-P.

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Jorge Bustamante, Juan J. Merino and José M. Quesada contributed equally to this work.

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Adell, J.A., Bustamante, J., Merino, J.J. et al. Generalized Jacobi–Weierstrass operators and Jacobi expansions. J Inequal Appl 2018, 153 (2018). https://doi.org/10.1186/s13660-018-1747-2