⁰⁰footnotetext: ; Institute of Mathematics, University of Rzeszów, Poland

Some exact constants for bilateral approximations in quasi-normed groups

\fnmOleh \surLopushansky

Abstract

We establish inverse and direct theorems on best approximations in quasi-normed Abelian groups in the form of bilateral Bernstein-Jackson inequalities with exact constants. Using integral representations for quasi-norms of functions $f$ in Lebesgue’s spaces by decreasing rearrangements $f^{*}$ with the help of approximation $E$ -functionals, error estimates are found. Examples of numerical calculations, spectral approximations of self-adjoint operators obtained by obtained estimates are given.

keywords:

Bernstein-Jackson inequalities, quasi-normed Abelian groups, exact approximation constants

pacs:

[

MSC Classification]42B35, 41A44, 41A17,46B70

1 Introduction and main results

This article is devoted to the proof of a bilateral version of Bernstein-Jackson inequalities for quasi-normed Abelian groups. The case of similar inequalities on Gaussian Hilbert spaces of random variables was analyzed in the previous publication Lopushansky2023 . In the studied case, we use the approximation scale

\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}^{1/\vartheta},\quad s+1=1/\vartheta,\quad\tau=\vartheta q

of compatible couple of quasi-normed Abelian groups $(\mathfrak{A}_{\imath},{|\cdot|_{\imath}})$ with ${0<\vartheta<1}$ and ${0<q\leq\infty}$ , defined by the Lions-Peetre method of real interpolation. It should be noted that in partial cases this scale coincides with the known scale of Besov spaces (see (bergh76, , Thm 7.2.4), DeVore1988 ; DeVore1998 ,(Nikolski75, , Thm 2.5.4)).

We also analyze the problem of accurate error estimates in quasi-normed spaces $L^{p}_{\mu}=L^{p}_{\mu}(\mathfrak{A})$ $(0\leq p\leq\infty)$ of $\mathfrak{A}$ -valued functions ${f\colon X\to\mathfrak{A}}$ on a measure space $(X,\mu)$ . Using the decreasing rearrangement method (see e.g. Burchard ; Grafakos2014 ), we reduce this problem to the simpler case for uniquely defined decreasing functions $f^{*}$ on $(0,\infty)$ .

The scale of quasi-normed spaces $\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ endowed with the quasinorm ${\|\cdot\|_{\mathcal{B}_{\tau}^{s}}}$ is determined by the best approximation $E$ -functional

E(t,a\mathchar 24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})=\inf\left\{|a-a_{0}|_{1}\colon|a_{0}|_{0}<t\right\},\quad a\in{\mathfrak{A}_{1}}

which fully characterizes the accuracy of error estimates (see e.g. bergh76 ; Maligranda1991 ; PeetreSparr1972 ). In Theorem 1 we establish the bilateral version of Bernstein-Jackson inequalities for quasi-normed groups with the exact constants $c_{s,\tau}$ depend only on basis parameters $\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$

\displaystyle t^{s}E(t,a)\leq{c_{s,\tau}}|a|_{\mathcal{B}_{\tau}^{s}}\leq 2^{1/2}|a|^{s}_{0}|a|_{1},\quad c_{s,\tau}=\bigg{[}\frac{s}{\tau(s+1)^{2}}\bigg{]}^{1/\tau}

for all $a\in\mathfrak{A}_{0}\bigcap\mathfrak{A}_{1}$ and $t,\tau>0$ . The left inequality allows the unique extension to the whole approximation spaces $\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ in form of the Jackson-type inequality

E(t,a)\leq{c_{s,\tau}t^{-s}}\,|a|_{\mathcal{B}_{\tau}^{s}}\quad\text{for all}\quad{a\in\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})}.

In Section 3 these results are applied to the quasi-normed Abelian groups $L^{p}_{\mu}=L^{p}_{\mu}(\mathfrak{A})$ with $0\leq p\leq\infty$ and a positive Radon measure $\mu$ on a measure space $X$ of measurable functions ${f\colon X\to\mathfrak{A}}$ endowed with the $\kappa$ -norm

\|f\|_{p}=\left\{\begin{array}[]{ll}\displaystyle\Big{(}\int|f(x)|^{p}\,\mu(dx)\Big{)}^{1/p}&\hbox{if }0<p<\infty\\ \mathop{\rm ess\,sup}_{x\in X}|f(x)|&\hbox{if }p=\infty\\[4.30554pt] \mu\left(\operatorname{\operatorname{supp}}f\right)&\hbox{if }p=0,\end{array}\right.

where $\operatorname{\operatorname{supp}}f\subset X$ is a measurable subset such that $f\mid_{X\setminus\operatorname{\operatorname{supp}}f}=0$ and ${f\neq 0}$ almost everywhere on $\operatorname{\operatorname{supp}}f$ with respect to the measure $\mu$ . In Theorem 7 the equality

f^{*}(t)=E\big{(}t,f\mathchar 24635\relax\;L_{\mu}^{0},L_{\mu}^{\infty}\big{)},\quad f\in{L_{\mu}^{\infty}},

where $f^{*}$ is the decreasing rearrangement of $\mathfrak{A}$ -valued functions $f\in L^{p}_{\mu}$ and, as a consequence, the following equalities

\|f\|_{\tau/s}=\left\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{s}f^{*}(t)\right]^{\tau}\frac{dt}{t}\right)^{1/\tau}&\hbox{if }\tau<\infty\\[8.61108pt] \sup\limits_{0<t<\infty}t^{s}f^{*}(t)&\hbox{if }\tau=\infty,\end{array}\right.

are established. We also compute exact constants in the Jackson-type inequalities

f^{*}(t)\leq t^{s}2^{(s+1)/2}c_{s,\tau}\|f\|_{\tau/s}\quad\text{for all}\quad{f\in L^{\tau/s}_{\mu}}

that estimate measurable errors for decreasing rearrangements $f^{*}$ , as well as the bilateral Bernstein-Jackson inequalities

\displaystyle t^{-s}2^{-(s+1)/2}f^{*}(t)\leq c_{s,\tau}\|f\|_{\tau/s}\leq\|f\|^{s}_{0}\|f\|_{\infty}\quad\text{for all}\quad{f\in L^{0}_{\mu}\cap L^{\infty}_{\mu}}.

In particular, the following inequality holds,

f^{*}(t)\leq\frac{2}{\pi t}\,\int|f|\,\mu(dx)\quad\text{for all}\quad f\in L^{1}_{\mu}.

The numerical algorithm resulting from the Theorem 7 was carried out on the example of inverse Gaussian distribution. Examples of applications for accurate estimates of spectral approximations of self-adjoint operators are given in Section 4.

Note that basic notations used in this work can be found in bergh76 ; Triebel78 .

2 Best approximation scales of quasi-normed Abelian groups

In what follows, we study the $\kappa$ -normed Abelian groups $(\mathfrak{A},{|\cdot|})$ relative to an operation $"+"$ , where $\kappa$ -norm ${|\cdot|}$ with the constant $\kappa\geq 1$ is determined by the assumptions (see e.g. bergh76 ; Maligranda1991 ):

$|0|=0$ and $|a|>0$ for all nonzero $a\in\mathfrak{A}$ ,
$|a|=|-a|$ for all $a\in\mathfrak{A}$ ,
$|a+b|\leq\kappa(|a|+|b|)$ with $\kappa$ independent on $a,b\in\mathfrak{A}$

As is known, each $\kappa$ -norm $|\cdot|$ can be replaced by an equivalent $1$ -norm $|\cdot|^{\prime}$ such that $|\cdot|^{\prime}\leq|\cdot|^{\rho}\leq 2|\cdot|^{\prime}$ by taking $(2\kappa)^{\rho}=2$ with a suitable $\rho>0$ (see e.g. PeetreSparr1972 ). We will not assume the completeness of groups.

Given a compatible couple $(\mathfrak{A}_{0},\mathfrak{A}_{1})$ of $\kappa_{\imath}$ -normed groups $(\mathfrak{A}_{\imath},{|\cdot|_{\imath}})$ with $\imath=\{0,1\}$ and $a=a_{0}+a_{1}$ in the sum ${\mathfrak{A}_{0}+\mathfrak{A}_{1}}$ such that $a_{\imath}\in\mathfrak{A}_{\imath}$ we define the best approximation $E$ -functional $E(t,a\mathchar 24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})$ with ${a\in\mathfrak{A}_{0}+\mathfrak{A}_{1}}$ and ${t>0}$ in the form (see (bergh76, , no.7))

E(t,a)=E(t,a\mathchar 24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})=\inf\left\{|a-a_{0}|_{1}\colon|a_{0}|_{0}<t\right\},\quad a\in{\mathfrak{A}_{1}}.

(1)

Definition 1.

For any $\tau=q\vartheta$ and $s+1=1/\vartheta$ with ${0<\vartheta<1}$ and ${0<q\leq\infty}$ the $\kappa$ -normed best approximation scales of Abelian groups are defined to be

\begin{split}\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})&=\left\{a\in\mathfrak{A}_{0}+\mathfrak{A}_{1}\colon|a|_{\mathcal{B}_{\tau}^{s}}<\infty\right\},\\ |a|_{\mathcal{B}_{\tau}^{s}}&=\left\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{s}E(t,a)\right]^{\tau}\frac{dt}{t}\right)^{1/\tau}&\!\!\!\!\!\!\hbox{if }\tau<\infty\\[8.61108pt] \sup\limits_{0<t<\infty}t^{s}E(t,a)&\!\!\!\!\!\!\hbox{if }\tau=\infty,\end{array}\right.\end{split}

(2)

where (by (bergh76, , Lemma 7.1.6)) $\kappa=2\kappa_{1}\max(\kappa_{0}^{s},\kappa_{1}^{s})\max(1,2^{-1/q^{\prime}})\max(1,2^{s-1})$ , $(1/q^{\prime}=1-1/q).$

Given a compatible couple $(\mathfrak{A}_{0},\mathfrak{A}_{1})$ , we also define the quadratic functional of Lions-Peetre’s type (see e.g. McLean2000 ; PeetreSparr1972 )

K_{2}(t,a)=K_{2}(t,a\mathchar 24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})=\inf\limits_{a=a_{0}+a_{1}}\left(|a_{0}|_{0}^{2}+t^{2}|a_{1}|_{1}^{2}\right)^{1/2},\quad t>0.

We will use the quadratically modified real interpolation method. Define the interpolation Abelian group of Lions-Peetre’s type $\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}$ endowed with the quasinorm ${\|\cdot\|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}}$ ,

	$\displaystyle K_{\theta,q}\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)$	$\displaystyle=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}=\big{\{}a\in\mathfrak{A}_{0}+\mathfrak{A}_{1}\big{\}},$
	$\displaystyle\|a\|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}$	$\displaystyle=\left\{\begin{array}[]{ll}\displaystyle{\bigg{(}\int_{0}^{\infty}\left[t^{-\vartheta}K_{2}(t,a)\right]^{q}\frac{dt}{t}\bigg{)}^{1/q}}&\!\!\!\!\hbox{if }q<\infty\\[8.61108pt] \sup\limits_{0<t<\infty}t^{-\vartheta}K_{2}(t,a)&\!\!\!\!\hbox{if }q=\infty.\end{array}\right.$

Following DL19 ; Lopushansky2023 , we extend the use of approximation constants $c_{s,\tau}$ to the case of arbitrary quasi-normed Abelian groups $\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ described in the classic works PeetreSparr1972 ; pietsch1981 , where

c_{s,\tau}:=\left\{\begin{array}[]{cl}\displaystyle\left[\frac{s}{\tau(s+1)^{2}}\right]^{1/\tau}&\hbox{if }\tau<\infty\\ \displaystyle 1&\hbox{if }\tau=\infty\end{array}\right.

(3)

is determined by the normalization factor $N_{\vartheta,q}$ from (bergh76, , Thm 3.4.1) to be

c_{s,\tau}=N^{1/q}_{\vartheta,q}\left(\vartheta q^{2}\right)^{-1/q\vartheta},\quad N_{\vartheta,q}:=[q\vartheta(1-\vartheta)]^{1/q}.

(4)

The following theorem establishes the bilateral form of approximation inequalities with exact constants for the case of quasi-normed Abelian groups.

Theorem 1.

The bilateral Bernstein-Jackson inequalities with the exact constant ${c}_{s,\tau}$

\displaystyle t^{s}E(t,a)

\displaystyle\leq{c}_{s,\tau}|a|_{\mathcal{B}_{\tau}^{s}}\leq 2^{1/2}|a|^{s}_{0}|a|_{1}\quad\text{for all}\quad a\in\mathfrak{A}_{0}\cap\mathfrak{A}_{1}

(5)

hold. There is a unique extension of Jackson’s inequality on the whole approximation scale

\displaystyle E(t,a)

\displaystyle\leq t^{-s}{c}_{s,\tau}|a|_{\mathcal{B}_{\tau}^{s}}\quad\text{for all}\quad{a\in\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})},\ \ r>0.

(6)

Proof: Let ${0<q<\infty}$ and $\alpha=|a|_{\mathfrak{A}_{1}}/|a|_{\mathfrak{A}_{0}}$ with a nonzero $a\in\mathfrak{A}_{0}\cap\mathfrak{A}_{1}$ . Since

\displaystyle K_{2}(t,a)^{2}

\displaystyle=\inf\limits_{a=a_{0}+a_{1}}\left(|a_{0}|_{0}^{2}+t^{2}|a_{1}|_{1}^{2}\right)\leq|a|^{2}_{0}\min(1,\alpha^{2}t^{2})={\min\left(|a|^{2}_{0},t^{2}|a|^{2}_{1}\right)}

or otherwise $K_{2}(t,a)\leq{|a|_{0}\min(1,\alpha t)}={\min\left(|a|_{0},t|a|_{1}\right)},$ we get the inequality

|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}\leq|a|_{1}^{q}\int_{0}^{\alpha}t^{-1+q(1-\vartheta)}dt+|a|^{q}_{0}\int_{\alpha}^{\infty}t^{-1-\vartheta q}dt.

Calculating the integrals and using that $\alpha=|a|_{1}/|a|_{0}$ , we obtain

\displaystyle|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}

\displaystyle\leq\frac{\alpha^{q(1-\vartheta)}}{q(1-\vartheta)}|a|_{1}^{q}+\frac{\alpha^{-\vartheta q}}{\vartheta q}|a|^{q}_{0}=\frac{1}{q\vartheta(1-\vartheta)}\left(|a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}\right)^{q}.

Let it be now $q=\infty$ . Then the inequality

t^{-\vartheta}K_{2}(t,a)\leq{\min\left(t^{-\vartheta}|a|_{0},t^{1-\vartheta}|a|_{1}\right)}

holds. Taking in this case $t=|a|_{0}/|a|_{1}$ , we obtain that $t^{-\vartheta}K_{2}(t,a)\leq|a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}.$ Combining previous inequalities and taking into account (4), we have

\begin{split}\|a\|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}&\leq\left\{\begin{array}[]{ll}N_{\vartheta,q}^{-1}|a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}&\hbox{if }q<\infty\\[6.45831pt] |a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}&\hbox{if }q=\infty\end{array}\right.,\quad a\in\mathfrak{A}_{0}\cap\mathfrak{A}_{1}.\end{split}

(7)

For further considerations, we need the functional

K_{\infty}(v,a):=\inf\limits_{a=a_{0}+a_{1}}\max\big{(}|a_{0}|_{\mathfrak{A}_{0}},v|a_{1}|_{\mathfrak{A}_{1}}\big{)}.

Now we will use the known properties of the considered functionals that

	$\displaystyle v^{-\theta}K_{\infty}(v,f)\to 0\quad\text{as \ }{v\to 0}\text{ or }{v\to\infty}$
	$\displaystyle{t^{-1+1/\theta}E(t,f)\to 0}\quad\text{as \ }{v\to 0}\text{ or }{v\to\infty}$

(see (bergh76, , Thm 7.1.7)). As a result, we get

\displaystyle\int_{0}^{\infty}\left(v^{-\vartheta}K_{\infty}(v,a)\right)^{q}\frac{dv}{v}

\displaystyle=-\frac{1}{\vartheta q}\int_{0}^{\infty}K_{\infty}(v,a)^{q}dv^{-\vartheta q}=\frac{1}{\vartheta q}\int_{0}^{\infty}v^{-\vartheta q}dK_{\infty}(v,a)^{q}.

On the other hand, integrating by parts with the change $v=t/E(t,a)$ , we get for $s+1=1/\vartheta$ that

\displaystyle\int_{0}^{\infty}\left(v^{-\vartheta}K_{\infty}(v,a)\right)^{q}\frac{dv}{v}

\displaystyle=\frac{1}{\vartheta q}\int_{0}^{\infty}\left(t/E(t,a)\right)^{-\vartheta q}dt^{q}=\frac{1}{\vartheta q^{2}}\int_{0}^{\infty}\left(t^{s}E(t,a)\right)^{\vartheta q}\frac{dt}{t}.

From the definition $K_{\infty}$ and $K_{2}$ , it follows that

K_{\infty}(t,a)\leq K_{2}(t,a)\leq 2^{1/2}K_{\infty}(t,a)

(8)

(see (PeetreSparr1972, , Remark 3.1)). Now, taking into account that

\frac{1}{\vartheta q^{2}}\|a\|^{\vartheta q}_{\mathcal{B}_{\tau}^{s}}=\frac{1}{\vartheta q^{2}}\int_{0}^{\infty}\left(t^{s}E(t,a)\right)^{\vartheta q}\frac{dt}{t}=\int_{0}^{\infty}\left(v^{-\vartheta}K_{\infty}(v,a)\right)^{q}\frac{dv}{v},

according to the left inequality from (8), we obtain

\frac{1}{\vartheta q^{2}}|a|^{\vartheta q}_{\mathcal{B}_{\tau}^{s}}\leq\int_{0}^{\infty}\left(v^{-\vartheta}K_{2}(v,a)\right)^{q}\frac{dv}{v}=|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}.

(9)

By using the right inequality from (8), we have

	$\displaystyle\|a\|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}=\int_{0}^{\infty}\left(v^{-\vartheta}K_{2}(v,a)\right)^{q}\frac{dv}{v}\leq 2^{q/2}\int_{0}^{\infty}\left(v^{-\vartheta}K_{\infty}(v,a)\right)^{q}\frac{dv}{v}$
	$\displaystyle=\frac{2^{q/2}}{\vartheta q^{2}}\int_{0}^{\infty}\left(t^{s}E(t,a)\right)^{\vartheta q}\frac{dt}{t}=\frac{2^{q/2}}{\vartheta q^{2}}\|a\|^{\vartheta q}_{\mathcal{B}_{\tau}^{s}}.$

As a result, by virtue of (9) we have the inequalities

|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}\leq\frac{2^{q/2}}{\vartheta q^{2}}|a|^{\vartheta q}_{\mathcal{B}_{\tau}^{s}}\leq 2^{q/2}|a|^{q}_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}\quad\text{with}\quad\tau=\vartheta q.

(10)

That gives the following isomorphism with equivalent quasinorms

\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}^{1/\vartheta}.

(11)

By (bergh76, , Lemma 7.1.2 and Thm 7.1.1) for each ${v>0}$ there is ${t>0}$ such that

(t^{s}E(t,a))^{\vartheta}\leq v^{-\vartheta}K_{\infty}(v,a)\leq\left(t^{s}E(t-0,a)\right)^{\vartheta}.

(12)

Since $|a|^{\vartheta}_{\mathcal{B}_{\infty}^{s}}\leq|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}$ for $q=\infty$ , the inequalities (12) for any ${v>0}$ yield

	$\displaystyle v^{-\vartheta}K_{2}(v,a)$	$\displaystyle\leq v^{-\vartheta}2^{1/2}K_{\infty}(v,a)\leq 2^{1/2}\big{(}t^{s}E(t-0,a)\big{)}^{\vartheta}$
		$\displaystyle\leq 2^{1/2}\Big{(}\sup_{t>0}\,t^{s}E(t,a)\Big{)}^{\vartheta}=2^{1/2}\|a\|^{\vartheta}_{\mathcal{B}_{\infty}^{s}}.$

As a result, $|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}\leq 2^{1/2}|a|^{\vartheta}_{\mathcal{B}_{\infty}^{s}}$ and the isomorphism (11) holds for $q=\infty$ .

Prove the Bernstein inequality (5). Combining (7) and (9), we obtain

|a|^{\vartheta}_{\mathcal{B}_{\tau}^{s}}\leq\left\{\begin{array}[]{ll}\displaystyle\left[\frac{q}{1-\vartheta}\right]^{1/q}|a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}&\hbox{if }q<\infty\\[8.61108pt] |a|^{1-\vartheta}_{0}|a|_{1}^{\vartheta}&\hbox{if }q=\infty.\end{array}\right.

(13)

Thus, the second inequality (5) we get by setting $s+1=1/\vartheta$ and $\tau=\vartheta q$ .

Prove the inequality (6). Integrating ${\min(1,v/t)K_{2}(t,a)\leq{K}_{2}(v,a)}$ , we obtain

\int_{0}^{\infty}\left(v^{-\vartheta}\min(1,v/t)\right)^{q}\frac{dv}{v}K_{2}(t,a)^{q}\leq\int_{0}^{\infty}\left(v^{-\vartheta}K_{2}(v,a)\right)^{q}\frac{dv}{v}=|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}^{q}.

Since the left integral can be rewritten as the following sum

\begin{split}\int_{0}^{\infty}\left(v^{-\vartheta}\min(1,v/t)\right)^{q}\frac{dv}{v}&=\int_{0}^{t}v^{(1-\vartheta)q-1}t^{-q}dv+\int_{t}^{\infty}v^{-\vartheta q-1}dv=\frac{1}{t^{\vartheta}N_{\vartheta,q}^{1/q}},\end{split}

we obtain the inequality

\begin{split}\bigg{(}\int_{0}^{t}\frac{v^{(1-\vartheta)q-1}}{t^{q}}dv+\int_{t}^{\infty}v^{-\vartheta q-1}dv\bigg{)}^{1/q}K_{2}(t,a)&=\frac{K_{2}(t,a)}{t^{\vartheta}N_{\vartheta,q}}\leq|x|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}.\end{split}

As a result, $K_{2}(t,a)\leq t^{\vartheta}N_{\vartheta,q}|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}$ . Hence, taking into account (8) and (12), we get

v^{1-\vartheta}E(v,a)^{\vartheta}\leq t^{-\vartheta}K_{\infty}(t,a)\leq N_{\vartheta,q}|a|_{(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}}.

Applying (10), we obtain

v^{1-\vartheta}E(v,a)^{\vartheta}\leq{2^{1/2}}N_{\vartheta,q}|a|^{\vartheta}_{\mathcal{B}_{\tau}^{s}}.

Taking into account that $\vartheta=1/(s+1)=q/\tau,$ we obtain the inequality (6) for all ${0<q<\infty}$ .

In the case $q=\infty$ , we have

t^{s}E(t,a)\leq\sup_{t>0}\,t^{s}E(t,a)=|a|_{\mathcal{B}_{\infty}^{s}}

for all ${a\in\mathcal{B}_{\infty}^{s}}$ . Thus, the inequality (6) holds for both cases.

Finally note that Bernstein-Jackson inequalities are achieved at $\tau=\infty$ , so they are sharp.

Corollary 2.

If the quasi-normed Abelian groups $\mathfrak{A}_{0}$ , $\mathfrak{A}_{1}$ are compatible then the following isomorphism with equivalent quasi-norms holds,

\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}^{1/\vartheta},\quad s+1=1/\vartheta,\quad\tau=\vartheta q.

(14)

If compatible groups $\mathfrak{A}_{0}$ and $\mathfrak{A}_{1}$ are complete, the approximation group $\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ is complete.

Proof: The isomorphism (14) immediately follows from (11) and Theorem 1. Note that if both compatible groups $\mathfrak{A}_{0}$ and $\mathfrak{A}_{1}$ are complete, the interpolation group $\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}$ is complete (bergh76, , Thm 3.4.2 & Lemma 3.10.2). By Theorem 1 $\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})$ is complete.

Corollary 3.

Let $(\mathfrak{A}_{0},\mathfrak{A}_{1})$ , $(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})$ be compatible. If $T\colon\mathfrak{A}_{\imath}\to\mathfrak{A}^{\prime}_{\imath}$ $(\imath=0,1)$ are quasi-Abelian mappings, i.e. ${T(a_{0}+a_{1})}={b_{0}+b_{1}}$ and $|Ta_{\imath}|_{\mathfrak{A}_{\imath}}\leq\kappa_{\imath}|b_{\imath}|_{\mathfrak{A}^{\prime}_{\imath}}$ with constants $\kappa_{\imath}$ (bergh76, , p.81), then there exists a constant $C>0$ such that

T\colon\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})\to\mathcal{B}_{\tau}^{s}(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})\quad\text{and}\quad|Ta|_{\mathcal{B}_{\tau}^{s}(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})}\leq C|a|_{\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})}.

Proof: It follows from (10), (11) and the boundness of

T\colon(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}\to(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})_{\vartheta,q},

where $C\leq 4^{1/\vartheta}(\vartheta q^{2})^{-1/\vartheta q}\kappa_{0}^{s}\,\kappa_{1}$ that gives the required inequality.

Corollary 4.

Let $0\leq\vartheta_{0}<\vartheta_{1}\leq 1$ , ${0<q\leq\infty}$ and $\vartheta=(1-\eta)\vartheta_{0}+\eta\vartheta_{1}$ ${(0<\eta<1)}$ . If the quasi-normed Abelian groups $(\mathfrak{A}_{0},\mathfrak{A}_{1})$ , $(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1})$ are compatible and the inclusions

(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta_{\imath},q}\subset\mathfrak{A}^{\prime}_{\imath}\subset(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta_{\imath},\infty},\quad(\imath=0,1)

are valid, the following isomorphism with equivalent quasinorms holds,

\mathcal{B}_{\vartheta q}^{(1-\vartheta)/\vartheta}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\mathcal{B}_{\eta q}^{(1-\vartheta)/\vartheta}(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1}).

(15)

Proof: Using (11) and the known equality $\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}=\left(\mathfrak{A}^{\prime}_{0},\mathfrak{A}^{\prime}_{1}\right)_{\eta,q}$ PeetreSparr1972 , (Komatsu1981, , Theorem 3) in the case ${r=1}$ , we get the reiteration identity (15) with accuracy to quasinorm equivalency.

Corollary 5.

The following continuous embedding hold

\mathcal{B}_{\vartheta q}^{(1-\vartheta)/\vartheta}(\mathfrak{A}_{0},\mathfrak{A}_{1})\looparrowright\mathcal{B}_{\vartheta p}^{(1-\vartheta)/\vartheta}(\mathfrak{A}_{0},\mathfrak{A}_{1})\quad\text{with}\quad q<p.

Proof: It follows from (11), since for arbitrary $q<p$ the following continuous embedding $(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,q}\looparrowright(\mathfrak{A}_{0},\mathfrak{A}_{1})_{\vartheta,p}$ is true by virtue of (Komatsu1981, , Lemma p. 385).

Corollary 6.

If $\vartheta_{0}<\vartheta_{1}$ then the embedding $\mathfrak{A}_{1}\subset\mathfrak{A}_{0}$ yields

\mathcal{B}_{\vartheta_{1}q}^{(1-\vartheta_{1})/\vartheta_{1}}(\mathfrak{A}_{0},\mathfrak{A}_{1})\subset\mathcal{B}_{\vartheta_{0}q}^{(1-\vartheta_{0})/\vartheta_{0}}(\mathfrak{A}_{0},\mathfrak{A}_{1}).

Proof: It follows from Corollary 4 and (bergh76, , Theorem 3.4.1(d)).

3 Exact constants in bilateral error estimates

Let ${(\mathfrak{A},|\cdot|)}$ be a $\kappa$ -normed Abelian group. Consider a quasi-normed space $L^{p}_{\mu}=L^{p}_{\mu}(\mathfrak{A})$ with a positive Radon measure $\mu$ on a measure space $(X,\mu)$ of measurable functions ${f\colon X\to\mathfrak{A}}$ endowed with the $\kappa$ -norm

\|f\|_{p}=\left\{\begin{array}[]{ll}\displaystyle\Big{(}\int|f(x)|^{p}\,\mu(dx)\Big{)}^{1/p}&\hbox{if }0<p<\infty\\ \mathop{\rm ess\,sup}_{x\in X}|f(x)|&\hbox{if }p=\infty\\[4.30554pt] \mu\left(\operatorname{\operatorname{supp}}f\right)&\hbox{if }p=0,\end{array}\right.

For a $\mu$ -measurable function ${f\colon X\to\mathfrak{A}}$ , we define the distribution function

m(\sigma,f)=\mu\left\{x\in X\colon{|f(x)|>\sigma}\right\}.

Denote by $f^{*}$ the decreasing rearrangement of $f$ , where

f^{*}(t):=\inf\left\{\sigma>0\colon m(\sigma,f_{\sigma})\leq t\right\}

with $f_{\sigma}(x)=f(x)$ if $|f(x)|>\sigma$ and $f_{\sigma}(x)=0$ otherwise (for details see (bergh76, , no 1.3)). The decreasing rearrangement $f^{*}$ is nonnegative nonincreasing continuous on the right function of $\sigma$ on $(0,\infty)$ which is equimeasurable with $f$ in the sense that

m(\sigma,f)=m(\sigma,f^{*})\quad\text{for all}\quad\sigma\geq 0.

In other words, the decreasing rearrangement of a function $f$ is a generalized inverse of its distribution function in the sense that if $m(\sigma,f)$ is one-to-one then $f^{*}$ is simply the inverse of $m(\sigma,f)$ .

Since $f$ is rearrangeable, $\left\{\sigma>0\colon m(\sigma,f_{\sigma})\leq t\right\}$ is nonempty for $t>0$ thus $f^{*}$ is finite on its domain.

In the case where $\mu(X)<\infty$ , we consider $f^{*}$ as a function on $(0,\mu(X))$ , since $f^{*}(t)=0$ for all $t>\mu(X)$ . The set $\left\{\sigma>0\colon m(\sigma,f_{\sigma})=0\right\}$ can be empty. If $f$ is bounded then $f^{*}\to\|f\|_{\infty}$ as $t\to 0_{+}$ , otherwise, $f^{*}$ is unbounded at the origin.

As is also know (see e.g. ONeilWeiss1963 ) for $f\in L^{p}_{\mu}$ with $p\geq 1$ that

\int|f|^{p}\,\mu(dx)=p\int_{0}^{\infty}\sigma^{p-1}m(\sigma,f)\,d\sigma,\quad\|f\|_{p}^{p}=\int_{0}^{\infty}f^{*}(t)^{p}\,dt.

The couple $\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}$ is compatible in the interpolation theory sense (see e.g. (bergh76, , Lemma 3.10.3) or (Komatsu1981, , no 1)). Given elements $f={f_{0}+f_{\infty}}$ in the algebraic sum ${L^{0}_{\mu}+L^{\infty}_{\mu}}$ , we define the best approximation $E$ -functional $E\left(t,f\mathchar 24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\right)$ with ${t>0}$ as (see e.g. (bergh76, , no 7))

\displaystyle E(t,f)

\displaystyle=E\big{(}t,f\mathchar 24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)}=\inf\left\{\|f-f_{0}\|_{\infty}\colon\|f_{0}\|_{0}<t\right\},\quad f\in{L^{\infty}_{\mu}}.

In what follows, we use the equivalent parameters ${0<\vartheta<1},\ {0<q\leq\infty}$ , where $s+1=1/\vartheta$ and $\tau=\vartheta q$ . Following Lopushansky2023 , the appropriate system of exact constants $C_{\theta,q}$ is defined by the normalization factor

N_{\theta,q}=\left(\int_{0}^{\infty}|t^{-\theta}\mathcal{N}(t)|^{q}\frac{dt}{t}\right)^{-1/q}\quad\text{with}\quad\mathcal{N}(t)=\frac{t}{\sqrt{1+t^{2}}}

in the known interpolation Lions-Peetre method to be

\begin{split}C_{\theta,q}&=\left\{\begin{array}[]{cl}\displaystyle\frac{2^{1/2\theta}}{(q^{2}\theta)^{1/q\theta}}N_{\theta,q}^{1/\theta}&\hbox{if }q<\infty,\\[8.61108pt] \displaystyle\left(\frac{\sin\pi\theta}{\pi\theta}\right)^{1/2\theta}&\hbox{if }q=2,\\[8.61108pt] \displaystyle 2^{1/2\theta}&\hbox{if }q=\infty.\end{array}\right.\end{split}

(16)

Remark 1.

For this equivalent parameter system the sharp constant $c_{s,\tau}$ takes an equivalent form

2^{-1/2\theta}C_{\theta,q}=\left\{\begin{array}[]{cl}c_{s,\tau}&\hbox{if }\tau<\infty\\ 1&\hbox{if }\tau=\infty\end{array}\right.,

where the following dependencies between indexes are performed,

2^{-1/2\theta}C_{\theta,q}=\left[(1-\theta)/q\right]^{1/q\theta}=(\theta q^{2})^{-1/q\theta}N_{\theta,q}^{1/\vartheta}.

A graphical image of the normalized sharp constant $c_{s,\tau}$ in variables $s,\tau$ shown on Fig. 1.

Refer to caption — Figure 1: 3D-graph of $c_{s,\tau}=(s/\tau)^{1/\tau}(s+1)^{-2/\tau}$ for ${s,\tau>0}$

Theorem 7.

(a) For any ${0<\vartheta<1}$ and ${0<q\leq\infty}$ the equality

f^{*}(t)=E\big{(}t,f\mathchar 24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)},\quad f\in{L^{\infty}_{\mu}}

(17)

and, as a consequence, the following equalities hold,

\|f\|_{q\theta^{2}/(1-\theta)}=\left\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{-1+1/\theta}f^{*}(t)\right]^{q\theta}\frac{dt}{t}\right)^{1/q\theta}&\hbox{if }q<\infty\\[8.61108pt] \sup\limits_{0<t<\infty}t^{-1+1/\theta}f^{*}(t)&\hbox{if }q=\infty.\end{array}\right.

(18)

(b) The Jackson-type inequality

f^{*}(t)\leq t^{1-1/\theta}C_{\theta,q}\|f\|_{q\theta^{2}/(1-\theta)}\quad\text{for all}\quad{f\in L_{\mu}^{q\theta^{2}/(1-\theta)}},

(19)

as well as, the following bilateral Bernstein-Jackson-type inequalities are valid,

\displaystyle t^{-1+1/\theta}f^{*}(t)

\displaystyle\leq{C}_{\theta,q}\|f\|_{q\vartheta^{2}/(1-\theta)}\leq 2^{1/2\theta}\|f\|^{-1+1/\theta}_{0}\|f\|_{\infty}\text{ for all }f\in L^{0}_{\mu}.

(20)

Proof: (a) By definition, the functional $E(t,f)$ is the infimum of $\|f-f_{0}\|_{\infty}$ such that ${\mu(\operatorname{\operatorname{supp}}f_{0})\leq t}$ . The next reasoning extends (bergh76, , Lemma 7.2.1) on the case of $\mathfrak{A}$ -valued functions.

Put $g_{0}(x)=f(x)$ on the subset $\operatorname{\operatorname{supp}}f$ and let $g_{0}(x)=0$ outside $\operatorname{\operatorname{supp}}f$ . Then we obtain ${\|f-g_{0}\|_{\infty}}\leq{\|f-g\|_{\infty}}$ .

In a similar way, let $g_{\sigma}(x)=f(x)$ if $|f(x)|>\sigma$ and ${g_{\sigma}(x)=0}$ otherwise. Then for the number $\tau={\sup\big{\{}|f(x)|\colon x\neq\operatorname{\operatorname{supp}}f\big{\}}}$ we get $\operatorname{\operatorname{supp}}f_{\tau}\subset\operatorname{\operatorname{supp}}f$ . It follows ${\mu(\operatorname{\operatorname{supp}}g_{\tau})\leq t}$ . Since $\|f-g_{\tau}\|_{\infty}\leq\tau$ and $\|f-g_{0}\|_{\infty}=\tau$ , we obtain

E\big{(}t,f\mathchar 24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)}={\inf}_{\sigma}\left\{\|f-g_{\sigma}\|_{\infty}\colon\mu\left(\operatorname{\operatorname{supp}}g_{\sigma}\right)\leq t\right\},

where the rigth hand side is equal to $f^{*}$ . As a result, the equality (17) holds.

Using the best approximation $E$ -functional, we define the quasi-normed space

	$\displaystyle E_{\theta,q}\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}$	$\displaystyle=\left\{f\in{L^{0}_{\mu}+L^{\infty}_{\mu}}\colon\\|f\\|_{E_{\theta,q}}<\infty\right\},$		(21)
	$\displaystyle\\|f\\|_{E_{\theta,q}}$	$\displaystyle=\left\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{-1+1/\theta}E(t,f)\right]^{q\theta}\frac{dt}{t}\right)^{1/q\theta}&\hbox{if }q<\infty,\\[8.61108pt] \sup\limits_{0<t<\infty}t^{-1+1/\theta}E(t,f)&\hbox{if }q=\infty.\end{array}\right.$		(24)

By the known approximation theorem (see (bergh76, , Thm 7.2.2)) the isometric isomorphism

E_{\theta,q}\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}=L_{\mu}^{q\theta^{2}/(1-\theta)},\quad\|f\|_{E_{\theta,q}}=\|f\|_{\vartheta^{2}q/(1-\theta)}

(25)

holds for all ${f\in L_{\mu}^{\infty}}.$ Combining equalities (17), (21) and (25), we get (18).

(b) In what follows, we use the classic integral of the real interpolation method

\|f\|_{\theta,q}=\left(\int_{0}^{\infty}\left|t^{-\theta}f(x)\right|^{q}\frac{dt}{t}\right)^{1/q},\quad{0<\theta<1},\ 0<q<\infty.

(26)

Let us consider the quadratic $K_{2}$ -functional (see e.g. (McLean2000, , App. B)) for the interpolation couple of quasi-normed groups $\big{(}L^{0},L^{\infty}\big{)}$ ,

	$\displaystyle K_{2}(t,f)$	$\displaystyle={K}_{2}\big{(}t,f\mathchar 24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)}$
		$\displaystyle=\inf_{f=f_{0}+f_{\infty}}\left\{\left(\\|f_{0}\\|_{0}^{2}+t^{2}\\|f_{\infty}\\|_{\infty}^{2}\right)^{1/2}\colon f_{0}\in L^{0}_{\mu},\ f_{\infty}\in L^{\infty}_{\mu}\right\},$

determining the real interpolation quasi-normed Abelian group

\displaystyle\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}_{\theta,q}:=K_{\theta,q}\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}=\left\{f=f_{0}+f_{\infty}\colon\|K_{2}(\cdot,f)\|_{\theta,q}<\infty\right\}

which is endowed with the norm

\displaystyle\|f\|_{K_{\theta,q}}

\displaystyle=\left\{\begin{array}[]{ll}N_{\theta,q}\|K_{2}(\cdot,f)\|_{\theta,q}&\hbox{if }q<\infty\\[6.45831pt] \sup\limits_{t\in(0,\infty)}t^{-\theta}K_{2}(t,f)&\hbox{if }q=\infty.\end{array}\right.

Note that for any ${0<\vartheta<1}$ and ${q=2}$ the following equalities hold,

\displaystyle N_{\theta,2}=\|\mathcal{N}\|_{\theta,2}^{-1}=\|K_{2}(\cdot,1)\|_{\theta,2}^{-1}=\left(\frac{2\sin\pi\theta}{\pi}\right)^{1/2}.

In fact, the relationship between the weight function $\mathcal{N}(t)^{2}=t^{2}/(1+t^{2})$ and the quadratic $K_{2}$ -functional is explained by the formula

N_{\theta,2}=\|\mathcal{N}\|_{\theta,2}^{-1}=\|K_{2}(\cdot,1)\|_{\theta,2}^{-1}

(see e.g. (McLean2000, , Ex. B.4)). It follows from

\min_{z=z_{0}+z_{1}}\left(\alpha_{0}|z_{0}|^{2}+\alpha_{1}|z_{1}|^{2}\right)={\alpha_{0}\alpha_{1}|z|^{2}}{\alpha_{0}+\alpha_{1}}

for a fixed $\alpha_{0},\alpha_{1}>0$ and a complex $z$ . This minimum is achieved when

\alpha_{0}z_{0}=\alpha_{1}z_{1}=\frac{\alpha_{0}\alpha_{1}z}{(\alpha_{0}+\alpha_{1})}.

Thus, $K_{2}(t,1)$ is minimized when $f_{0}$ , $f_{1}$ are such that

f_{0}=t^{2}f_{1}=\frac{t^{2}}{1+t^{2}}.

By integrating the above functions (see e.g. (McLean2000, , Ex. B.5, Thm B.7)) it follows that the normalization factor $N_{\theta,2}$ is equal to $N_{\theta,2}=\left({2\sin\pi\theta}/{\pi}\right)^{1/2}.$

In particular, the equalities (17),

	$\displaystyle f^{*}(t)$	$\displaystyle=\inf\big{\{}\sigma\colon m(\sigma,f_{\sigma})\leq t\big{\}}$
		$\displaystyle=\inf\left\{\\|f-f_{0}\\|_{\infty}\colon\\|f_{0}\\|_{0}<t\right\}=E\big{(}t,f\mathchar 24635\relax\;L^{0}_{\mu},L^{\infty}_{\mu}\big{)},\quad f\in{L_{\mu}^{\infty}},$

as well as, the isometric isomorphism (25),

\displaystyle E_{\theta,q}\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}=L_{\mu}^{q\theta^{2}/(1-\theta)},\quad\|f\|_{E_{\theta,q}}=\|f\|_{\vartheta^{2}q/(1-\theta)}

allows calculating the exact form of best constants (16). Now, applying Theorem 1 to $\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}$ , we obtain inequalities (19) and (20).

Corollary 8.

The decreasing rearrangement for $q=2$ has the estimation

\displaystyle f^{*}(t)

\displaystyle\leq t^{1-1/\theta}\left(\frac{\sin\pi\theta}{\pi\theta}\right)^{1/2\theta}\!\|f\|_{2\theta^{2}/(1-\theta)}\quad\text{for all}\quad{f\in L_{\mu}^{q\theta^{2}/(1-\theta)}}

(27)

which for $\theta=1/2$ can be written as follows

\displaystyle f^{*}(t)

\displaystyle\leq\frac{2}{\pi t}\,\int|f|\,\mu(dx)\quad\text{for all}\quad f\in L^{1}_{\mu}.

(28)

Note that if $\theta\to 1$ then $\sin\pi\theta\to 0$ thus above inequalities disappear. One has a sense only for the quasi-normed group $L_{\mu}^{2\theta^{2}/(1-\theta)}$ with $0<\theta<1$ .

Corollary 9.

The equality (18) with $s+1=1/\vartheta$ and $\tau=\vartheta q$ takes the form

\|f\|_{\tau/s}=\left\{\begin{array}[]{ll}\displaystyle\left(\int_{0}^{\infty}\left[t^{s}f^{*}(t)\right]^{\tau}\frac{dt}{t}\right)^{1/\tau}&\hbox{if }\tau<\infty\\[8.61108pt] \sup\limits_{0<t<\infty}t^{s}f^{*}(t)&\hbox{if }\tau=\infty\end{array}\right.

for any $f\in L^{\tau/s}_{\mu}$ . Then the Jackson-type inequality takes the form

f^{*}(t)\leq t^{-s}2^{(s+1)/2}\,c_{s,\tau}\|f\|_{\tau/s}\quad\text{for all}\quad{f\in L^{\tau/s}_{\mu}}

(29)

and bilateral Bernstein-Jackson-type inequalities take the form

\displaystyle t^{s}2^{-(s+1)/2}f^{*}(t)

\displaystyle\leq c_{s,\tau}\|f\|_{\tau/s}\leq\|f\|^{s}_{0}\|f\|_{\infty}\quad\text{for all}\quad{f\in L^{0}_{\mu}\cap L^{\infty}_{\mu}}.

(30)

It instantly follows from the relations

C_{\theta,q}=\left\{\begin{array}[]{cl}2^{(s+1)/2}\,c_{s,\tau}&\hbox{if }\tau<\infty\\ 2^{(s+1)/2}&\hbox{if }\tau=\infty\end{array}\right.,\quad\text{where}\quad\vartheta=\frac{1}{s+1},\quad 2^{1/2\theta}=2^{(s+1)/2}.

Remark 2.

The inequalities (5) is an extension on the case measurable functions the known best approximation Bernstein-Jackson inequalities. Whereas the inequality (6) is an extension of the approximation Jackson inequality. The scale of approximation quasi-normed Abelian groups is often denoted as

\mathcal{B}_{\tau}^{s}(L^{0}_{\mu},L^{\infty}_{\mu}):=E_{\theta,q}(L^{0}_{\mu},L^{\infty}_{\mu}),

where the space $\mathcal{B}_{\tau}^{s}$ coincides with a suitable extension of the Besov type quasi-normed groups (see e.g. (bergh76, , Thm 7.2.4), (Triebel78, , Def. 4.2.1/1)). The scale of quasi-normed Besov Abelian groups for another approximation couples is described in DL19 ; Feichtinger2016 ; Pesenson2024 .

4 Examples of bilateral estimates with exact constants

Taking into account the previous statements, we can give typical examples of Bernstein-Jackson inequalities with explicit constants for different types of best approximations.

Example 1 (Applications to classic Besov scales).

Let ${\mathfrak{A}_{0}=L^{p}(\mathbb{R}^{n})}$ with ${(1<p<\infty)}$ . As is known (see e.g. Nikolski75 ) each real-valued function ${a\in L^{p}(\mathbb{R}^{n})}$ can be approximated by entire analytic functions $g\in\mathcal{E}_{p}^{t}(\mathbb{C}^{n})$ , where $\mathcal{E}_{p}^{t}(\mathbb{C}^{n})$ means the space of entire analytic functions on $\mathbb{C}^{n}$ of an exponential type ${t>0}$ with restrictions to $\mathbb{R}^{n}$ belonging to $L^{p}(\mathbb{R}^{n})$ . The best approximations can be characterized by the best approximation functional

E\left(t,a\mathchar 24635\relax\;\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\right)=\inf_{t>0}\left\{\|a-g\|_{L^{p}(\mathbb{R}^{n})}\colon g\in\mathcal{E}_{p}^{t}(\mathbb{C}^{n})\right\},

where the subspace ${\mathfrak{A}_{1}=\mathcal{E}_{p}}$ with $\mathcal{E}_{p}={\bigcup}_{t>0}\mathcal{E}_{p}^{t}(\mathbb{C}^{n})$ is endowed with the quasinorm

|g|_{\mathcal{E}_{p}}=\|g\|_{L^{p}(\mathbb{R}^{n})}+\left\{\sup{|\zeta|\colon\zeta\in\operatorname{\operatorname{supp}}\hat{g}}\right\},\quad g\in\mathcal{E}_{p}

defined using the support $\operatorname{\operatorname{supp}}\hat{g}$ of the Fourier-image $\hat{g}$ . Then according to Theorem 1 the corresponding approximation inequalities take the form

	$\displaystyle t^{s}E\left(t,a\mathchar 24635\relax\;\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\right)$	$\displaystyle\leq c_{s,\tau}\\|f\\|_{B_{p,\tau}^{s}(\mathbb{R}^{n})}\leq 2^{1/2}\|a\|^{s}_{\mathcal{E}_{p}}\\|a\\|_{L^{p}},\ {a\in\mathcal{E}_{p}\cap L^{p}(\mathbb{R}^{n})},$
	$\displaystyle E\left(t,a\mathchar 24635\relax\;\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\right)$	$\displaystyle\leq{t^{-s}c_{s,\tau}}\,\\|a\\|_{B_{p,\tau}^{s}(\mathbb{R}^{n})},\quad{a\in{B_{p,\tau}^{s}(\mathbb{R}^{n})}},$

In this case the approximation scale

\mathcal{B}_{\tau}^{s}\big{(}\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\big{)}=\left(\mathcal{E}_{p},L^{p}(\mathbb{R}^{n})\right)_{\vartheta,q}^{1/\vartheta}

exactly coincides with the classic Besov scale denoted by $B_{p,\tau}^{s}(\mathbb{R}^{n})$ (see (Triebel78, , p.197)).

Example 2 (Applications to scales of periodic functions).

Following e.g. (bergh76, , no 1.5), Prestin we can write Bernstein-Jackson inequalities in a more general form. Let $X=\mathbb{T}$ be the $1$ -dimensional torus and the Hilbert space $\mathfrak{A}_{1}=L^{2}(\mathbb{T})$ has the orthonormal basis $\left\{\mathfrak{e}_{k}=e^{2\pi\mathrm{i}kt}\colon k\in\mathbb{Z},\,t\in\mathbb{T}\right\}$ . Let $\mathfrak{A}_{0}=\{\text{trigonometric polynomials \ }a_{0}\}$ with the quasi-norm $|a_{0}|_{0}=\operatorname{\operatorname{deg}}a_{0}$ , and $\mathfrak{A}_{1}=\{\text{$2\pi$-periodic functions \ }a\}$ with the norm $|a|_{1}=\|a\|_{L^{2}(\mathbb{T})}$ , as well as, $E(n,a\mathchar 24635\relax\;\mathfrak{A}_{0},\mathfrak{A}_{1})=\inf\left\{\|a-a_{0}\|_{L^{2}(\mathbb{T})}\colon\operatorname{\operatorname{deg}}a_{0}<n\right\}$ for all ${a\in L^{2}(\mathbb{T})}$ . Let $c_{j,\tau}:=\left\{\begin{array}[]{cl}\displaystyle\left[\frac{j}{\tau(j+1)^{2}}\right]^{1/\tau}&\hbox{if }\tau<\infty\\ \displaystyle 1&\hbox{if }\tau=\infty\end{array}\right.$ with $j\in\mathbb{N}$ . Then Bernstein-Jackson inequalities have the form

	$\displaystyle n^{j}E(n,a)$	$\displaystyle\leq{c}_{j,\tau}\|a\|_{\mathcal{B}_{\tau}^{j}}\leq 2^{1/2}\|a\|^{j}_{0}\|a\|_{1}\quad\text{for all}\quad a\in\mathfrak{A}_{0}\cap\mathfrak{A}_{1},$
	$\displaystyle E(n,a)$	$\displaystyle\leq n^{-j}{c}_{j,\tau}\|a\|_{\mathcal{B}_{\tau}^{j}}\quad\text{for all}\quad{a\in\mathcal{B}_{\tau}^{j}(\mathfrak{A}_{0},\mathfrak{A}_{1})},$

where $\mathcal{B}_{\tau}^{s}(\mathfrak{A}_{0},\mathfrak{A}_{1})=\left(\mathfrak{A}_{0},\mathfrak{A}_{1}\right)_{\vartheta,q}^{1/\vartheta}$ with $j+1=1/\vartheta\in\mathbb{N}$ , $\tau=\vartheta q$ .

For $\mathfrak{A}_{1}=L^{\infty}(\mathbb{T})$ with $|a|_{1}=\|a\|_{L^{\infty}(\mathbb{T})}$ and $\mathfrak{A}_{0}=L^{0}(\mathbb{T})$ with $|a_{0}|_{0}=\operatorname{\operatorname{deg}}a_{0}$ , we consider the Banach space $L^{\infty}(\mathbb{T})$ -valued functions ${f\colon X\to L^{\infty}(\mathbb{T})}$ . Then bilateral Bernstein-Jackson inequalities have the form

	$\displaystyle n^{-j}2^{-(j+1)/2}f^{*}(n)$	$\displaystyle\leq c_{j,\tau}\\|f\\|_{\tau/j}\leq\\|f\\|^{j}_{0}\\|f\\|_{\infty},\quad{f\in L^{0}\cap L^{\infty}},$
	$\displaystyle f^{*}(n)$	$\displaystyle\leq n^{j}2^{(j+1)/2}c_{j,\tau}\\|f\\|_{\tau/j},\quad{f\in L^{\tau/j}},$

where $f^{*}(n)=E\big{(}n,f\mathchar 24635\relax\;L^{0},L^{\infty}\big{)}=\inf\big{\{}\sigma\colon m(\sigma,f_{\sigma})\leq t\big{\}}$ for all $f\in{L^{\infty}}$ .

Example 3 (Applications to operator spectral approximations).

Consider the example commonly used in foundations of quantum systems for describing quantum states (see e.g. (Hall2013, , no 3.4)). Similar estimates of spectral approximations were early analyzed in the paper DL19 . In what follows, we consider the spaces $L^{p}=L^{p}(\mathbb{R}\mathchar 24635\relax\;\mathbb{R})$ of real-valued functions.

Let $H$ be a Hilbert complex space with the norm $\|\cdot\|_{H}={\langle\cdot\mid\cdot\rangle^{1/2}}$ and a self-adjoint unbounded linear operator $T$ with the dense domain $\mathcal{D}(T)\subset H$ is given. By spectral theorem the measurable function ${f}$ from $T$ can be well defined using the following spectral expansions (see e.g. Hall2013 )

T=\int_{\sigma(T)}\lambda\,\mu(d\lambda),\quad{f}(T)=\int_{\sigma(T)}{f}(\lambda)\,\mu(d\lambda),

where $\mu$ is a unique projection-valued measure determined on its spectrum $\sigma(T)\subset\mathbb{R}$ with values in the Banach space of bounded linear operators $\mathcal{L}(H)$ , which can be extended on $\mathbb{R}\setminus\sigma(A)$ as zero.

At beginning, we consider the case when for any Borel set $\varSigma\subset\mathbb{R}$ and any $\psi\in\mathcal{D}(A)$ the $\mathcal{L}(H)$ -valued measurable function

\varSigma\longmapsto f(\varSigma)=\int_{\varSigma}{f}(\lambda)\,\mu_{\psi}(d\lambda)

belongs to $L^{\infty}$ with respect to the positive measure $\mu_{\psi}(d\lambda):=\big{\langle}\mu(d\lambda)\psi\mid\psi\big{\rangle}$ , where $\|\psi\|_{H}=1$ .

Consider the corresponding quadratic forms $f_{\psi}(\varSigma)=\big{\langle}f(\varSigma)\psi\mid\psi\big{\rangle}$ as measurable functions from $L^{2}$ and let $f^{*}_{\psi}$ be the suitable decreasing rearrangement. By Theorem 11 the inequalities (6) and (5) in this case take the form

	$\displaystyle f_{\psi}^{*}(t)$	$\displaystyle\leq t^{1-1/\theta}C_{\theta,q}\\|f_{\psi}\\|_{q\theta^{2}/(1-\theta)},\quad{f_{\psi}\in L_{\mu}^{q\theta^{2}/(1-\theta)}},$		(31)
	$\displaystyle t^{-1+1/\theta}f_{\psi}^{*}(t)$	$\displaystyle\leq{C}_{\theta,q}\\|f_{\psi}\\|_{q\vartheta^{2}/(1-\theta)}\leq 2^{1/2\theta}\\|f_{\psi}\\|^{-1+1/\theta}_{0}\\|f_{\psi}\\|_{\infty}$		(32)

for all $f_{\psi}\in L^{0}_{\mu}\cap L^{\infty}_{\mu}$ , where $\|f_{\psi}\|_{\infty}=\sup_{\varSigma\subset\mathbb{R}}|f_{\psi}(\varSigma)|$ and $\|f_{\psi}\|_{0}=\mu_{\psi}\left(\operatorname{\operatorname{supp}}f_{\psi}\right)$ . Applying the estimation (28) from Corollary 8, we obtain

\displaystyle f_{\psi}^{*}(t)

\displaystyle\leq\frac{2}{\pi t}\,\int|f(\lambda)|\,\mu(d\lambda)\quad\text{for all}\quad f_{\psi}\in L^{1}_{\mu}.

(33)

The inequality (31), (33) give error estimations (in quadratic forms terms) of spectral approximations $f_{\psi}(T)={\langle f(T)\psi\mid\psi\rangle}$ by $f_{\psi}(\varSigma)$ with $\mu\left(\varSigma\right)\leq t$ . The inequalities (20) characterise the approximation accuracy.

In another case, using the $\mathcal{L}(H)$ -valued measurable uniformly bounded functions

\tilde{f}\colon\varSigma\longmapsto\int_{\varSigma}f(\lambda)\,\mu(d\lambda),\quad f\in L^{1}_{\mu}

belonging to $L^{1}_{\mu}(\mathbb{R},\mathcal{L}(H))$ , we same as above obtain the inequality

\displaystyle\tilde{f}^{*}(t)

\displaystyle\leq\frac{2}{\pi t}\Big{\|}\int f(\lambda)\,\mu(d\lambda)\Big{\|}_{\mathcal{L}(H)}.

Example 4 (Samples of numerical calculations).

The inequality (6) together with the exact value of the normalized constant ${c}_{s,\tau}$ are suitable for direct numerical calculations. Illustrate this on several graphs, taking as an example the inverse Gaussian distribution function.

Consider in $L_{\mu}^{\infty}$ with the Gaussian measure $\mu(d\tau)=(2\pi)^{-1/2}e^{-\tau^{2}/2}$ $(\tau\in\mathbb{R})$ two-parameter inverse Gaussian distribution with support on $(0,\infty)$ that is given by

f(t)=C\sqrt{\frac{l}{2\pi t^{3}}}\exp\left(-\frac{l(t-m)^{2}}{2m^{2}t}\right),\quad C,t>0

with the mean $m>0$ and the shape parameter $l>0$ . By Theorem 11

	$\displaystyle f^{*}(u)=E(u,f)$	$\displaystyle=\inf\left\{\\|f-f_{\sigma}\\|_{\infty}\colon m(\sigma,f_{\sigma})\leq u\right\},$
	$\displaystyle m(\sigma,f_{\sigma})$	$\displaystyle=\mu\left\{u\in(0,\infty)\colon{\|f_{\sigma}(u)\|>\sigma}\right\}$

coincides with the non-increasing rearrangement of $f$ on $(0,\infty)$ . As a result, we obtain

\displaystyle f^{*}(u)=E(u,f)

\displaystyle\leq{c}_{s,\tau}u^{-s}\,\|f\|_{\tau/s}\quad\text{for all}\quad{f\in L_{\mu}^{\tau/s}},

where $L_{\mu}^{\tau/s}=\mathcal{B}_{\tau}^{s}\big{(}L^{0}_{\mu},L^{\infty}_{\mu}\big{)}$ in accordance with Remark 2 and the equality (18).

The obtained results of numerical calculations are illustrated on Fig. 2,3.

Statements and Declarations: There are no conflicts and potential competing of interest to disclose.

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