# PIP-Space Valued Reproducing Pairs of Measurable Functions

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Remark**

**1.**

## 3. The Case $\mathit{Y}=\mathcal{H}$ and $\mathit{Z}={\mathit{L}}^{\mathbf{2}}(\mathit{X},\mathit{\mu})$, Both Hilbert Spaces

**Proposition**

**1.**

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

**Theorem**

**2.**

## 4. The Case $\mathit{Y}=\mathcal{H}$, a Hilbert Space, and $\mathit{Z}=\mathit{V}$, a PIP-Space

#### 4.1. The Case Where Z Is a Rigged Hilbert Space

**Theorem**

**3.**

#### 4.2. The Case Where Z Is a LHS/LBS

- ($\mathsf{p}$)
- $\exists \phantom{\rule{0.166667em}{0ex}}p\in J$ such that ${C}_{\psi}f=\langle f|{\psi}_{\xb7}\rangle \in {V}_{p}$ and ${C}_{\varphi}g=\langle g|{\varphi}_{\xb7}\rangle \in {V}_{\overline{p}},\forall \phantom{\rule{0.166667em}{0ex}}f,g\in \mathcal{H}$.

**Theorem**

**4.**

**Proposition**

**2.**

## 5. The Case Where $\mathit{Y}$ and $\mathit{Z}$ Are Both PIP-Spaces

**Definition**

**1.**

#### 5.1. Construction of Coefficient Spaces

**Proposition**

**3.**

**Proof.**

#### 5.2. Compatible Pairs

**Definition**

**2.**

**Theorem**

**5.**

- (i)
- ($\psi ,\varphi $) is a compatible pair;
- (ii)
- The operator ${S}_{\psi ,\varphi}:{Y}_{\overline{u}}\to {Y}_{v}$ is bounded with bounded inverse, that is, ($\psi ,\varphi $) is a reproducing pair.

**Proof.**

#### 5.3. The General Case

#### 5.4. Comparison with the Case $Y=\mathcal{H}$

- ($\mathsf{k}$)
- If ${\xi}_{n}\to \xi $ in ${V}_{r}$, then, for every compact subset $K\subset X$, there exists a subsequence $\{{\xi}_{n}^{K}\}$ of $\{{\xi}_{n}\}$ which converges to $\xi $ almost everywhere in K.

**Proposition**

**4.**

- (i)
- ${C}_{\psi}:{D}_{r,\overline{u}}({C}_{\psi})\to {V}_{r}$ is a closed linear map.
- (ii)
- If for some $r\in J$, ${C}_{\psi}({Y}_{\overline{u}})\subset {V}_{r}$, then ${C}_{\psi}:{Y}_{\overline{u}}\to {V}_{r}$ is continuous.
- (iii)
- If ${C}_{\psi}({Y}_{\overline{u}})\subset {V}_{p}$ and ${C}_{\varphi}({Y}_{\overline{v}})\subset {V}_{\overline{p}}$, the form Ω is bounded on ${Y}_{\overline{u}}\times {Y}_{\overline{v}}$, that is, $|{\mathsf{\Omega}}_{\psi ,\varphi}(f,g)|\le c{\u2225f\u2225}_{\overline{u}}\phantom{\rule{0.166667em}{0ex}}{\u2225g\u2225}_{\overline{v}}$.

**Proof.**

## 6. Examples

#### 6.1. A Purely Hilbertian Reproducing Pair

#### 6.2. A Reproducing Pair with a PIP-Space Target

#### 6.3. A Reproducing Pair with Two PIP-Spaces

- ${\psi}_{x}\in {\mathcal{H}}_{l}$ and ${\varphi}_{x}\in {\mathcal{H}}_{\overline{l}}$; for instance,$$\begin{array}{c}\hfill {\left(\right)}_{{\psi}_{x}}^{}l2\\ ={\int}_{X}|{\psi}_{x}{|}^{2}\phantom{\rule{0.166667em}{0ex}}{m}^{2l}(x)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{\mu}_{K}(x)={\int}_{X}{|{k}_{x}|}^{2}\phantom{\rule{0.166667em}{0ex}}{m}^{-2n}(x)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{\mu}_{K}(x)\hfill \end{array}$$
- $\forall f\in {\mathcal{H}}_{\overline{l}},\phantom{\rule{0.166667em}{0ex}}{C}_{\psi}f={\langle f|{\psi}_{x}\rangle}_{K}=f\phantom{\rule{0.166667em}{0ex}}{m}^{-(l+n})\in {\mathcal{H}}_{n}$.
- $\forall g\in {\mathcal{H}}_{l},\phantom{\rule{0.166667em}{0ex}}{C}_{\varphi}g={\langle g|{\varphi}_{x}\rangle}_{K}=g\phantom{\rule{0.166667em}{0ex}}{m}^{(l+n)}\in {\mathcal{H}}_{\overline{n}}\phantom{\rule{0.166667em}{0ex}}.$

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Lattices of Banach or Hilbert Spaces and Operators on Them

#### Appendix A.1. Lattices of Banach or Hilbert Spaces

. involution: | ${V}_{r}$ | $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\leftrightarrow \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$ | ${V}_{\overline{r}}={V}_{r}^{\times}$, the conjugate dual of ${V}_{r}$ |

. infimum: | ${V}_{p\wedge q}$ | $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}:=\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$ | ${V}_{p}\wedge {V}_{q}={V}_{p}\cap {V}_{q}$ |

. supremum: | ${V}_{p\vee q}$ | $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}:=\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$ | ${V}_{p}\vee {V}_{q}={V}_{p}+{V}_{q}$. |

- (i)
- $\mathcal{I}$ contains a unique self-dual, Hilbert subspace ${V}_{o}={V}_{\overline{o}}$.
- (ii)
- for every ${V}_{r}\in \mathcal{I}$, the norm ${\parallel \xb7\parallel}_{\overline{r}}$ on ${V}_{\overline{r}}={V}_{r}^{\times}$ is the conjugate of the norm ${\parallel \xb7\parallel}_{r}$ on ${V}_{r}$.

- .
- $({A}_{\mathrm{p}}f)(x)={(1+{x}^{2})}^{1/2}f(x)$ yields the Fourier transform of the Sobolev spaces ${H}^{s}(\mathbb{R}),\phantom{\rule{0.166667em}{0ex}}s\in \mathbb{Z}$.
- .
- $({A}_{\mathrm{m}}f)(x)={(1-\frac{{d}^{2}}{d{x}^{2}})}^{1/2}f(x)$ yields the Sobolev spaces ${H}^{s}(\mathbb{R}),\phantom{\rule{0.166667em}{0ex}}s\in \mathbb{Z}$.
- .
- $({A}_{\mathrm{osc}}f)(x)=(1+{x}^{2}-\frac{{d}^{2}}{d{x}^{2}})f(x)$ yields the harmonic oscillator representation of the Schwartz space $\mathcal{S}(\mathbb{R})$ of smooth functions of fast decay and its conjugate dual ${\mathcal{S}}^{\times}(\mathbb{R})$, the space of of tempered distributions.

#### Appendix A.2. Operators on LBSs and LHSs

- (i)
- $\mathcal{D}(A)={\bigcup}_{q\in \mathsf{d}(A)}{V}_{q}$, where $\mathsf{d}(A)$ is a nonempty subset of J;
- (ii)
- For every $q\in \mathsf{d}(A)$, there exists $p\in J$ such that the restriction of A to ${V}_{q}$ is a continuous linear map into ${V}_{p}$ (we denote this restriction by ${A}_{pq})$;
- (iii)
- A has no proper extension satisfying (i) and (ii).

- (i)
- Adjoint: every $A\in \mathrm{Op}({V}_{J})$ has a unique adjoint ${A}^{\times}\in \mathrm{Op}{V}_{J})$, defined by$$\langle {A}^{\times}y|x\rangle =\langle y|Ax\rangle ,\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.166667em}{0ex}}x\in {V}_{q},\phantom{\rule{0.166667em}{0ex}}y\in {V}_{\overline{p}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}{A}_{pq}\phantom{\rule{0.166667em}{0ex}}\mathrm{exists}.$$
- (ii)
- Partial multiplication: Let $A,B\in \mathrm{Op}({V}_{J})$. The product $BA$ is defined if and only if there is a continuous factorization through some ${V}_{r}$:$${V}_{q}\phantom{\rule{0.277778em}{0ex}}\stackrel{A}{\to}\phantom{\rule{0.277778em}{0ex}}{V}_{r}\phantom{\rule{0.277778em}{0ex}}\stackrel{B}{\to}\phantom{\rule{0.277778em}{0ex}}{V}_{p},\phantom{\rule{1.em}{0ex}}\mathrm{i}.\mathrm{e}.,\phantom{\rule{1.em}{0ex}}{(BA)}_{pq}={B}_{pr}{A}_{rq}.$$

- (i)
- Symmetric operators, defined as those operators satisfying the relation ${A}^{\times}=A$, since these are the ones that may generate self-adjoint operators in the central Hilbert space [7] (Section 3.3).
- (ii)
- Invertible operators: A is invertible if it has at least one invertible representative; in that case, A has a unique inverse operator ${A}^{-1}\in \mathrm{Op}({V}_{J})$ [28].

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Antoine, J.-P.; Trapani, C.
PIP-Space Valued Reproducing Pairs of Measurable Functions. *Axioms* **2019**, *8*, 52.
https://doi.org/10.3390/axioms8020052

**AMA Style**

Antoine J-P, Trapani C.
PIP-Space Valued Reproducing Pairs of Measurable Functions. *Axioms*. 2019; 8(2):52.
https://doi.org/10.3390/axioms8020052

**Chicago/Turabian Style**

Antoine, Jean-Pierre, and Camillo Trapani.
2019. "PIP-Space Valued Reproducing Pairs of Measurable Functions" *Axioms* 8, no. 2: 52.
https://doi.org/10.3390/axioms8020052