<p>In this paper, we develop an overcomplete family of states (OFS) on a separable Hilbert space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> using a subgroup <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of the affine group <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$E_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> from a square integrable representation of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$F_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. By utilizing the unitary equivalence of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> with a closed subspace <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^{2}(F_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we show that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation> is a reproducing kernel Hilbert space (RKHS) consisting of complex valued, bounded, continuous and square integrable functions. An appropriate choice of fiducial vector is made using position and momentum operators to ensure the smoothness of elements of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {K}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">K</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> It is also shown that any bounded operator on <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> is an integral operator on <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation> upto unitary equivalence.</p>

Continuous representation of a Hilbert space using a subgroup of the affine group of $\mathbb {R}^n$

期刊论文
发表日期：

2026年3月12日

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K. J. Vaishakh,
Noufal Asharaf

摘要

In this paper, we develop an overcomplete family of states (OFS) on a separable Hilbert space $\mathcal {H}$ $H$ using a subgroup $F_{n}$ $F_{n}$ of the affine group $E_{n}$ $E_{n}$ from a square integrable representation of $F_{n}$ $F_{n}$ . By utilizing the unitary equivalence of $\mathcal {H}$ $H$ with a closed subspace $\mathcal {K}$ $K$ of $L^{2}(F_n)$ $L^{2} (F_{n})$ , we show that $\mathcal {K}$ $K$ is a reproducing kernel Hilbert space (RKHS) consisting of complex valued, bounded, continuous and square integrable functions. An appropriate choice of fiducial vector is made using position and momentum operators to ensure the smoothness of elements of $\mathcal {K}.$ $K .$ It is also shown that any bounded operator on $\mathcal {H}$ $H$ is an integral operator on $\mathcal {K}$ $K$ upto unitary equivalence.

Continuous representation of a Hilbert space using a subgroup of the affine group of \(\mathbb {R}^n\)

摘要