The k-Hankel transform <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$F_{k,1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mrow> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> (or the (k, 1)-generalized Fourier transform) is the Dunkl analogue of the unitary inversion operator in the minimal representation of a conformal group initiated by T. Kobayashi and G. Mano. It is one of the two most significant cases in (k, a)-generalized Fourier transforms. We will establish a positive radial product formula for the integral kernels of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$F_{k,1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mrow> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>. Such a product formula is equivalent to a representation of the generalized spherical mean operator in terms of the probability measure <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma _{x,t}^{k,1}(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>σ</mi> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We will then study the representing measure <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma _{x,t}^{k,1}(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>σ</mi> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and analyze the support of this measure, and derive a weak Huygens’s principle for the deformed wave equation in (k, 1)-generalized Fourier analysis.

A positive product formula of integral kernels of k-Hankel transforms

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2026年5月15日

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Wentao Teng

摘要

The k-Hankel transform $F_{k,1}$ $F_{k, 1}$ (or the (k, 1)-generalized Fourier transform) is the Dunkl analogue of the unitary inversion operator in the minimal representation of a conformal group initiated by T. Kobayashi and G. Mano. It is one of the two most significant cases in (k, a)-generalized Fourier transforms. We will establish a positive radial product formula for the integral kernels of $F_{k,1}$ $F_{k, 1}$ . Such a product formula is equivalent to a representation of the generalized spherical mean operator in terms of the probability measure $\sigma _{x,t}^{k,1}(\xi )$ $σ_{x, t}^{k, 1} (ξ)$ . We will then study the representing measure $\sigma _{x,t}^{k,1}(\xi )$ $σ_{x, t}^{k, 1} (ξ)$ and analyze the support of this measure, and derive a weak Huygens’s principle for the deformed wave equation in (k, 1)-generalized Fourier analysis.