A multivariate version of the disk convolution

详细信息    查看全文

• 作者：Margit Rö ; sler^a ; ^{roesler@math.upb.de" class="auth_mail" title="E-mail the corresponding author} ; Michael Voit^b ; ^{michael.voit@math.tu-dortmund.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Hypergeometric functions associated with root systems ; Heckman&ndash ; Opdam theory ; Disk hypergroups ; Positive product formulas ; Compact Grassmann manifolds ; Spherical functions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：435
• 期：1
• 页码：701-717
• 全文大小：435 K

文摘

We present an explicit product formula for the spherical functions of the compact Gelfand pairs class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010057&_mathId=si1.gif&_user=111111111&_pii=S0022247X15010057&_rdoc=1&_issn=0022247X&md5=fc69afd712f9851329b8944c301ffe39" title="Click to view the MathML source">(G,K₁)=(SU(p+q),SU(p)×SU(q))class="mathContainer hidden">class="mathCode">$(G,K_1)=(\mathit{SU}(p+q),\mathit{SU}(p)\times \mathit{SU}(q))$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010057&_mathId=si167.gif&_user=111111111&_pii=S0022247X15010057&_rdoc=1&_issn=0022247X&md5=31bf8352d2789a5a5a98ce6ca9b279d6" title="Click to view the MathML source">p≥2qclass="mathContainer hidden">class="mathCode">$p\ge 2q$, which can be considered as the elementary spherical functions of one-dimensional K  -type for the Hermitian symmetric spaces class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010057&_mathId=si3.gif&_user=111111111&_pii=S0022247X15010057&_rdoc=1&_issn=0022247X&md5=effd0e61a7544b28841f8869bbb2b802" title="Click to view the MathML source">G/Kclass="mathContainer hidden">class="mathCode">$G/K$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010057&_mathId=si4.gif&_user=111111111&_pii=S0022247X15010057&_rdoc=1&_issn=0022247X&md5=59cf13451ea617b1cc2a5317fa079a98" title="Click to view the MathML source">K=S(U(p)×U(q))class="mathContainer hidden">class="mathCode">$K=S(U(p)\times U(q))$. Due to results of Heckman, they can be expressed in terms of Heckman–Opdam Jacobi polynomials of type class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010057&_mathId=si5.gif&_user=111111111&_pii=S0022247X15010057&_rdoc=1&_issn=0022247X&md5=ab6ddbc31938d3d205c7b57b54b12d48" title="Click to view the MathML source">BC_qclass="mathContainer hidden">class="mathCode">${\mathit{BC}}_q$ with specific half-integer multiplicities. By analytic continuation with respect to the multiplicity parameters we obtain positive product formulas for the extensions of these spherical functions as well as associated compact and commutative hypergroup structures parametrized by real class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010057&_mathId=si236.gif&_user=111111111&_pii=S0022247X15010057&_rdoc=1&_issn=0022247X&md5=e7bf33f4ba2e0c48dcb2d6191786f6b8" title="Click to view the MathML source">p∈]2q−1,∞[class="mathContainer hidden">class="mathCode">$p\in ]2q-1,\infty [$. We also obtain explicit product formulas for the involved continuous two-parameter family of Heckman–Opdam Jacobi polynomials with regular, but not necessarily positive multiplicities. The results of this paper extend well known results for the disk convolutions for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15010057&_mathId=si7.gif&_user=111111111&_pii=S0022247X15010057&_rdoc=1&_issn=0022247X&md5=e5cc0b6bbc4fcd068153324320bbcd6d" title="Click to view the MathML source">q=1class="mathContainer hidden">class="mathCode">$q=1$ to higher rank.