The subject matter of this paper is an integral with exponential oscillation of phase f（x） weighted by g（x） on a finite interval [α, β]. When the phase f（x） has a single stationary point in （α,β）, an nth-o...The subject matter of this paper is an integral with exponential oscillation of phase f（x） weighted by g（x） on a finite interval [α, β]. When the phase f（x） has a single stationary point in （α,β）, an nth-order asymptotic expansion of this integral is proved for n ≥ 2. This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f（x） and g（x） are smooth and g（x） is compactly supported on R. In the present paper, however, these functions are only assumed to be continuously differentiable on [α,β] 2n ＋ 3 and 2n ＋ 1 times, respectively. Because there are no requirements on the vanishing of g（x） and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.展开更多
Let W be a real symplectic space and(G,G')an irreducible dual pair in Sp(W),in the sense of Howe,with G compact.Let G be the preimage of G in the metaplectic group Sp(W).Given an irreducible unitary representation...Let W be a real symplectic space and(G,G')an irreducible dual pair in Sp(W),in the sense of Howe,with G compact.Let G be the preimage of G in the metaplectic group Sp(W).Given an irreducible unitary representation II of G that occurs in the restriction of the Weil representation to G,let On denote its character.We prove that,for a suitable embedding T of Sp(W)in the space of tempered distributions on W,the distribution T(On)admits an asymptotic limit,and the limit is a nilpotent orbital integral.As an application,we compute the wave front set of II',the representation of G dual to II,by elementary means.展开更多
基金The authors would like to thank anonymous referees for critical readings and thoughtful comments. Gratitude is also due to Xiumin Ren who gave the authors many helpful suggestions. The second author was partially supported by the National Natural Science Foundation of China （Grant No. 11601271）.
文摘The subject matter of this paper is an integral with exponential oscillation of phase f（x） weighted by g（x） on a finite interval [α, β]. When the phase f（x） has a single stationary point in （α,β）, an nth-order asymptotic expansion of this integral is proved for n ≥ 2. This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f（x） and g（x） are smooth and g（x） is compactly supported on R. In the present paper, however, these functions are only assumed to be continuously differentiable on [α,β] 2n ＋ 3 and 2n ＋ 1 times, respectively. Because there are no requirements on the vanishing of g（x） and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.
基金the University of Oklahoma for hospitality and financial supporthospitality and financial support from the Université de Lorraine+1 种基金partial support from NSA (Grant No. H98230-13-1-0205)NSF (Grant No. DMS-2225892)
文摘Let W be a real symplectic space and(G,G')an irreducible dual pair in Sp(W),in the sense of Howe,with G compact.Let G be the preimage of G in the metaplectic group Sp(W).Given an irreducible unitary representation II of G that occurs in the restriction of the Weil representation to G,let On denote its character.We prove that,for a suitable embedding T of Sp(W)in the space of tempered distributions on W,the distribution T(On)admits an asymptotic limit,and the limit is a nilpotent orbital integral.As an application,we compute the wave front set of II',the representation of G dual to II,by elementary means.
