Mathematical Sciences: Symmetry and Orthogonal Polynomials
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Several areas within the theory of special functions will be               investigated.  Particular emphasis will be placed on the                        development of a mathematical theory of orthogonal polynomials of               several variables in which there will be a concentration on                     functions defined on the n-sphere.  These polynomials are taken                 to be orthogonal with respect to a weight function which is                     invariant under a finite reflection group.  The weight functions                are powers of products of linear functions whose zero sets are                  the mirrors for the reflections in the group.  A form of                        Selberg's integral, corresponding to the octahedral group, is an                important special case.                                                              By means of a commutative algebra of first-order                           differential-difference operators, which have an action analogous               to that of partial derivatives in the theory of spherical                       harmonics, work will focus on three problems.  The first is to                  construct a nonabelian fractional integral operator which                       intertwines ordinary differentiation and the differential-                      difference operators.  This gives a mapping from harmonic                       functions and other eigenfunctions of the Laplacian to the                      functions under investigation.  Secondly, by use of these                       operators and their adjoints, work will be done constructing                    explicit orthogonal bases and norms for functions on the sphere.                Emphasis will be placed first on three and four dimensional                     examples.  Thirdly, work will be done extending a recent                        positivity result on the Poisson kernel for the ball to finding                 this kernel explicitly for the general reflection group.                             This work is expected to have application to theoretical                   physics.  It has already demonstrated its utility by playing a                  major role in the grinding of mirrors for the new Keck 10-meter                 telescope.
{'FirstName': 'Charles', 'LastName': 'Dunkl', 'PI_MID_INIT': 'F', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Charles F Dunkl', 'EmailAddress': 'cfd5z@virginia.edu', 'NSF_ID': '000335813', 'StartDate': '05/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Virginia Main Campus', 'CityName': 'CHARLOTTESVILLE', 'ZipCode': '229034833', 'PhoneNumber': '4349244270', 'StreetAddress': '1001 EMMET ST N', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Virginia', 'StateCode': 'VA', 'CONGRESSDISTRICT': '05', 'CONGRESS_DISTRICT_ORG': 'VA05', 'ORG_UEI_NUM': 'JJG6HU8PA4S5', 'ORG_LGL_BUS_NAME': 'RECTOR & VISITORS OF THE UNIVERSITY OF VIRGINIA', 'ORG_PRNT_UEI_NUM': None}
{'Name': None, 'CityName': None, 'StateCode': None, 'ZipCode': None, 'StreetAddress': None, 'CountryCode': None, 'CountryName': None, 'StateName': None, 'CountryFlag': None, 'CONGRESSDISTRICT': None, 'CONGRESS_DISTRICT_PERF': '""'}
{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8802400.xml'}