Mathematical Sciences: Abstract Spectral Decompositions and Transference Methods in Modern Analysis
{'Code': '03040103', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': None, 'PO_EMAI': None, 'PO_PHON': None}
Professor Berkson conducts research on the interface of                 harmonic analysis, ergodic theory, and operator theory.  All                    three are important subdisciplines of Modern Analysis.  Harmonic                analysis, roughly speaking, embodies the interrelationships of                  mathematical analysis and group actions.  Historically, harmonic                analysis has its origins in Fourier series and partial                          differential equations.  The related field of ergodic theory                    originated in statistical mechanics.  Its thrust is in the                      understanding and exploitation of the concept of recurrence.                    Both ergodic theory and harmonic analysis, as throughout all of                 mathematical analysis, utilize methods and operations that are                  susceptible to broad abstract principles.  Modern operator                      theory, which arose in the development of the mathematical                      foundation of quantum physics, seeks to identify and explicate                  these abstract principles so as to unify and expand the frontiers               of science.  The subject matter of this proposal is operator                    theory in this conceptual sense, with emphasis on applications to               ergodic theory and harmonic analysis.                                                   Professor Berkson proposes to continue his development of an            abstract operator theory that encompasses weakened forms of                     orthogonality (such as Fourier inversion and triangular                         truncation in noncommutative analysis).  The approach is based                  upon the notions of trigonometrically well-bounded operator and                 spectral family of projections which are used in the general                    settings of analysis to replace, respectively, the unitary                      operators and projection-valued measures central to Hilbert space               operator theory and its physical applications.  In prior                        collaborative research, Professor Berkson has developed these                   notions, along with extensive applications to commutative and                   noncommutative analysis.  The current investigations will expand                and sharpen the abstract operator theory along these lines and                  its companion vector-valued transference methods, while extending               its scope to such topics as representations of locally compact                  abelian groups, Fourier multiplier transference, ergodic theory,                and generalized analyticity.
{'FirstName': 'Earl', 'LastName': 'Berkson', 'PI_MID_INIT': 'R', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Earl R Berkson', 'EmailAddress': 'berkson@math.uiuc.edu', 'PI_PHON': '2173331261', 'NSF_ID': '000450831', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Illinois at Urbana-Champaign', 'CityName': 'URBANA', 'ZipCode': '618013620', 'PhoneNumber': '2173332187', 'StreetAddress': '506 S WRIGHT ST', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Illinois', 'StateCode': 'IL', 'CONGRESSDISTRICT': '13', 'CONGRESS_DISTRICT_ORG': 'IL13', 'ORG_UEI_NUM': 'Y8CWNJRCNN91', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF ILLINOIS', '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': '1262', 'Text': 'MODERN ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1987&All=true', 'xml': '8701627.xml'}