{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T15:45:24Z","timestamp":1771947924885,"version":"3.50.1"},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12611,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,9]]},"abstract":"<jats:p>Let <jats:italic>G<\/jats:italic> be a locally compact abelian group, \u0393 its dual group, and \u03bc its Haar measure. For a function <jats:italic>f<\/jats:italic>: <jats:italic>G<\/jats:italic> \u2192 <jats:italic>C<\/jats:italic>, the Fourier transform <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048337_inline01\"\/> of <jats:italic>f<\/jats:italic> is defined by<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200048337_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>for every \u03b3 \u2208 \u0393.<\/jats:p><jats:p>We extend this definition for a function <jats:italic>f<\/jats:italic> on <jats:italic>G<\/jats:italic>, whose values are pairwise commutable normal operators in a Hilbert space. Then we study harmonic analysis for this extended Fourier transform.<\/jats:p><jats:p>Our method is Boolean valued analysis as introduced in [11] and [12]. Instead of developing the theory in a step-by-step manner, we shall develop a general machinery showing how to transform classical theorems to theorems in our situation.<\/jats:p><jats:p>In Chapter 1, we summarize the basic knowledge on Hilbert space, on Boolean valued model of set theory, and on Boolean valued analysis. In Chapter 2, we develop the theory of integration. Since we deal with unbounded operators as well as bounded operators, we need a new theory of integration. For a separable Hilbert space, the value of our integration coincides with the value of the usual integration with an adequate generalization for unbounded operators.<\/jats:p><jats:p>In Chapter 3, we establish machinery for our transfer principle and carry out the transformation of many classical theorems to theorems for our case.<\/jats:p>","DOI":"10.2307\/2273134","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:50:56Z","timestamp":1146952256000},"page":"417-440","source":"Crossref","is-referenced-by-count":20,"title":["A transfer principle in harmonic analysis"],"prefix":"10.1017","volume":"44","author":[{"given":"Gaisi","family":"Takeuti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048337_ref014","volume-title":"L'int\u00e9gration dans les groupes topologiques et ses applications","author":"Weil","year":"1938"},{"key":"S0022481200048337_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-8751-0"},{"key":"S0022481200048337_ref010","article-title":"Note on integration","volume":"34","author":"Stone","year":"1948","journal-title":"Proceedings of the National Academy of Sciences of the United States of America, I, II, III"},{"key":"S0022481200048337_ref009","doi-asserted-by":"publisher","DOI":"10.2307\/1970860"},{"key":"S0022481200048337_ref005","volume-title":"Functional analysis and semi-groups","author":"Hille","year":"1957"},{"key":"S0022481200048337_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9440-2"},{"key":"S0022481200048337_ref001","volume-title":"Boolean-valued models and independence proofs in set theory","author":"Bell","year":"1978"},{"key":"S0022481200048337_ref012","volume-title":"Proceedings of the 1977 Durham Research Symposium on Applications of Sheaf Theory to Logic, Algebra and Analysis, Lecture Notes in Mathematics","author":"Takeuti"},{"key":"S0022481200048337_ref007","volume-title":"The Haar integral","author":"Nachbin","year":"1965"},{"key":"S0022481200048337_ref003","volume-title":"Fourier transforms","author":"Goldberg","year":"1970"},{"key":"S0022481200048337_ref011","volume-title":"Two applications of logic to mathematics","author":"Takeuti","year":"1978"},{"key":"S0022481200048337_ref008","volume-title":"Fourier analysis on groups","author":"Rudin","year":"1962"},{"key":"S0022481200048337_ref006","volume-title":"An introduction to abstract harmonic analysis","author":"Loomis","year":"1953"},{"key":"S0022481200048337_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/1967495"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048337","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T20:52:40Z","timestamp":1558903960000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048337\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,9]]},"references-count":14,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1979,9]]}},"alternative-id":["S0022481200048337"],"URL":"https:\/\/doi.org\/10.2307\/2273134","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,9]]}}}