{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,26]],"date-time":"2025-03-26T14:19:44Z","timestamp":1742998784970,"version":"3.40.3"},"publisher-location":"Cham","reference-count":54,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783031052958"},{"type":"electronic","value":"9783031052965"}],"license":[{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022]]},"DOI":"10.1007\/978-3-031-05296-5_3","type":"book-chapter","created":{"date-parts":[[2022,7,27]],"date-time":"2022-07-27T20:28:34Z","timestamp":1658953714000},"page":"51-125","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Whittaker Convolution"],"prefix":"10.1007","author":[{"given":"R\u00faben","family":"Sousa","sequence":"first","affiliation":[]},{"given":"Manuel","family":"Guerra","sequence":"additional","affiliation":[]},{"given":"Semyon","family":"Yakubovich","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,19]]},"reference":[{"key":"3_CR7","doi-asserted-by":"publisher","first-page":"124010","DOI":"10.1103\/PhysRevD.59.124010","volume":"59","author":"C Barrab\u00e8s","year":"1999","unstructured":"C. Barrab\u00e8s, V. Frolov, R. Parentani, Metric fluctuation corrections to Hawking radiation. Phys. Rev. D 59, 124010 (1999)","journal-title":"Phys. Rev. D"},{"key":"3_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-66128-0","volume-title":"Potential Theory on Locally Compact Abelian Groups","author":"C Berg","year":"1975","unstructured":"C. Berg, G. Forst, Potential Theory on Locally Compact Abelian Groups (Springer, Berlin, 1975)"},{"key":"3_CR19","volume-title":"Harmonic Analysis of Probability Measures on Hypergroups","author":"WR Bloom","year":"1994","unstructured":"W.R. Bloom, H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups (Walter de Gruyter, Berlin, 1994)"},{"key":"3_CR21","doi-asserted-by":"crossref","unstructured":"A.N. Borodin, Stochastic Processes (Birkh\u00e4user, Cham, 2017)","DOI":"10.1007\/978-3-319-62310-8"},{"key":"3_CR30","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/BFb0060609","volume-title":"Th\u00e9orie du Potentiel et Analyse Harmonique","author":"H Chebli","year":"1974","unstructured":"H. Chebli, Op\u00e9rateus de translation g\u00e9n\u00e9ralis\u00e9e et semi-groupes de convolution, in Th\u00e9orie du Potentiel et Analyse Harmonique, ed. by J. Faraut (Springer, Berlin, 1974), pp. 35\u201359"},{"key":"3_CR39","unstructured":"W.C. Connett, A.L. Schwartz, Subsets of $$\\mathbb {R}$$ which support hypergroups with polynomial characters. J. Comput. Appl. Math. 65, 73\u201384 (1995)"},{"key":"3_CR42","volume-title":"One-parameter Semigroups","author":"EB Davies","year":"1980","unstructured":"E.B. Davies, One-parameter Semigroups (Academic Press, London, 1980)"},{"issue":"1","key":"3_CR47","doi-asserted-by":"publisher","first-page":"179","DOI":"10.4171\/RMI\/292","volume":"17","author":"C Donati-Martin","year":"2001","unstructured":"C. Donati-Martin, R. Ghomrasni, M. Yor, On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options. Rev. Mat. Iberoam. 17(1), 179\u2013193 (2001)","journal-title":"Rev. Mat. Iberoam."},{"key":"3_CR55","volume-title":"Higher Transcendental Functions","author":"A Erd\u00e9lyi","year":"1953","unstructured":"A. Erd\u00e9lyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, vol. I (McGraw-Hill, New York, 1953)"},{"key":"3_CR56","volume-title":"Higher Transcendental Functions","author":"A Erd\u00e9lyi","year":"1953","unstructured":"A. Erd\u00e9lyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. II (McGraw\u2013Hill, New York, 1953)"},{"key":"3_CR60","volume-title":"An Introduction to Probability Theory and its Applications","author":"W Feller","year":"1971","unstructured":"W. Feller, An Introduction to Probability Theory and its Applications, 2nd edn. (Wiley, New York, 1971)","edition":"2"},{"issue":"1\u20132","key":"3_CR61","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/BF02388521","volume":"11","author":"M Flensted-Jensen","year":"1973","unstructured":"M. Flensted-Jensen, T. Koornwinder, The convolution structure for Jacobi function expansions. Ark. Mat. 11(1\u20132), 245\u2013262 (1973)","journal-title":"Ark. Mat."},{"key":"3_CR71","volume-title":"Commutative Normed Rings","author":"I Gelfand","year":"1964","unstructured":"I. Gelfand, D. Raikov, G. Shilov, Commutative Normed Rings (Chelsea, New York, 1964)"},{"key":"3_CR77","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0063135","volume-title":"Bessel Polynomials","author":"E Grosswald","year":"1978","unstructured":"E. Grosswald, Bessel Polynomials (Springer, Berlin, 1978)"},{"issue":"8","key":"3_CR85","doi-asserted-by":"publisher","first-page":"590","DOI":"10.1080\/10652469.2017.1330825","volume":"28","author":"PV Hoang","year":"2017","unstructured":"P.V. Hoang, T. Tuan, N.X. Thao, V.K. Tuan, Boundedness in weighted L p spaces for the Kontorovich\u2013Lebedev\u2013Fourier generalized convolutions and applications. Integr. Transf. Spec. Funct. 28(8), 590\u2013604 (2017)","journal-title":"Integr. Transf. Spec. Funct."},{"issue":"1","key":"3_CR86","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1016\/j.jmaa.2016.03.052","volume":"440","author":"NT Hong","year":"2016","unstructured":"N.T. Hong, P.V. Hoang, V.K. Tuan, The convolution for the Kontorovich-Lebedev transform revisited. J. Math. Anal. Appl. 440(1), 369\u2013378 (2016)","journal-title":"J. Math. Anal. Appl."},{"key":"3_CR97","first-page":"53","volume":"2","author":"VA Kakichev","year":"1967","unstructured":"V.A. Kakichev, On the convolution for integral transforms. Izv. Vyssh. Uchebn. Zaved. Mat. 2, 53\u201362 (1967) (in Russian)","journal-title":"Izv. Vyssh. Uchebn. Zaved. Mat."},{"key":"3_CR102","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-56402-5","volume-title":"Probability Theory\u2014A Comprehensive Course","author":"A Klenke","year":"2020","unstructured":"A. Klenke, Probability Theory\u2014A Comprehensive Course, 3rd edn. (Springer, London, 2020)","edition":"3"},{"key":"3_CR109","doi-asserted-by":"publisher","first-page":"100","DOI":"10.1090\/S0002-9947-1949-0028473-1","volume":"65","author":"HL Krall","year":"1949","unstructured":"H.L. Krall, O. Frink, A new class of orthogonal polynomials: the Bessel polynomials. Trans. Am. Math. Soc. 65, 100\u2013115 (1949)","journal-title":"Trans. Am. Math. Soc."},{"key":"3_CR111","unstructured":"M.G. Krein, Integral equations on a half-line with kernel depending on the difference of the arguments. Am. Math. Soc. Transl. (2) 22, 163\u2013288 (1962)"},{"key":"3_CR113","doi-asserted-by":"crossref","unstructured":"B.J. Laurenzi, Derivatives of Whittaker functions W \u03ba,1\u22152 and M \u03ba,1\u22152 with respect to order \u03ba. Math. Comp. 27(121), 129\u2013132 (1973)","DOI":"10.1090\/S0025-5718-1973-0364694-3"},{"key":"3_CR114","first-page":"621","volume":"65","author":"NN Lebedev","year":"1949","unstructured":"N.N. Lebedev, Some singular integral equations connected with the integral representations of mathematical physics. Dokl. Akad. Nauk SSSR 65, 621\u2013624 (1949) (in Russian)","journal-title":"Dokl. Akad. Nauk SSSR"},{"issue":"6","key":"3_CR124","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1287\/opre.1040.0113","volume":"52","author":"V Linetsky","year":"2004","unstructured":"V. Linetsky, Spectral expansions for Asian (average price) options. Oper. Res. 52(6), 856\u2013867 (2004)","journal-title":"Oper. Res."},{"key":"3_CR125","unstructured":"J.V. Linnik, I.V. Ostrovskii\u030c, Decomposition of Random Variables and Vectors (American Mathematical Society, Providence, 1977)"},{"key":"3_CR128","volume-title":"The Special Functions and Their Approximations","author":"YL Luke","year":"1969","unstructured":"Y.L. Luke, The Special Functions and Their Approximations, vol. I (Academic Press, New York and London, 1969)"},{"key":"3_CR132","unstructured":"P. Maroni, Une th\u00e9orie alg\u00e9brique des polyn\u00f4mes orthogonaux. Application aux polyn\u00f4mes orthogonaux semi-classiques, in Orthogonal Polynomials and Their Applications. IMACS Ann. Comput. Appl. Math., vol. 9 (Baltzer, Basel, 1991), pp. 95\u2013130"},{"issue":"1\u20132","key":"3_CR133","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1016\/0377-0427(93)90319-7","volume":"48","author":"P Maroni","year":"1993","unstructured":"P. Maroni, Variations around classical orthogonal polynomials. Connected problems. J. Comput. Appl. Math. 48(1\u20132), 133\u2013155 (1993)","journal-title":"Connected problems. J. Comput. Appl. Math."},{"key":"3_CR135","unstructured":"L. Mattner, Complex differentiation under the integral. Nieuw Arch. Wiskd. (5) 2(1), 32\u201335 (2001)"},{"key":"3_CR138","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780199215256.001.0001","volume-title":"Stochastic Integration Theory","author":"P Medvegyev","year":"2007","unstructured":"P. Medvegyev, Stochastic Integration Theory (Oxford University Press, Oxford, 2007)"},{"volume-title":"NIST Handbook of Mathematical Functions","year":"2010","key":"3_CR145","unstructured":"F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (eds.), NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010)"},{"key":"3_CR146","doi-asserted-by":"publisher","DOI":"10.1016\/B978-1-4832-0022-4.50006-5","volume-title":"Probability Measures on Metric Spaces","author":"KR Parthasarathy","year":"1967","unstructured":"K.R. Parthasarathy, Probability Measures on Metric Spaces (American Mathematical Society, Providence, 1967)"},{"issue":"2","key":"3_CR147","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1215\/ijm\/1255644642","volume":"7","author":"KR Parthasarathy","year":"1963","unstructured":"K.R. Parthasarathy, R. Ranga Rao, S.R.S. Varadhan, Probability distributions on locally compact abelian groups. Illinois J. Math. 7(2), 337\u2013369 (1963)","journal-title":"Illinois J. Math."},{"key":"3_CR149","first-page":"536","volume-title":"From Stochastic Calculus to Mathematical Finance\u2014The Shiryaev Festschrift","author":"G Peskir","year":"2006","unstructured":"G. Peskir, On the fundamental solution of the Kolmogorov\u2013Shiryaev equation, in From Stochastic Calculus to Mathematical Finance\u2014The Shiryaev Festschrift, ed. by Yu. Kabanov, R.Liptser, J. Stoyanov (Springer, Berlin, 2006), pp. 536\u2013546"},{"issue":"4","key":"3_CR152","doi-asserted-by":"publisher","first-page":"1153","DOI":"10.1007\/s11009-016-9478-7","volume":"18","author":"AS Polunchenko","year":"2016","unstructured":"A.S. Polunchenko, G. Sokolov, An analytic expression for the distribution of the generalized Shiryaev\u2013Roberts diffusion. Methodol. Comput. Appl. Probab. 18(4), 1153\u20131195 (2016)","journal-title":"Methodol. Comput. Appl. Probab."},{"issue":"11","key":"3_CR154","doi-asserted-by":"publisher","first-page":"878","DOI":"10.1080\/10652469.2016.1227330","volume":"27","author":"A Prasad","year":"2016","unstructured":"A. Prasad, U.K. Mandal, Composition of pseudo-differential operators associated with Kontorovich\u2013Lebedev transform. Integr. Transf. Spec. Funct. 27(11), 878\u2013892 (2016)","journal-title":"Integr. Transf. Spec. Funct."},{"key":"3_CR155","unstructured":"A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series, Vol. I: Elementary Functions (Gordon and Breach, New York and London, 1986)"},{"key":"3_CR156","unstructured":"A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series, Vol. II: Special Functions (Gordon and Breach, New York and London, 1986)"},{"key":"3_CR157","unstructured":"A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series, Vol. III: More Special Functions (Gordon and Breach, New York and London, 1989)"},{"key":"3_CR163","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1090\/conm\/261\/04135","volume-title":"Probability on Algebraic Structures: AMS Special Session on Probability on Algebraic Structures, March 12\u201313, 1999, Gainesville, Florida","author":"C Rentzsch","year":"2000","unstructured":"C. Rentzsch, M. Voit, L\u00e9vy Processes on Commutative Hypergroups, in Probability on Algebraic Structures: AMS Special Session on Probability on Algebraic Structures, March 12\u201313, 1999, Gainesville, Florida, ed. by G. Budzban, P. Feinsilver, A. Mukherjea (American Mathematical Society, Providence, 2000), pp. 83\u2013105"},{"key":"3_CR166","volume-title":"L\u00e9vy Processes and Infinitely Divisible Distributions","author":"K Sato","year":"1999","unstructured":"K. Sato, L\u00e9vy Processes and Infinitely Divisible Distributions (Cambridge University Press, Cambridge, 1999)"},{"key":"3_CR168","first-page":"795","volume":"2","author":"AN Shiryaev","year":"1961","unstructured":"A.N. Shiryaev, The problem of the most rapid detection of a disturbance in a stationary process. Soviet Math. Dokl. 2, 795\u2013799 (1961)","journal-title":"Soviet Math. Dokl."},{"key":"3_CR170","volume-title":"Operator Theory\u2014A Comprehensive Course in Analysis, Part 4","author":"B Simon","year":"2015","unstructured":"B. Simon, Operator Theory\u2014A Comprehensive Course in Analysis, Part 4 (American Mathematical Society, Providence, 2015)"},{"issue":"2","key":"3_CR180","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1093\/qmathj\/49.3.375","volume":"49","author":"HM Srivastava","year":"1998","unstructured":"H.M. Srivastava, Y.V. Vasil\u2019ev, S. Yakubovich, A class of index transforms with Whittaker\u2019s function as the kernel. Q. J. Math. 49(2), 375\u2013394 (1998)","journal-title":"Q. J. Math."},{"key":"3_CR184","volume-title":"Functional Equations on Hypergroups","author":"L Sz\u00e9kelyhidi","year":"2013","unstructured":"L. Sz\u00e9kelyhidi, Functional Equations on Hypergroups (World Scientific, Singapore, 2013)"},{"key":"3_CR190","doi-asserted-by":"publisher","first-page":"217","DOI":"10.4064\/sm-23-3-217-245","volume":"23","author":"K Urbanik","year":"1964","unstructured":"K. Urbanik, Generalized convolutions. Stud. Math. 23, 217\u2013245 (1964)","journal-title":"Stud. Math."},{"key":"3_CR191","doi-asserted-by":"publisher","first-page":"57","DOI":"10.4064\/sm-45-1-57-70","volume":"45","author":"K Urbanik","year":"1973","unstructured":"K. Urbanik, Generalized convolutions II. Stud. Math. 45, 57\u201370 (1973)","journal-title":"Stud. Math."},{"issue":"4","key":"3_CR196","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1007\/BF01083639","volume":"40","author":"VE Volkovich","year":"1988","unstructured":"V.E. Volkovich, Infinitely divisible distributions in algebras with stochastic convolution. J. Sov. Math. 40(4), 459\u2013467 (1988)","journal-title":"J. Sov. Math."},{"issue":"3","key":"3_CR198","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1007\/BF01060427","volume":"5","author":"VE Volkovich","year":"1992","unstructured":"V.E. Volkovich, On symmetric stochastic convolutions. J. Theoret. Probab. 5(3), 417\u2013430 (1992)","journal-title":"J. Theoret. Probab."},{"key":"3_CR204","doi-asserted-by":"publisher","DOI":"10.1142\/2707","volume-title":"Index Transforms","author":"S Yakubovich","year":"1996","unstructured":"S. Yakubovich, Index Transforms (World Scientific, Singapore, 1996)"},{"issue":"2","key":"3_CR205","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/j.jat.2004.10.007","volume":"131","author":"S Yakubovich","year":"2004","unstructured":"S. Yakubovich, On the least values of L p-norms for the Kontorovich\u2013Lebedev transform and its convolution. J. Approx. Theory 131(2), 231\u2013242 (2004)","journal-title":"J. Approx. Theory"},{"key":"3_CR207","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-1196-6","volume-title":"The Hypergeometric Approach to Integral Transforms and Convolutions","author":"S Yakubovich","year":"1994","unstructured":"S. Yakubovich, Y.F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions (Kluwer Academic Publishers, Dordrecht, 1994)"},{"key":"3_CR208","doi-asserted-by":"crossref","unstructured":"H. Zeuner, Laws of Large Numbers of Hypergroups on $$\\mathbb {R}^+$$ . Math. Ann. 283(4), 657\u2013678 (1989)","DOI":"10.1007\/BF01442859"},{"issue":"1","key":"3_CR210","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/BF02571436","volume":"211","author":"H Zeuner","year":"1992","unstructured":"H. Zeuner, Moment functions and laws of large numbers on hypergroups. Math. Z. 211(1), 369\u2013407 (1992)","journal-title":"Math. Z."},{"issue":"2","key":"3_CR211","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1515\/JAA.1995.213","volume":"1","author":"H Zeuner","year":"1995","unstructured":"H. Zeuner, Domains of attraction with inner norming on Sturm\u2013Liouville hypergroups. J. Appl. Anal. 1(2), 213\u2013221 (1995)","journal-title":"J. Appl. Anal."}],"container-title":["Lecture Notes in Mathematics","Convolution-like Structures, Differential Operators and Diffusion Processes"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-05296-5_3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,25]],"date-time":"2023-11-25T01:33:33Z","timestamp":1700876013000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-05296-5_3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022]]},"ISBN":["9783031052958","9783031052965"],"references-count":54,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-05296-5_3","relation":{},"ISSN":["0075-8434","1617-9692"],"issn-type":[{"type":"print","value":"0075-8434"},{"type":"electronic","value":"1617-9692"}],"subject":[],"published":{"date-parts":[[2022]]},"assertion":[{"value":"19 April 2022","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}