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Celebratio Mathematica

Donald L. Burkholder

Complete Bibliography

[1] D. L. Burk­hold­er: “On a class of stochast­ic ap­prox­im­a­tion pro­cesses,” Ann. Math. Stat. 27 : 4 (1956), pp. 1044–​1059. This ap­pears to be an abridged ver­sion of the au­thor’s 1956 PhD thes­is. MR 85653 Zbl 0075.​13803 article

[2] D. L. Burk­hold­er: On a cer­tain class of stochast­ic ap­prox­im­a­tion pro­cesses. Ph.D. thesis, The Uni­versity of North Car­o­lina at Chapel Hill, 1956. Ad­vised by W. Höff­d­ing. An abridged ver­sion of this was pub­lished in Ann. Math. Stat. 27:4 (1956). MR 2938781 phdthesis

[3] D. L. Burk­hold­er: “On the ex­ist­ence of a best ap­prox­im­a­tion of one dis­tri­bu­tion func­tion by an­oth­er of a giv­en type,” Ann. Math. Stat. 30 : 3 (1959), pp. 738–​742. MR 106522 Zbl 0096.​13102 article

[4] D. L. Burk­hold­er: “Ef­fect on the min­im­al com­plete class of tests of changes in the test­ing prob­lem,” Ann. Math. Stat. 31 : 2 (1960), pp. 325–​331. MR 117846 Zbl 0119.​15305 article

[5] D. L. Burk­hold­er: “Suf­fi­ciency in the un­dom­in­ated case,” Ann. Math. Stat. 32 : 4 (1961), pp. 1191–​1200. MR 131287 Zbl 0221.​62003 article

[6] D. L. Burk­hold­er and Y. S. Chow: “It­er­ates of con­di­tion­al ex­pect­a­tion op­er­at­ors,” Proc. Am. Math. Soc. 12 : 3 (1961), pp. 490–​495. MR 142144 Zbl 0106.​33201 article

[7] D. L. Burk­hold­er: “On the or­der struc­ture of the set of suf­fi­cient sub­fields,” Ann. Math. Stat. 33 : 2 (1962), pp. 596–​599. MR 137227 Zbl 0127.​34808 article

[8] D. L. Burk­hold­er: “Semi-Gaus­si­an sub­spaces,” Trans. Am. Math. Soc. 104 : 1 (1962), pp. 123–​131. MR 138986 Zbl 0131.​35001 article

[9] D. L. Burk­hold­er: “Suc­cess­ive con­di­tion­al ex­pect­a­tions of an in­teg­rable func­tion,” Ann. Math. Stat. 33 : 3 (1962), pp. 887–​893. MR 143246 Zbl 0128.​12602 article

[10] D. L. Burk­hold­er and R. A. Wijs­man: “Op­tim­um prop­er­ties and ad­miss­ib­il­ity of se­quen­tial tests,” Ann. Math. Stat. 34 : 1 (1963), pp. 1–​17. MR 148160 Zbl 0113.​13005 article

[11] D. L. Burk­hold­er: “Max­im­al in­equal­it­ies as ne­ces­sary con­di­tions for al­most every­where con­ver­gence,” Z. Wahr­schein­lich­keit­s­the­or. Verw. Geb. 3 : 1 (March 1964), pp. 75–​88. MR 189110 Zbl 0134.​14602 article

[12] D. L. Burk­hold­er: “Mar­tin­gale trans­forms,” Ann. Math. Stat­ist. 37 (1966), pp. 1494–​1504. MR 0208647 Zbl 0306.​60030 article

[13] D. L. Burk­hold­er: “In­de­pend­ent se­quences with the Stein prop­erty,” Ann. Math. Stat. 39 : 4 (1968), pp. 1282–​1288. MR 228045 Zbl 0162.​49404 article

[14] D. L. Burk­hold­er and R. F. Gundy: “Ex­tra­pol­a­tion and in­ter­pol­a­tion of quasi-lin­ear op­er­at­ors on mar­tin­gales,” Acta Math. 124 (1970), pp. 249–​304. MR 0440695 Zbl 0223.​60021 article

[15] D. L. Burk­hold­er, R. F. Gundy, and M. L. Sil­ver­stein: “A max­im­al func­tion char­ac­ter­iz­a­tion of the class \( H^{p} \),” Trans. Amer. Math. Soc. 157 (1971), pp. 137–​153. MR 0274767 Zbl 0223.​30048 article

[16] D. L. Burk­hold­er: “Mar­tin­gale in­equal­it­ies,” pp. 1–​8 in Mar­tin­gales (Ober­wolfach, Ger­many, 17–23 May 1970). Lec­ture Notes in Math­em­at­ics 190. Spring­er (Ber­lin), 1971. MR 372985 incollection

[17] D. L. Burk­hold­er: “In­equal­it­ies for op­er­at­ors on mar­tin­gales,” pp. 551–​557 in Act­es du Con­grès In­ter­na­tion­al des Mathématiciens (Nice, France, 1–10 Septem­ber 1970), vol. 2. Gau­th­i­er-Vil­lars (Par­is), 1971. MR 420830 Zbl 0245.​60041 incollection

[18] D. L. Burk­hold­er and R. F. Gundy: “Ex­tra­pol­a­tion and in­ter­pol­a­tion of quasi-lin­ear op­er­at­ors on mar­tin­gales,” Matem­atika 15 : 3 (1971), pp. 91–​141. Rus­si­an trans­la­tion of art­icle pub­lished in Acta Math. 124 (1970). Zbl 0223.​60022 article

[19] D. L. Burk­hold­er and R. F. Gundy: “Dis­tri­bu­tion func­tion in­equal­it­ies for the area in­teg­ral,” pp. 527–​544 in Col­lec­tion of art­icles hon­or­ing the com­ple­tion by Ant­oni Zyg­mund of 50 years of sci­entif­ic activ­ity, VI, published as Stu­dia Math. 44 : 6. Państ­wowe Wydawn­ict­wo Naukowe (Warsaw), 1972. MR 340557 Zbl 0219.​31009 incollection

[20] D. L. Burk­hold­er, B. J. Dav­is, and R. F. Gundy: “In­teg­ral in­equal­it­ies for con­vex func­tions of op­er­at­ors on mar­tin­gales,” pp. 223–​240 in Pro­ceed­ings of the sixth Berke­ley sym­posi­um on math­em­at­ic­al stat­ist­ics and prob­ab­il­ity (Berke­ley, CA, 21 June–18 Ju­ly 1970), vol. 2: Prob­ab­il­ity the­ory. Edi­ted by L. Le Cam, J. Ney­man, and E. L. Scott. Uni­versity of Cali­for­nia Press (Berke­ley and Los Angeles, CA), 1972. MR 400380 Zbl 0253.​60056 incollection

[21] D. L. Burk­hold­er and R. F. Gundy: “Bound­ary be­ha­viour of har­mon­ic func­tions in a half-space and Browni­an mo­tion,” Ann. Inst. Four­i­er (Gren­oble) 23 : 4 (1973), pp. 195–​212. MR 365691 Zbl 0253.​31010 article

[22] D. L. Burk­hold­er: “Dis­tri­bu­tion func­tion in­equal­it­ies for mar­tin­gales,” Ann. Probab. 1 : 1 (1973), pp. 19–​42. MR 365692 Zbl 0301.​60035 article

[23] D. L. Burk­hold­er: “\( H^p \) spaces and exit times of Browni­an mo­tion,” Bull. Am. Math. Soc. 81 : 3 (May 1975), pp. 556–​558. MR 362519 Zbl 0321.​60059 article

[24] D. L. Burk­hold­er: “One-sided max­im­al func­tions and \( H^p \),” J. Funct. Anal. 18 : 4 (April 1975), pp. 429–​454. MR 365693 Zbl 0294.​31005 article

[25] D. L. Burk­hold­er: “Har­mon­ic ana­lys­is and prob­ab­il­ity,” pp. 136–​149 in Stud­ies in har­mon­ic ana­lys­is (Chica­go, 1974). Edi­ted by J. M. Ash. MAA Stud­ies in Math­em­at­ics 13. Math­em­at­ic­al As­so­ci­ation of Amer­ica (Wash­ing­ton, DC), 1976. MR 463788 Zbl 0339.​42011 incollection

[26] D. L. Burk­hold­er: “Browni­an mo­tion and clas­sic­al ana­lys­is,” pp. 5–​14 in Prob­ab­il­ity (Urb­ana, IL, March 1976). Edi­ted by J. L. Doob. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 31. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1977. MR 474524 Zbl 0365.​60075 incollection

[27] D. L. Burk­hold­er: “Exit times of Browni­an mo­tion, har­mon­ic ma­jor­iz­a­tion, and Hardy spaces,” Adv. Math. 26 : 2 (November 1977), pp. 182–​205. MR 474525 Zbl 0372.​60112 article

[28] D. L. Burk­hold­er and T. Shintani: “Ap­prox­im­a­tion of \( L^1 \)-bounded mar­tin­gales by mar­tin­gales of bounded vari­ation,” Proc. Am. Math. Soc. 72 : 1 (October 1978), pp. 166–​169. MR 494472 Zbl 0387.​60048 article

[29] D. L. Burk­hold­er: “Bound­ary value es­tim­a­tion of the range of an ana­lyt­ic func­tion,” Mich. Math. J. 25 : 2 (1978), pp. 197–​211. MR 501568 Zbl 0372.​30026 article

[30] D. L. Burk­hold­er: “A sharp in­equal­ity for mar­tin­gale trans­forms,” Ann. Probab. 7 : 5 (1979), pp. 858–​863. MR 542135 Zbl 0416.​60047 article

[31] D. L. Burk­hold­er: “Mar­tin­gale the­ory and har­mon­ic ana­lys­is in Eu­c­lidean spaces,” pp. 283–​301 in Har­mon­ic ana­lys­is in Eu­c­lidean spaces (Wil­li­am­stown, MA, 10–28 Ju­ly 1978), Part 2. Edi­ted by G. Weiss and S. Wainger. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 35. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1979. MR 545315 Zbl 0417.​60055 incollection

[32] D. L. Burk­hold­er: “Weak in­equal­it­ies for exit times and ana­lyt­ic func­tions,” pp. 27–​34 in Prob­ab­il­ity the­ory (Warsaw, 6 Feb­ru­ary–11 June 1976). Edi­ted by Z. Ciesiel­ski. Banach Cen­ter Pub­lic­a­tions 5. PWN (Warsaw), 1979. MR 561465 Zbl 0416.​60048 incollection

[33] D. L. Burk­hold­er: “Browni­an mo­tion and the Hardy spaces \( H^p \),” pp. 97–​118 in As­pects of con­tem­por­ary com­plex ana­lys­is (Durham, UK, 1–20 Ju­ly 1979). Edi­ted by D. A. Bran­nan and J. Clunie. Aca­dem­ic Press (Lon­don and New York), 1980. MR 623466 Zbl 0497.​30028 incollection

[34] D. L. Burk­hold­er: “Some new in­equal­it­ies in com­plex ana­lys­is (and what they do),” Real Anal. Exch. 5 : 1 (1980), pp. 124–​127. Zbl 0423.​30023 article

[35] D. L. Burk­hold­er: “A geo­met­ric­al char­ac­ter­iz­a­tion of Banach spaces in which mar­tin­gale dif­fer­ence se­quences are un­con­di­tion­al,” Ann. Probab. 9 : 6 (1981), pp. 997–​1011. MR 632972 Zbl 0474.​60036 article

[36] D. L. Burk­hold­er: “Mar­tin­gale trans­forms and the geo­metry of Banach spaces,” pp. 35–​50 in Prob­ab­il­ity in Banach spaces, III (Med­ford, MA, 4–16 Au­gust 1980). Edi­ted by A. Beck. Lec­ture Notes in Math­em­at­ics 860. Spring­er (Ber­lin), 1981. MR 647954 Zbl 0471.​60012 incollection

[37] D. L. Burk­hold­er: “A non­lin­ear par­tial dif­fer­en­tial equa­tion and the un­con­di­tion­al con­stant of the Haar sys­tem in \( L^p \),” Bull. Am. Math. Soc. (N.S.) 7 : 3 (1982), pp. 591–​595. Re­search an­nounce­ment. MR 670133 Zbl 0504.​46022 article

[38] D. L. Burk­hold­er: “A geo­met­ric con­di­tion that im­plies the ex­ist­ence of cer­tain sin­gu­lar in­teg­rals of Banach-space-val­ued func­tions,” pp. 270–​286 in Con­fer­ence on har­mon­ic ana­lys­is in hon­or of Ant­oni Zyg­mund (Chica­go, 23–28 March 1981), vol. 1. Edi­ted by W. Beck­ner, A. P. Calder­ón, R. Fef­fer­man, and P. W. Jones. Wadsworth Math­em­at­ics Series. Wadsworth (Bel­mont, CA), 1983. MR 730072 incollection

[39] D. L. Burk­hold­er: “Bound­ary value prob­lems and sharp in­equal­it­ies for mar­tin­gale trans­forms,” Ann. Probab. 12 : 3 (1984), pp. 647–​702. MR 744226 Zbl 0556.​60021 article

[40] D. L. Burk­hold­er: “An ele­ment­ary proof of an in­equal­ity of R. E. A. C. Pa­ley,” Bull. Lon­don Math. Soc. 17 : 5 (September 1985), pp. 474–​478. MR 806015 Zbl 0566.​46014 article

[41] D. L. Burk­hold­er: “An ex­ten­sion of a clas­sic­al mar­tin­gale in­equal­ity,” pp. 21–​30 in Prob­ab­il­ity the­ory and har­mon­ic ana­lys­is (New York). Edi­ted by J.-A. Chao and W. Woyczyński. Pure and Ap­plied Math­em­at­ics 98. Dek­ker, 1986. MR 830228 Zbl 0594.​60020 incollection

[42] D. L. Burk­hold­er: “Mar­tin­gales and Four­i­er ana­lys­is in Banach spaces,” pp. 61–​108 in Prob­ab­il­ity and ana­lys­is (Var­enna, Italy, 31 May–8 June 1985). Edi­ted by G. Letta and M. Pra­telli. Lec­ture Notes in Math­em­at­ics 1206. Spring­er (Ber­lin), 1986. MR 864712 Zbl 0605.​60049 incollection

[43] D. L. Burk­hold­er: “A sharp and strict \( L^p \)-in­equal­ity for stochast­ic in­teg­rals,” Ann. Probab. 15 : 1 (1987), pp. 268–​273. MR 877602 Zbl 0617.​60042 article

[44] D. L. Burk­hold­er: “A proof of Pełczyński’s con­jec­ture for the Haar sys­tem,” Stu­dia Math. 91 : 1 (1988), pp. 79–​83. MR 957287 Zbl 0652.​42012 article

[45] D. L. Burk­hold­er: “Sharp in­equal­it­ies for mar­tin­gales and stochast­ic in­teg­rals,” pp. 75–​94 in Col­loque Paul Lévy sur les pro­ces­sus stochastiques [Paul Lévy col­loqui­um on stochast­ic pro­cesses] (Pal­aiseau, France, 22–26 June 1987). Edi­ted by L. Schwartz. Astérisque 157–​158. Sociét’e Mathématique de France (Par­is), 1988. MR 976214 Zbl 0656.​60055 incollection

[46] D. L. Burk­hold­er: “Dif­fer­en­tial sub­or­din­a­tion of har­mon­ic func­tions and mar­tin­gales,” pp. 1–​23 in Har­mon­ic ana­lys­is and par­tial dif­fer­en­tial equa­tions (El Escori­al, Spain, 9–13 June 1987). Edi­ted by J. Gar­cía-Cuerva. Lec­ture Notes in Math­em­at­ics 1384. Spring­er (Ber­lin), 1989. MR 1013814 Zbl 0675.​31003 incollection

[47] D. L. Burk­hold­er: “On the num­ber of es­capes of a mar­tin­gale and its geo­met­ric­al sig­ni­fic­ance,” pp. 159–​178 in Al­most every­where con­ver­gence: Pro­ceed­ings of the in­ter­na­tion­al con­fer­ence on al­most every­where con­ver­gence in prob­ab­il­ity and er­god­ic the­ory (Colum­bus, OH, 11–14 June 1988). Edi­ted by G. A. Edgar and L. Suche­ston. Aca­dem­ic Press (Bo­ston, MA), 1989. MR 1035244 Zbl 0685.​60049 incollection

[48] D. L. Burk­hold­er, É. Par­doux, and A. Szn­it­man: École d’été de prob­ab­ilités de Saint-Flour XIX–1989 [12th Saint-Flour prob­ab­il­ity sum­mer school–1989] (Saint-Flour, France, 16 Au­gust–2 Septem­ber 1989). Edi­ted by P. L. Hen­nequin. Lec­ture Notes in Math­em­at­ics 1464. Spring­er (Ber­lin), 1991. MR 1108182 Zbl 0722.​00029 book

[49] D. L. Burk­hold­er: “Ex­plor­a­tions in mar­tin­gale the­ory and its ap­plic­a­tions,” pp. 1–​66 in D. L. Burk­hold­er, E. Par­doux, and A.-S. Szn­it­man: École d’été de prob­ab­ilités de Saint-Flour XIX–1989 [12th Saint-Flour prob­ab­il­ity sum­mer school–1989] (Saint-Flour, France, 16 Au­gust–2 Septem­ber 1989). Edi­ted by P. L. Hen­nequin. Lec­ture Notes in Math­em­at­ics 1464. Spring­er (Ber­lin), 1991. MR 1108183 Zbl 0771.​60033 incollection

[50] D. L. Burk­hold­er: “Book re­view: G. Pis­i­er, ‘The volume of con­vex bod­ies and Banach space geo­metry’,” Bull. Am. Math. Soc. (N.S.) 25 : 1 (1991), pp. 140–​145. MR 1567936 article

[51]59 are chosen for Na­tion­al Academy of Sci­ences,” NYT (10 May 1992), pp. 1001024. article

[52] D. L. Burk­hold­er: “Strong dif­fer­en­tial sub­or­din­a­tion and stochast­ic in­teg­ra­tion,” Ann. Probab. 22 : 2 (1994), pp. 995–​1025. MR 1288140 Zbl 0816.​60046 article

[53] D. L. Burk­hold­er: “Sharp norm com­par­is­on of mar­tin­gale max­im­al func­tions and stochast­ic in­teg­rals,” pp. 343–​358 in Pro­ceed­ings of the Norbert Wien­er Cen­ten­ary Con­gress, 1994 (East Lans­ing, MI, 27 Novem­ber–3 Decem­ber 1994). Edi­ted by V. Man­drekar and P. R. Mas­ani. Pro­ceed­ings of Sym­po­sia in Ap­plied Math­em­at­ics 52. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1997. MR 1440921 Zbl 0899.​60040 incollection

[54] D. L. Burk­hold­er: “Some ex­tremal prob­lems in mar­tin­gale the­ory and har­mon­ic ana­lys­is,” pp. 99–​115 in Har­mon­ic ana­lys­is and par­tial dif­fer­en­tial equa­tions: Es­says in hon­or of Al­berto P. Calderón (Chica­go, Feb­ru­ary 1996). Edi­ted by M. Christ, C. E. Kenig, and C. Sa­d­osky. Chica­go Lec­tures in Math­em­at­ics. Uni­versity of Chica­go Press, 1999. Es­says in hon­or of Calderón’s 75th birth­day. MR 1743857 Zbl 0954.​60034 incollection

[55] D. L. Burk­hold­er: “Mar­tin­gales and sin­gu­lar in­teg­rals in Banach spaces,” Chapter 6, pp. 233–​269 in Hand­book of the geo­metry of Banach spaces, vol. 1. Edi­ted by W. B. John­son and J. Linden­strauss. El­sevi­er (Am­s­ter­dam), 2001. MR 1863694 Zbl 1029.​46007 incollection

[56] D. L. Burk­hold­er: “The best con­stant in the Dav­is in­equal­ity for the ex­pect­a­tion of the mar­tin­gale square func­tion,” Trans. Am. Math. Soc. 354 : 1 (2002), pp. 91–​105. MR 1859027 Zbl 0984.​60041 article

[57] D. Burk­hold­er and P. Prot­ter: “Joseph Leo Doob, 1910–2004,” Stochast­ic Pro­cess. Ap­pl. 115 : 7 (July 2005), pp. 1061–​1072. MR 2147241 Zbl 1073.​01516 article

[58] D. Burk­hold­er: “Fore­word,” pp. v in Joseph Doob: A col­lec­tion of math­em­at­ic­al art­icles in his memory, published as Ill. J. Math. 50 : 1–​4. Duke Uni­versity Press (Durham, NC), 2006. MR 2247820 incollection

[59] D. Burk­hold­er: “Joseph Leo Doob (1910–2004),” pp. vii–​viii in Joseph Doob: A col­lec­tion of math­em­at­ic­al art­icles in his memory, published as Ill. J. Math. 50 : 1–​4. Issue edi­ted by D. Burk­hold­er. Duke Uni­versity Press (Durham, NC), 2006. MR 2247821 Zbl 1130.​01303 incollection

[60] Joseph Doob: A col­lec­tion of math­em­at­ic­al art­icles in his memory, published as Ill. J. Math. 50 : 1–​4. Issue edi­ted by D. Burk­hold­er. Duke Uni­versity Press (Durham, NC), 2007. Zbl 1120.​60002 book

[61] D. L. Burk­hold­er: “Com­ments on (5), (14), (22), (30), and (31),” pp. 553–​558 in Se­lec­ted pa­pers of Al­berto P. Calderón with com­ment­ary. Edi­ted by A. Bel­low, C. E. Kenig, and P. Malliavin. AMS Col­lec­ted Works 21. Amer­ic­an Math­ema­tial So­ci­ety (Provid­ence, RI), 2008. MR 2435333 incollection

[62] A col­lec­tion of art­icles in hon­or of Don Burk­hold­er, published as Ill. J. Math. 54 : 4. Issue edi­ted by R. Bauer, R. Song, and R. Sowers. Uni­versity of Illinois, De­part­ment of Math­em­at­ics (Urb­ana, IL), 2010. Zbl 1328.​01047 book

[63] L. Ahl­berg: Illinois math­em­atician elec­ted fel­low of AAAS, 11 January 2011. Illinois News Bur­eau on­line art­icle. misc

[64] D. L. Burk­hold­er: Se­lec­ted works of Don­ald L. Burk­hold­er. Edi­ted by B. Dav­is and R. Song. Se­lec­ted Works in Prob­ab­il­ity and Stat­ist­ics. Spring­er (Ber­in), 2011. MR 2850314 Zbl 1271.​01046 book

[65]Ob­it­u­ary: Don­ald Burk­hold­er,” The News-Gaz­ette (17 April 2013). article

[66] Me­mori­al is­sue for Don­ald Burk­hold­er (1927–2013), published as Ann. Probab. 45 : 1. Issue edi­ted by R. Bañuelos and B. Dav­is. In­sti­tute of Math­em­at­ic­al Stat­ist­ics (Beach­wood, OH), 2017. book

[67] R. Bañuelos and B. Dav­is: “In­tro­duc­tion to me­mori­al is­sue for Don­ald Burk­hold­er (1927–2013),” pp. 1–​3 in Me­mori­al is­sue for Don­ald Burk­hold­er (1927–2013), published as Ann. Probab. 45 : 1. Issue edi­ted by R. Bañuelos and B. Dav­is. In­sti­tute of Math­em­at­ic­al Stat­ist­ics (Beach­wood, OH), 2017. MR 3601643 Zbl 1359.​01025 incollection