Celebratio Mathematica

Donald L. Burkholder

Don Burkholder

by Burgess Davis

Don was born shortly be­fore the Great De­pres­sion in­to a farm­ing fam­ily in east­ern Neb­raska. Though the farm was not en­gulfed in the Dust Bowl, he would have seen huge dust clouds far away, as my fath­er did from a Mis­souri farm. Per­haps grow­ing up in all this trouble had something to do with Don’s fe­ro­cious work eth­ic, which was passed on to his two sons, now both out­stand­ingly suc­cess­ful aca­dem­ics.

Joe Doob was at the Uni­versity of Illinois when Don ar­rived there, and he cer­tainly in­flu­enced Don’s in­terest in mar­tin­gales. Don worked with only a few co-au­thors, but he and Dick Gundy made a great team. They began their joint work by find­ing com­par­is­ons of \( L_p \) norms of mar­tin­gales’ square and max­im­al func­tions. This had been an open prob­lem and their solu­tion, called good-lambda, is re­mark­ably easy to un­der­stand and now used to com­pare norms all over math­em­at­ics. Not much later Don with Dick Gundy and Marty Sil­ver­stein used prob­ab­il­ist­ic meth­ods to com­plete res­ults of Hardy and Lit­tle­wood by giv­ing a max­im­al func­tion char­ac­ter­iz­a­tion of the class of \( H^p \) of har­mon­ic func­tions in the up­per half plane. Be­fore this pa­per, but not after it, many math­em­aticians did not think prob­ab­il­ity was real math. This was the last re­search pa­per Don pub­lished with a col­lab­or­at­or, but he wrote many more alone. Some of the best in­volved solv­ing a tough mar­tin­gale prob­lem he had thought up and then voilà his solu­tion is im­port­ant not just for mar­tin­gales. I en­cour­age the read­er to google “Bell­man func­tions in har­mon­ic ana­lys­is A. Vol­berg” for a great short dis­cus­sion of an ex­ample.

Don was very kind and en­cour­aging to young math­em­aticians. He was an ex­cel­lent dis­ser­ta­tion ad­visor and a good friend. He re­ceived many hon­ors, but the only one he ever talked about (and loved to) is that he was pres­id­ent of his seni­or class in high school be­cause the oth­er three stu­dents in his class had already been pres­id­ent.

Don’s wife Jean was elec­ted to the Urb­ana School Board and served for 22 years, and also was first chair of the Urb­ana Hu­man Re­la­tions Com­mit­tee. In ad­di­tion to that and more civic en­gage­ment, Jean found time to be a strong sup­port to Don and his ca­reer. They were an amaz­ing couple.

Bur­gess Dav­is was an un­der­gradu­ate at Ohio State and Don Burk­hold­er was his doc­tor­al ad­visor at the Uni­versity of Illinois, Urb­ana-Cham­paign. He worked at Rut­gers and then Purdue and is now Emer­it­us Pro­fess­or of Math­em­at­ics and Stat­ist­ics at Purdue.