By J. Garnett
Booklet by way of Garnett, J.
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4) hold for f. ;0 Uo is the "'-neighborhood ot where e > 0, there is, for any such that if ~ Q 21B. Since is of bounded varia,tion a. 0 V(f,U~) > EO, is a. recte,ngle for which R f is covered by this collection of lines. ble number of lines. pproximate identity introduced above, and write Xp * XP E: 00 C. ,. Let f f(z)dz R € ~ -1 rr f P is the Cauchy transform of M -2 dxdy . 5). • ,R n ~~, then \' I L j Consequently Let R €: IhJ) 1 S v( f) . l Let be so small thllt nd let U5 is the o-neighborhood of dR.
Is there a non-zero complex measure Probably this is false. independent of and E, I1fJ. l, on ~ If it is true there and e non-zero measure ~ on E although the proof of this is II counterexample would give a negative solution to the problem on the "approximate Garabedian function" posed in Chapter I. 2: Describe the sets whose characteristic functions are cauchy transforms almost everywhere. II solution of this problem would yield information on the Gleason ReK). A SWiss cheese is a compact set parts for the uniform algebra obtained by deleting from the closed unit disc ~ e sequence of ~j pairwise disjoint open discs whose radii sum and whose union is dense in ~.
Is the <'let of one point - 56- parts (peak points) and each is a measure on then on each -dz/27ri area. zero. e. " 1 R(K) , v is orthogonal to functi on of and Utv f( Z) < n 1. \J m \) and \J P can meet m 00, 0(z) -1o , d6 and on has = 2','" \J n~l nJ is t he characteristic n should tell us something about the structure of the parts PH v Thus an answer to Problem 4. 2 (almost everywhere) • not known whether If so that F. and M. Riesz Uleorem, By the abstract vn a non-trivial part. n be found in An unpublished result of John Wermer answers n I- m.