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INTEGRATION ON PRODUCT SPACES 173For each x E X, XE(X, t) = 1 if t
174 REAL AND COMPLEX ANALYSISWe now use Theorem 8.16, with X = Rk, J1. = m, cp(t) = tP, to calculater (M.f)Pdm=p rOOm{Mf>t}tP-ldt~~ r OO tP- 2 dt r fdmJRk Jo 1 - c Jo JEt= -- Ap i fdm iJ/C t p - 2 dt = Apc 1 - ip fP dm.1 - c Rk 0 (1 - c)(p - 1) RkThis proves the theorem. However, to get a good constant, let us choose c soas to minimize that last expression. This happens when c = (p - 1)/p = l/q,where q is the exponent conjugate to p. For this c,and the preceding computation yields1 )P-lp-lc 1 - P = ( 1 + -- < e,Note that C p -+ 1 as p-+ 00, which agrees with formula 8.17(1), and thatCp-+.oo as p-+ 1.(6)IIIIExercises1 Find a monotone class !lJI in R I which is not a u-algebra, even though R I E !lJI and R I - A E !lJI forevery A E!lJl.2 SupposeJis a Lebesgue measurable nonnegative real function on RI and A(f) is the ordinate set off This is the set of all points (x, y) E R2 for which 0 < Y 0, and show that there existJ E LI(RI) and g E I!(RI) such thatIIJ * gllp > (1 - €)IIJlllligllp.
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REAL AND COMPLEX ANALYSIS
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REAL AND COMPLEX ANALYSISINTERNATIO
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CONTENTSPrefacexiiiPrologue: The Ex
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CONTENTS ixChapter 10Elementary Pro
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PROLOGUETHE EXPONENTIAL FUNCTIONThi
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PROLOGUE: THE EXPONENTIAL FUNCTION
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CHAPTERONEABSTRACT INTEGRATIONTowar
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ABSTRACT INTEGRA nON 7If no two mem
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ABSTRACT INTEGRA nON 9It would perh
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ABSTRACT INTEGRATION 111.8 Theorem
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ABSTRACT INTEGRATION 131.12 Theorem
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ABSTRACT INTEGRATION 15Corollaries(
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ABSTRACT INTEGRA nON 17As the proof
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ABSTRACT INTEGRATION 19The cancella
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ABSTRACT INTEGRATION 21Next, let s
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Corollary If aij :2= ° for i and j
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ABSTRACT INTEGRATION 25each of thes
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ABSTRACT INTEGRA nON 27PROOF Since
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ABSTRACT INTEGRA TlON 291.37 The fa
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ABSTRACT INTEGRATION 31PROOF Let a
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CHAPTERTWOPOSITIVE BOREL MEASURESVe
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POSITIVE BOREL MEASURES 35[The conv
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POSITIVE BOREL MEASURES 372.6 Theor
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POSITIVE BOREL MEASURES 39will mean
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POSITIVE BOREL MEASURES 41(b) I-'(K
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POSITIVE BOREL MEASURES 43Put V = U
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POSITIVE BOREL MEASURES 45PROOF If
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POSITIVE BOREL MEASURES 47Since hi
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POSITIVE BOREL MEASURES 49PROOF Put
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POSITIVE BOREL MEASURES 51(c) m is
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POSITIVE BOREL MEASURES 532.21 Rema
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POSITIVE BOREL MEASURES 55If T is o
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POSITIVE BOREL MEASURES 57Sincethe
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POSITIVE BOREL MEASURES S913 Is it
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CHAPTERTHREELP-SPACESConvex Functio
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If-SPACES 63(4) becomesexp U (Xl +
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I!'-SPACES 65Hence the left side of
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H-SPACES 67Suppose j, g, and h are
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I!-SPACES 69The simple functions pl
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l!'-SPACES 71Given / E Co(X) and
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If-SPACES 73(b) Prove that equality
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I!'-SPACES 75(b) If 1 :5 P < 00 and
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ELEMENTARY HILBERT SPACE THEORY 774
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ELEMENTARY HILBERT SPACE THEORY 79I
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ELEMENTARY HILBERT SPACE THEORY 81P
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ELEMENTARY HILBERT SPACE THEORY 834
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ELEMENTARY IDLBERT SPACE THEORY 8S4
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ELEMENTARY HILBERT SPACE THEORY 87(
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ELEMENTARY HILBERT SPACE THEORY 89W
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ELEMENTARY HILBERT SPACE THEORY 91T
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ELEMENTARY HILBERT SPACE THEORY 93M
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CHAPTERFIVEEXAMPLES OF BANACH SPACE
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EXAMPLES OF BANACH SPACE lECHNIQUES
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EXAMPLES OF BANACH SPACE TECHNIQUES
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EXAMPLFS OF BANACH SPACE TECHNIQUES
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COMPLEX MEASURES 117This notation i
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COMPLEX MEASURES 119We have thus sp
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COMPLEX MEASURES 121(f) Since A2 .1
Page 138 and 139: COMPLEX MEASURES 123Hence g(x) E [0
Page 140 and 141: COMPLEX MEASURES 125By analogy with
Page 142 and 143: COMPLEX MEASURES 127then
Page 144 and 145: COMPLEX MEASURES 129The first part
Page 146 and 147: COMPLEX MEASURES 131Once we have th
Page 148 and 149: COMPLEX MEASURES 1334 Suppose 1 :5
Page 150 and 151: CHAPTERSEVENDIFFERENTIATIONIn eleme
Page 152 and 153: DIFFERENTIATION 1377.3 Lemma If W i
Page 154 and 155: DIFFERENTIATION 139PROOF Definefor
Page 156 and 157: DIFFERENTIA nON 141PROOF Let x be a
Page 158 and 159: DIFFERENTIATION 1437.14 Theorem Sup
Page 160 and 161: DIFFERENTIATION 1452- n - 1
Page 162 and 163: DIFFERENTIATION 147Assume next that
Page 164 and 165: DIFFERENTIATION 149The next theorem
Page 166 and 167: DIFFERENTIATION 151stronger hypothe
Page 168 and 169: DIFFERENTIATION, 153Since r" = m(B(
Page 170 and 171: DIFFERENTIATION 155Theorem 7.8 tell
Page 172 and 173: DIFFERENTIATION 1577 Construct a co
Page 174 and 175: DIFFERENTIATION 15921 Iffis a real
Page 176 and 177: INTEGRATION ON PRODUCT SPACES 161If
Page 178 and 179: INTEGRATION ON PRODUCT SPACES 163Pr
Page 180 and 181: INTEGRATION ON PRODUCT SPACES 165(c
Page 182 and 183: INTEGRATION ON PRODUCT SPACES 167To
Page 184 and 185: INTEGRATION ON PRODUCT SPACES 169Th
Page 186 and 187: INTEGRATION ON PRODUCT SPACES 171wh
Page 190 and 191: INTEGRATION ON PRODUCT SPACES 175S
Page 192 and 193: INTEGRATION ON PRODUCT SPACES 177Su
Page 194 and 195: FOURIER TRANSFORMS 179The formal pr
Page 196 and 197: FOURIER TRANSFORMS 181Let us see wh
Page 198 and 199: FOURIER TRANSFORMS 183The integrand
Page 200 and 201: FOURIER lRANSFORMS 185and Theorem 3
Page 202 and 203: FOURIER lRANSFORMS 187Theorem 9.2(d
Page 204 and 205: FOURIER TRANSFORMS 189orthogonal pr
Page 206 and 207: FOURIER TRANSFORMS 191follows: We k
Page 208 and 209: FOURIER TRANSFORMS 193Thenf'! (d (X
Page 210 and 211: FOURIER TRANSFORMS 195What does (*)
Page 212 and 213: ELEMENTARY PROPERTIES OF HOLOMORPHI
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ELEMENTARY PROPERTIES OF HOLOMORPHI
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ELEMENTARY PROPERTIES OF HOLOMORPlD
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CHAPTERELEVENHARMONIC FUNCTIONSThe
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HARMONIC FUNCTIONS 233The Poisson I
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HARMONIC FUNCTIONS 235Note: This th
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HARMONIC FUNCTIONS 237The Mean Valu
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HARMONIC FUNCTIONS 239Hence
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HARMONIC FUNCTIONS 241The regions n
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HARMONIC FUNCTIONS 243Hence, settin
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HARMONIC FUNCTIONS 245(a) If P. is
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HARMONIC FUNCTIONS 247PROOF To say
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HARMONIC FUNCTIONS 249As before, Lo
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HARMONIC FUNCTIONS 251If {u.} is a
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CHAPTERTWELVETHE MAXIMUM MODULUS PR
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THE MAXIMUM MODULUS PRINCIPLE 255Th
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THE MAXIMUM MODULUS PRINCIPLE 257Fi
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THE MAXIMUM MODULUS PRINCIPLE 259Fi
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THE MAXIMUM MODULUS PRINCIPLE 26112
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THE MAXIMUM MODULUS PRINCIPLE 263No
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THE MAXIMUM MODULUS PRINCIPLE 265Su
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APPROXIMATIONS BY RATIONAL FUNCTION
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APPROXIMA nONS BY RA nONAL FUNCTION
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APPROXIMA TIO~S BY RATIONAL FUNCTIO
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CONFORMAL MAPPING 279Here /'(zo) =
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CONFORMAL MAPPING 281Let us discuss
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CONFORMAL MAPPING 283Under these co
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CONFORMAL MAPPING 28514.9 Remarks T
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CONFORMAL MAPPING 287Divide Eqs. (5
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CONFORMAL MAPPING 289PROOF Iff= I/(
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CONFORMAL MAPPING 291Theorem 14.18(
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CONFORMAL MAPPING 293Put y(t) = .JR
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CONFORMAL MAPPING 29518 Suppose 0 i
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CONFORMAL MAPPING 297(d) Let ex be
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ZEROS OF HOLOMORPHIC FUNCTIONS 299I
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ZEROS OF HOLOMORPHIC FUNCTIONS 301c
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ZEROS OF HOLOMORPHIC FUNCTIONS 303I
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ZEROS OF HOLOMORPlDC FUNCTIONS 305a
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ZEROS OF HOWMORPHIC FUNCTIONS 307Je
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ZEROS OF HOLOMORPIDC FUNCTIONS 309a
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ZEROS OF HOWMORPHIC FUNCTIONS 311is
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ZEROS OF HOLOMORPHIC FUNCTIONS 313t
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ZEROS OF HOLOMORPHIC FUNCTIONS 315i
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ZEROS OF HOLOMORPHIC FUNCTIONS 317P
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CHAPTERSIXTEENANALYTIC CONTINUATION
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ANALYTIC CONTINUATION 321integers.
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ANALYTIC CONTINUATION 323the radius
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ANALYTIC CONTINUATION 325There are
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ANALYTIC CONTINUATION 32716.15 Tbeo
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ANALYTIC CONTINUATION 329We claim t
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ANALYTIC CONTINUATION 331PROOF Let
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ANALYTIC CONTINUATION 333P(f1' 01)
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CHAPTERSEVENTEENThis chapter is dev
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HP-SPACES 337some r > 0 we have D(z
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HP-SPACES 339PROOF Note first thatI
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For each z E U, 11 - e-ilz I and P(
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log I 9 I are 0 a.e. (Theorem 11.22
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Since Ilog+ u - log+ V I ~ I u - v
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X n , the one-dimensional spaces sp
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So there exists a qJ E Y such that
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HP-SPACES 351Let us recall that eve
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5 Suppose fe HP, qJ e H(U), and qJ(
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ffI' -SPACES 355exists (and is fini
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ELEMENTARY TIIEORY OF BANACH ALGEBR
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ELEMENTARY THEORY OF BANACH ALGEBRA
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CHAPTERNINETEENHOLOMORPHIC FOURIER
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HOLOMORPHIC FOURIER TRANSFORMS 373a
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HOLOMORPHIC FOURIER TRANSFORMS 3751
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HOLOMORPHIC FOURIER TRANSFORMS 377i
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HOLOMORPHIC FOURIER TRANSFORMS 379C
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HOLOMORPHIC FOURIER TRANSFORMS 381T
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HOLOMORPHIC FOURIER TRANSFORMS 383F
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HOLOMORPHIC FOURIER TRANSFORMS 385h
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UNIFORM APPROXIMATION BY POLYNOMIAL
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UNIFORM APPROXIMATION BY pOLYNOMIAL
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APPENDIXHAUSDORFF'S MAXIMALITY THEO
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NOTES AND COMMENTSChapter 1The firs
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NOTES AND COMMENTS 399Chapter 6Sec.
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NOTES AND COMMENTS 401orthonormal s
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NOTES AND COMMENTS 403CbapterlOSee
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406 BIBLIOGRAPHY24. F. Riesz and B.
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408 LIST OF SPECIAL SYMBOLS AND ABB
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410 INDEXCartesian product, 7,160Ca
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412 INDEXHalmos, P. R., 398,403Hard
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414 INDEXOuter regular set, 47Overc
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416 INDEXUniform continuity, 51Unif
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