Making the case for Cauchy transforms

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(Kµ)(z) = = (1 − r)(Kµ)(re it ) = µ = c1δθ1 + c2δθ2 + · · · + cnδθn 2π 1 0 1 − e−iθ z dµ(θ) c1 c2 cn 1 − e−iθ1z + 1 − e−iθ2z + · · · + 1 − e−iθnz (1 − r)c1 1 − e (1 − r)c2 (1 − r)cn −iθ1re + it −iθ2re + · · · + it −iθnreit 1 − e 1 − e William T. Ross (University of Richmond) Making the case for Cauchy transforms 12 / 42
(Kµ)(z) = = (1 − r)(Kµ)(re it ) = µ = c1δθ1 + c2δθ2 + · · · + cnδθn 2π 1 0 1 − e−iθ z dµ(θ) c1 c2 cn 1 − e−iθ1z + 1 − e−iθ2z + · · · + 1 − e−iθnz (1 − r)c1 1 − e (1 − r)c2 (1 − r)cn −iθ1re + it −iθ2re + · · · + it −iθnreit 1 − e 1 − e William T. Ross (University of Richmond) Making the case for Cauchy transforms 12 / 42
Page 1 and 2: Making the case for Cauchy transfor
Page 3 and 4: Cauchy transform of a measure µ d
Page 5 and 6: Theorem (Cauchy - 1831) If f is ana
Page 7 and 8: 2π f (z) = 0 f (eiθ ) 1 − e−i
Page 9 and 10: Example µ = δ0 (point mass at θ
Page 11 and 12: Example µ = δ0 (point mass at θ
Page 13 and 14: What I am going to talk about: Boun
Page 19 and 20: What I’m not going to talk about:
Page 21 and 22: What I’m not going to talk about:
Page 23 and 24: Boundary values William T. Ross (Un
Page 25: (Kµ)(z) = = (1 − r)(Kµ)(re it )
Page 29 and 30: (1 − r)(Kµ)(re it ) = Thus (1
Page 31 and 32: (1 − r)(Kµ)(re it ) = Thus (1
Page 33 and 34: For a general measure µ, lim r→1
Page 35 and 36: f (z) = ∞ j=1 cj 1 − e −iθj
Page 37 and 38: Theorem (Smirnov - 1929) For almost
Page 39 and 40: Theorem (Fatou - 1906) For almost e
Page 41 and 42: Theorem (Privalov - 1919) For almos
Page 43 and 44: Theorem (Privalov - 1919) For almos
Page 45 and 46: There are a multitude of results wh
Page 47 and 48: Theorem (M. Riesz - 1927) If 1 < p
Page 49 and 50: We began this talk with the Cauchy
Page 51 and 52: An extension of the CIF f (z) = 2
Page 53 and 54: Distribution function William T. Ro
Page 55 and 56: We know that the function is define
Page 57 and 58: For any f (θ) ∈ L 1 [0, 2π], we
Page 59 and 60: Example |Kδ0(e iθ )| = 1 |1 − e
Page 61 and 62: Boole’s idea: Suppose c1, · ·
Page 63 and 64: Theorem (Boole - 1857) Suppose c1,
Page 65 and 66: g(x) = c1 + x − a1 c2 + · · ·
Page 67 and 68: g(x) = c1 + x − a1 c2 + · · ·
Page 69 and 70: g(x) = c1 + x − a1 c2 + · · ·
Page 71 and 72: Theorem (Kolmogorov - 1925) m(|Kµ|
Page 73 and 74: Kolmogorov says lim sup ym(|Kµ| >
Page 75 and 76: Geometric conditions William T. Ros
Page 77 and 78:
Theorem (Herglotz - 1911) If f (D)
Page 79 and 80:
Example f (z) = 2z 1 = 1 − z2 1
Page 81 and 82:
Example f (z) = 2z 1 = 1 − z2 1
Page 83 and 84:
Consider the normalized Cauchy tran
Page 85:
Theorem (Poltoratski - 1993) lim r
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Making the case for Cauchy transforms

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