Kenneth Kuttler

1806 CHAPTER 57. INFINITE PRODUCTS

Corollary 57.5.3 Suppose f is an analytic function on B(0,1) and | f (z)| ≤ M for allz ∈ B(0,1) . Suppose also that the nonzero zeros4 of f are {αk}∞

k=1 , listed according tomultiplicity. Then ∑

∞k=1 (1−|αk|)< ∞.

Proof: Suppose f has a zero of order m at 0. Then consider the analytic function,g(z) ≡ f (z)/zm which has the same zeros except for 0. The argument goes the sameway except here you use g instead of f and only consider r > r0 > 0. Thus from Jensen’sequation,

ln |g(0)|+n(r)

∑i=1

lnr− ln |α i|

2π

∫ 2π

0ln∣∣∣g(reiθ

)∣∣∣dθ

2π

∫ 2π

0ln∣∣∣ f (reiθ

)∣∣∣dθ − 12π

∫ 2π

0m ln(r)

≤ M+1

2π

∫ 2π

0m ln

(r−1)

≤ M+m ln(

1r0

Now the rest of the argument is the same.An interesting restatement yields the following amazing result.

Corollary 57.5.4 Suppose f is analytic and bounded on B(0,1) having zeros {αn} . Thenif ∑

∞k=1 (1−|αn|) = ∞, it follows f is identically equal to zero.

57.5.1 The Müntz-Szasz Theorem Again

Corollary 57.5.4 makes possible an easy proof of a remarkable theorem named abovewhich yields a wonderful generalization of the Weierstrass approximation theorem. Inwhat follows b > 0. The Weierstrass approximation theorem states that linear combina-tions of 1, t, t2, t3, · · · (polynomials) are dense in C ([0,b]) . Let λ 1 < λ 2 < λ 3 < · · · be anincreasing list of positive real numbers. This theorem tells when linear combinations of1, tλ 1 , tλ 2 , · · · are dense in C ([0,b]). The proof which follows is like the one given in Rudin[113]. There is a much longer one in Cheney [33] which discusses more aspects of thesubject. See also Page 533 where the version given in Cheney is presented. This otherapproach is much more elementary and does not depend in any way on the theory of func-tions of a complex variable. There are those of us who automatically prefer real variabletechniques. Nevertheless, this proof by Rudin is a very nice and insightful application ofthe preceding material. Cheney refers to the theorem as the second Müntz theorem. I guessSzasz must also have been involved.

4This is a fun thing to say: nonzero zeros.

1806 CHAPTER 57. INFINITE PRODUCTSCorollary 57.5.3 Suppose f is an analytic function on B(0,1) and |f (z)| <M for allz € B(0,1). Suppose also that the nonzero zeros* of f are {a}, , listed according tomultiplicity. Then Ye_, (1 — |Ox|) <°%.Proof: Suppose f has a zero of order m at 0. Then consider the analytic function,g(z) =f (z)/z” which has the same zeros except for 0. The argument goes the sameway except here you use g instead of f and only consider r > ro > 0. Thus from Jensen’sequation,n(r)In|g (0)|+ }° Inr—In|a;|i=l~ ae vle() |1 ) 20= ah In|f (re )\co-= | mln(rQnmin (r')M —+ 27 Jo1< M-+min({—}.roNow the rest of the argument is the same.An interesting restatement yields the following amazing result.IACorollary 57.5.4 Suppose f is analytic and bounded on B(0,1) having zeros {,}. Thenif Ye_; (1 —|Qn|) = ©, it follows f is identically equal to zero.57.5.1 The Miintz-Szasz Theorem AgainCorollary 57.5.4 makes possible an easy proof of a remarkable theorem named abovewhich yields a wonderful generalization of the Weierstrass approximation theorem. Inwhat follows b > 0. The Weierstrass approximation theorem states that linear combina-tions of 1,t,t7,1°,--- (polynomials) are dense in C([0,b]). Let A; < Az <A3 <-:- beanincreasing list of positive real numbers. This theorem tells when linear combinations of1,171,142,--. are dense in C ([0,b]). The proof which follows is like the one given in Rudin[113]. There is a much longer one in Cheney [33] which discusses more aspects of thesubject. See also Page 533 where the version given in Cheney is presented. This otherapproach is much more elementary and does not depend in any way on the theory of func-tions of a complex variable. There are those of us who automatically prefer real variabletechniques. Nevertheless, this proof by Rudin is a very nice and insightful application ofthe preceding material. Cheney refers to the theorem as the second Miintz theorem. I guessSzasz must also have been involved.4This is a fun thing to say: nonzero zeros.