Rudin's Theorem of Complex Analysis.

Rudin's Theorem
of Complex Analysis.

by 'Literka'.

There is a theorem of complex analysis, which is in a common use and almost nobody knows its origin. Literka analyzed all threads and realized that it was proved by Professor Walter Rudin. Since nobody indicates the origin of this theorem, Literka decided to write a page about it.

Theorem uses only basic definitions, but for completeness let us remind that:
Two functions f and g with values in a complex plane are called equidistributed, if for any measurable set A of this plane the sets

have the same measures.
Let

be a unit disk of a complex plane and T be a unit circle of U (set of all z with the absolute value 1).
We do not want to introduce whole theory - let us mention only that interesting us analytic functions defined on U have boundary values on T and these functions are determined by the boundary values on T.
Analytic function f defined on U, with absolute values bounded by 1 on U, is called inner if almost all boundary values of f (hence we regard f as function on T) have absolute values 1 (hence belong to T). That is why we can say that an inner function is a function defined on T with the range in T.
On T there is defined "normalized length" measure

in the ordinary way, assuming measure of T is equal 1. The simplest inner function is the identity function f(z)=z or any power of it.

Theorem (Rudin). All inner functions

such that f(0)=0 are equidistributed on T, with respect to the measure

.

Before we prove this theorem let us write few comments.
Of course, from this theorem it follows that all such inner functions must be equidistributed with the function I(z)=z. Since I is a measure-preserving function, all inner funtions f with f(0)=0 must be measure-preserving.

Some time ago, when Literka did not know Rudin's theorem, Literka took a part in a private conversation, when somebody asked about possible values of the expression Rozmiar: 3007 bajtów

where B(z) is a Blaschke product and we integrate with respect to the "normalized length" measure on T (Blaschke products are special cases of inner functions). After some computations they reached a proper answer. Literka writes about it because Rudin's theorem gives an immediate answer that there is only one value equal to (we replace z instead of z*B(z)) Rozmiar: 8067 bajtów

The real advantage of this theorem is that it has a nice and straightforward proof:

Proof of Rudin's theorem.
Let f be an inner function satisfying the assumptions of the Theorem and let A be a subset of T. We'll prove that f is a measure-preserving function. We'll denote

to be a characteristic function of a set A. We have Rozmiar: 3066 bajtów

Hence it is enough to prove that if h is a characteristic function of any set, then Rozmiar: 4198 bajtów

We'll prove more: that it is true for any measurable function h. Since countinuous functions are dense in the space of all measurable functions, it is enough to prove it for the case h is a continuous function.
The above equality is true for the functions

Assume first that n is positive. Then Rozmiar: 7311 bajtów

For n=0 we have Rozmiar: 5560 bajtów

Let us notice that

for z belonging to T.
Finally for negative n we have Rozmiar: 13482 bajtów

Finally let us make a remark that if the equality holds for these functions, it must hold for linear combinations of these functions. But linear combinations of these functions are dense in the space of continuous functions (Weierstrass theorem), which proves our statement.

Notice: Andreas Sauer wrote Literka that the above proof is in the book of Professor Walter Rudin "New constructions of functions holomorphic in the unit ball of C^n", AMS, 1986, (Theorem 1.3)

See the list and descriptions of mathematical pages from Mathematical Countryside.

See other pages of 'Literka' from Mathematical Countryside:

Weierstrass Approximation Theorem. Bernstein's Polynomials.
Monotonic subsequences.
A remarkable monotonic property of the gamma function .
Weight centers of simple geometrical figures.
An elementary problem can be unsolvable.
Roots of cubic equation. Cardano's formula.
Exact values of trigonometric functions of angles (n*pi)/17.
Equalities for values of trigonometric functions of angles (n*pi)/17.
Factorization of a polynomial, which defines values of sine function (angles n*pi/17).
Polynomials with roots cos(2*k*pi/n).
Factorization of polynomials with roots cos(2*k*pi/n), where n is Fermat number.
Values of trigonometric functions of angles (n*pi)/257. Part I.
Values of trigonometric functions of angles (n*pi)/257. Part II, Part III, Part IV, Part V, Part VI, Part VII.
Values of trigonometric functions of angles (n*pi)/65537. Part I.
Values of trigonometric functions of angles (n*pi)/65537. Part II, Part III, Part IV and Part V, Part VI, Part VII, Part VIII, Part IX, Part X, Part XI, Part XII, Part XIII, Part XIV.
Construction of a regular pentagon.
Construction of a regular heptadecagon.
Construction of a regular polygon with 257 sides.

Software that you really need.

Return to the list of pages of 'Literka' about polytopes.
Return to the main geometrical page of 'Literka'.
Return to the main page of 'Literka'.