Beurling's Theorem Research Articles

Extrapolation and Moving Average Representation for Stationary Random Fields and Beurling's Theorem

Strong regularity for stationary discrete random fields is discussed. An extension of the classical Beurling's Theorem to functions of several variables is given. Necessary and sufficient conditions for the moving average representation of stationary random fields are obtained. A recipe formula for the best linear extrapolator is also given.

Notes on Beurling's theorem

Notes on Beurling's theorem

On Fatou's and Beurling's Theorems

On Fatou's and Beurling's Theorems

On stochastic realization theory

Using the Beurling-Lax Theorem and properties of the backward shift operator a stochastic realization theory is developed. First it is shown that there is a one to one correspondence between the set of all purely nondeterministic Markov process in L 2 and the set of all functions {⊘,F}, such that <⊘> is inner and F is rigid. Using this classification, the set of all Markov representations for a wide sense stationary process is obtained. In particular minimal, constructible, and observable Markov representations are studied in detail.

The Szegö and Beurling theorems and operator factorizations

The Szegö and Beurling theorems and operator factorizations

On spectral analysis in locally compact motion groups

Beurling's theorem in spectral analysis of bounded functions on the real line is generalized to a class of locally compact motion groups and to Heisenberg groups.

On Beurling's Theorem for Locally Compact Groups

On Beurling's Theorem for Locally Compact Groups

Correction of a Generalization of a Theorem of Beurling and Livingston

A generalization of a Riesz-Fischer theorem proved by Beurling and Livingston for smooth uniformly convex Banach spaces also holds for smooth, strictly convex, reflexive Banach spaces. Theorem. Let B be a smooth, strictly convex, reflexive Banach space. Let $L:B \to {B^\ast }$ be a duality map, C a closed subspace of $B,h \in B,k \in {B^\ast }$. Then $T(C + h) \cap ({C^ \bot } + k)$ is a single point. A two-dimensional counterexample shows that $T(C + h) \cap ({C^ \bot } + k) = \emptyset$ is possible if B is not smooth, contrary to the claim of Theorem 4 of Browder, On a theorem of Beurling and Livingston, Canad. J. Math. 17 (1965), 367-372.

Harmonic estimation in certain slit regions and a theorem of Beurling and Malliavin

Harmonic estimation in certain slit regions and a theorem of Beurling and Malliavin

Correction of a generalization of a theorem of Beurling and Livingston

A generalization of a Riesz-Fischer theorem proved by Beurling and Livingston for smooth uniformly convex Banach spaces also holds for smooth, strictly convex, reflexive Banach spaces. Theorem. Let B be a smooth, strictly convex, reflexive Banach space. Let L : B → B ∗ L:B \to {B^\ast } be a duality map, C a closed subspace of B , h ∈ B , k ∈ B ∗ B,h \in B,k \in {B^\ast } . Then T ( C + h ) ∩ ( C ⊥ + k ) T(C + h) \cap ({C^ \bot } + k) is a single point. A two-dimensional counterexample shows that T ( C + h ) ∩ ( C ⊥ + k ) = ∅ T(C + h) \cap ({C^ \bot } + k) = \emptyset is possible if B is not smooth, contrary to the claim of Theorem 4 of Browder, On a theorem of Beurling and Livingston, Canad. J. Math. 17 (1965), 367-372.

A Theorem of Beurling and Tsuji is Best Possible

A Theorem of Beurling and Tsuji is Best Possible

A theorem of Beurling and Tsuji is best possible

We shall show that Beurling-Tsuji’s theorem (see Theorem A) is, in a sense, best possible. For each pair a , b ∈ ( 0 , + ∞ ) a, b \in (0, + \infty ) there exists a function f holomorphic in { | z | &gt; 1 } \{ |z| &gt; 1\} such that the Euclidean area of the Riemannian image of each non-Euclidean disk of non-Euclidean radius a, is bounded by b, and such that f has finite angular limit nowhere on the unit circle.

One-Parameter Semigroups Holomorphic Away from Zero

Suppose $T$ is a one-parameter semigroup of bounded linear operators on a Banach space, strongly continuous on $[0,\infty )$. It is known that $\lim {\sup _{x \to 0}}|T(x) - I| < 2$ implies $T$ is holomorphic on $(0,\infty )$. Theorem I is a generalization of this as follows: Suppose $M > 0,0 < r < s$, and $\rho$ is in (1,2). If $|{(T(h) - I)^n}| \leq M{\rho ^n}$ whenever $nh$ is in $[r,s],n = 1,2, \cdots ,h > 0$, then there exists $b > 0$ such that $T$ is holomorphic on $[b,\infty )$. Theorem II shows that, in some sense, $b \to 0$ as $r \to 0$. Theorem I is an application of Theorem III: Suppose $M > 0,0 < r < s,\rho$ is in (1,2), and $f$ is continuous on $[ - 4s,4s]$. If $|\sum _{q = 0}^n \binom {n}{q}( - 1)^{n - q} f(t + qh)| \leq M \rho ^n$ whenever $nh$ is in $[r,s]$, $n = 1$, $2$, …, $h > 0$, $[t,t + nh] \subset [ - 4s,4s]$, then $f$ has an analytic extension to an ellipse with center zero. Theorem III is a generalization of a theorem of Beurling in which the inequality on the differences is assumed for all $nh$. An example is given to show the hypothesis of Theorem I does not imply $T$ holomorphic on $(0,\infty )$.

One-parameter semigroups holomorphic away from zero

Suppose T T is a one-parameter semigroup of bounded linear operators on a Banach space, strongly continuous on [ 0 , ∞ ) [0,\infty ) . It is known that lim sup x → 0 | T ( x ) − I | &gt; 2 \lim {\sup _{x \to 0}}|T(x) - I| &gt; 2 implies T T is holomorphic on ( 0 , ∞ ) (0,\infty ) . Theorem I is a generalization of this as follows: Suppose M &gt; 0 , 0 &gt; r &gt; s M &gt; 0,0 &gt; r &gt; s , and ρ \rho is in (1,2). If | ( T ( h ) − I ) n | ≤ M ρ n |{(T(h) - I)^n}| \leq M{\rho ^n} whenever n h nh is in [ r , s ] , n = 1 , 2 , ⋯ , h &gt; 0 [r,s],n = 1,2, \cdots ,h &gt; 0 , then there exists b &gt; 0 b &gt; 0 such that T T is holomorphic on [ b , ∞ ) [b,\infty ) . Theorem II shows that, in some sense, b → 0 b \to 0 as r → 0 r \to 0 . Theorem I is an application of Theorem III: Suppose M &gt; 0 , 0 &gt; r &gt; s , ρ M &gt; 0,0 &gt; r &gt; s,\rho is in (1,2), and f f is continuous on [ − 4 s , 4 s ] [ - 4s,4s] . If | ∑ q = 0 n ( n q ) ( − 1 ) n − q f ( t + q h ) | ≤ M ρ n |\sum _{q = 0}^n \binom {n}{q}( - 1)^{n - q} f(t + qh)| \leq M \rho ^n whenever n h nh is in [ r , s ] [r,s] , n = 1 n = 1 , 2 2 , …, h &gt; 0 h &gt; 0 , [ t , t + n h ] ⊂ [ − 4 s , 4 s ] [t,t + nh] \subset [ - 4s,4s] , then f f has an analytic extension to an ellipse with center zero. Theorem III is a generalization of a theorem of Beurling in which the inequality on the differences is assumed for all n h nh . An example is given to show the hypothesis of Theorem I does not imply T T holomorphic on ( 0 , ∞ ) (0,\infty ) .

On Beurling's theorem

On Beurling's theorem

Representing measures for R( X) and their Green's functions

Let X be a compact subset of the complex plane with a nonempty interior, R( X) the uniform closure in C( X) of the rational functions with poles off X, and m a representing measure on ∂X for the functional on R( X) of evaluation at a point a in int X. Let N 2 be the space of functions f in L 2( m) satisfying ∝ f dm = ∝ f K dm = 0 for all h in R( X), and let T be the operator on N 2 of multiplication by z followed by projection onto N 2. The spectral properties of T are investigated and shown to depend in part on the behavior of the so-called Green's function of m. In case m is the harmonic measure on ∂X for a the latter function is the classical Green's function for int X with singularity at a. Special attention is paid to the case where X is the closure of a finitely connected Jordan domain whose boundary curves are analytic. In that context, new proofs are given of Beurling's invariant subspace theorem and of Forelli's theorem on extreme points in the unit ball of the Hardy space H 1( m).

On a theorem of beurling and Carleson

On a theorem of beurling and Carleson

On generalized Beurling's theorem and symmetry of $L_1$-group algebras

On generalized Beurling's theorem and symmetry of $L_1$-group algebras

A set of generalized numbers showing Beurling's theorem to be sharp

A set of generalized numbers showing Beurling's theorem to be sharp

Characterization of certain invariant subspaces ofHpandLpspaces derived from logmodular algebras

Let A = A(X) be a logmodular algebra and m a representing measure on X associated with a nontrivial Gleason part. For 1 ^ p ^ oo, let Hp(dm) denote the closure of A in Lp(dm) (w* closure for p — oo). A closed subspace M of Hp(dm) or Lp(dm) is called invariant if fe M and g e A imply that fg e M. The main result of this paper is a characterization of the invariant subspaces which satisfy a weaker hypothesis than that required in the usual form of the generalized Beurling theorem, as given by Hoffman or Srinivasan. For 1 ^ p ^ oo, let Ip be the subspace of functions in Hp(dm) vanishing on the Gleason part of m and let Am = ifeA: fdm — ok THEOREM. Let M be a closed invariant subspace of L 2(dm) such that the linear span of A mM is dense in M but the subspace R = {feM:fA_I°°M} is nontrivial and has the same support set E as M. Then M has the form χE F'(P)L for some unimodular function F. A modified form of the result holds for 1 <^ p <Ξ oo. This theorem is applied to give a complete characterization of the invariant subspaces of Lp(dm) when A is the standard algebra on the torus associated with a lexicographic ordering of the dual group and m is normalized Haar measure.