Fejér-Riesz theorem

A trigonometric polynomial is an expression in one of the equivalent forms $ a _ {0} + \sum _ {1} ^ {n} [ a _ {j} \cos ( jt ) + b _ {j} \sin ( jt ) ] $ or $ \sum _ {- n } ^ {n} c _ {j} e ^ {ijt } $. When the values of a trigonometric polynomial are real for all real $ t $, the coefficients $ a _ {j} $, $ b _ {j} $ in the first form are necessarily real, and those in the second form satisfy $ {\overline{ {c _ {j} }}\; } = c _ {- j } $ for all indices $ j $. L. Fejér [a2] was the first to note the importance of the class of trigonometric polynomials that assume only non-negative real values. His conjecture on the form of such a function was proved by F. Riesz, and is nowadays known as the Fejér–Riesz theorem: A trigonometric polynomial $ w ( e ^ {it } ) = \sum _ {- n } ^ {n} c _ {j} e ^ {ijt } $ that assumes only non-negative real values for all real $ t $ is expressible in the form

$$ w ( e ^ {it } ) = \left | {p ( e ^ {it } ) } \right | ^ {2} $$

for some polynomial $ p ( z ) = \sum _ {0} ^ {n} a _ {j} z ^ {j} $. The polynomial can be chosen to have no roots in $ D = \{ z : {| z | < 1 } \} $, and then it is unique except for a multiplicative constant of modulus one.

The proof is based on the observation that $ w ( z ) = \sum _ {- n } ^ {n} c _ {j} z ^ {j} $ satisfies $ \overline{ {w ( 1/ {\overline{z}\; } ) } } = w ( z ) $ as a function of the complex variable $ z $. If $ c _ {- n } \neq 0 $, then $ q ( z ) = z ^ {n} w ( z ) $ is a polynomial of degree $ 2n $ with $ q ( 0 ) \neq 0 $. The roots of $ q ( z ) $ of modulus $ \neq 1 $ occur in pairs $ \alpha,1/ {\overline \alpha \; } $ having equal multiplicity. Roots of unit modulus have even multiplicity. It follows that $ w ( z ) = c \prod _ {1} ^ {r} [ ( z - \alpha _ {j} ) ( z ^ {- 1 } - {\overline{ {\alpha _ {j} }}\; } ) ] $, where $ \alpha _ {1} \dots \alpha _ {r} $ have modulus $ \geq 1 $ and $ c $ is a positive constant. The desired representation is obtained with $ p ( z ) = \sqrt c \prod _ {1} ^ {r} ( z - \alpha _ {j} ) $. See [a3] for a variation of this method and an application in spectral theory.

A generalization of the Fejér–Riesz theorem plays an important role in the theory of orthogonal polynomials.

Szegő's theorem: Let $ w ( e ^ {it } ) $ be a non-negative function which is integrable with respect to the normalized Lebesgue measure $ d \sigma = dt/ ( 2 \pi ) $ on the unit circle $ \partial D = \{ {e ^ {it } } : {0 \leq t < 2 \pi } \} $. If

$$ \int\limits _ {\partial D } { { \mathop{\rm log} } w ( e ^ {it } ) } {d \sigma } > - \infty, $$

then

$$ w ( e ^ {it } ) = \left | {h ( e ^ {it } ) } \right | ^ {2} \sigma - \textrm{ a.e. } , $$

where $ h ( e ^ {it } ) $ is the boundary function of an outer function $ h ( z ) $ on $ D $ of Hardy class $ H ^ {2} $( cf. also Hardy classes). Such a function is unique up to a multiplicative constant of modulus one.

The asymptotic properties of the polynomials which are orthogonal with respect to such a weight function $ w ( e ^ {it } ) $ have been described in terms of the function $ h ( z ) $( G. Szegő, [a6], Chapt. 12). Here, the term outer means that the set of functions of the form $ h ( z ) k ( z ) $, where $ k ( z ) $ is a polynomial, is dense in $ H ^ {2} $. The log-integrability hypothesis is automatically satisfied when $ w ( e ^ {it } ) $ is a trigonometric polynomial, and then the outer function $ h ( z ) $ is the polynomial $ p ( z ) $ in the Fejér–Riesz theorem.

The Fejér–Riesz and Szegő theorems are prototypes for two kinds of hypotheses which assure the existence of similar representations of non-negative functions. One type stipulates algebraic or analytical structure, the other that the given function is not too small. Non-negativity on the unit circle is often replaced by non-negativity on the real line. For example, the Akhiezer theorem states that an entire function $ w ( z ) $ of exponential type $ \tau $ that is non-negative on the real axis and satisfies

$$ \int\limits _ {- \infty } ^ \infty { { \frac{ { \mathop{\rm log} } ^ {+} w ( x ) }{1 + x ^ {2} } } } {dx } < \infty $$

can be written as $ w ( x ) = | {f ( x ) } | ^ {2} $ for $ x $ real, where $ f ( z ) $ is an entire function of exponential type $ \tau/2 $ which has no zeros in the open upper half-plane (see [a1]).

Related problems arise in linear prediction theory, but there the functions to be factored are operator valued. Such problems date back to the 1940{}s and 1950{}s (see [a7], [a8], [a5], [a9], [a10]). In this context, the term spectral factorization is used to describe the representation of non-negative operator-valued functions. Operator extensions of the Fejér–Riesz theorem were proved in special cases by several authors, the final form being that given by M. Rosenblum (operator version of the Fejér–Riesz theorem): Let $ W ( e ^ {it } ) = \sum _ {- n } ^ {n} C _ {j} e ^ {ijt } $ be a trigonometric polynomial whose coefficients are operators on a Hilbert space $ {\mathcal K} $ and which assumes non-negative selfadjoint values for all real $ t $. Then

$$ W ( e ^ {it } ) = P ( e ^ {it } ) ^ {*} P ( e ^ {it } ) $$

for some outer polynomial $ P ( z ) = \sum _ {0} ^ {n} A _ {j} z ^ {j} $ whose coefficients are operators on $ {\mathcal K} $.

Here, the term outer is relative to the Hardy class $ H _ {\mathcal K} ^ {2} $ of functions with values in $ {\mathcal K} $: the meaning is that the set of functions $ P ( z ) k ( z ) $, where $ k ( z ) $ is a polynomial with coefficients in $ {\mathcal K} $, is dense in a subspace of $ H _ {\mathcal K} ^ {2} $ of the form $ H _ {\mathcal M} ^ {2} $ for some subspace $ {\mathcal M} $ of $ {\mathcal K} $.

Analogous theorems hold for operator-valued functions satisfying Szegő-type hypotheses and for polynomials, rational functions, and entire functions of exponential type. However, the techniques used to prove the representation theorems in the scalar case are usually not applicable in the operator extensions. For example, the fundamental theorem of algebra (cf. Complex number), which is used in the proof of the Fejér–Riesz theorem, has no counterpart for operator-valued polynomials. A method due to D. Lowdenslager allows a unified approach to the operator extensions. Many results based on the method, including a proof of the operator Fejér–Riesz theorem, may be found in [a4].

References

[a1]	R P. Boas, "Entire functions" , Acad. Press (1954)
[a2]	L. Fejér, "Über trigonometrische Polynome" J. Reine Angew. Math. , 146 (1916) pp. 53–82
[a3]	F. Riesz, B. Sz.-Nagy, "Functional analysis" , F. Ungar (1955)
[a4]	M. Rosenblum, J. Rovnyak, "Hardy classes and operator theory" , Dover, reprint (1997)
[a5]	Yu.A. Rozanov, "Spectral theory of $n$-dimensional stationary stochastic processes with discrete time" Selected Transl. in Math. Statistics and Probab. , 1 (1961) pp. 253–306 Uspekhi Mat. Nauk , 13 : 2 (80) (1958) pp. 93–142
[a6]	G. Szegő, "Orthogonal polynomials" , Colloq. Publ. , 23 , Amer. Math. Soc. (1975) (Edition: Fourth)
[a7]	H. Helson, D. Lowdenslager, "Prediction theory and Fourier series in several variables I" Acta Math. , 99 (1958) pp. 165–202
[a8]	H. Helson, D. Lowdenslager, "Prediction theory and Fourier series in several variables II" Acta Math. , 106 (1961) pp. 175–213
[a9]	N. Wiener, P. Masani, "The prediction theory of multivariate stochastic processes I" Acta Math. , 98 (1957) pp. 111–150
[a10]	N. Wiener, P. Masani, "The prediction theory of multivariate stochastic processes II" Acta Math. , 99 (1958) pp. 93–137