View source for Fejér-Riesz theorem

A [[Trigonometric polynomial|trigonometric polynomial]] is an expression in one of the equivalent forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f1100501.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f1100502.png" />. When the values of a trigonometric polynomial are real for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f1100503.png" />, the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f1100504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f1100505.png" /> in the first form are necessarily real, and those in the second form satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f1100506.png" /> for all indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f1100507.png" />. L. Fejér [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f1100508.png" /> that assumes only non-negative real values for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f1100509.png" /> is expressible in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005010.png" /></td> </tr></table>

for some polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005011.png" />. The polynomial can be chosen to have no roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005012.png" />, and then it is unique except for a multiplicative constant of modulus one.

The proof is based on the observation that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005013.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005014.png" /> as a function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005017.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005018.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005019.png" />. The roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005020.png" /> of modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005021.png" /> occur in pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005022.png" /> having equal multiplicity. Roots of unit modulus have even multiplicity. It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005024.png" /> have modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005026.png" /> is a positive constant. The desired representation is obtained with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005027.png" />. See [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005028.png" /> be a non-negative function which is integrable with respect to the normalized [[Lebesgue measure|Lebesgue measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005029.png" /> on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005030.png" />. If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005031.png" /></td> </tr></table>

then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005032.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005033.png" /> is the boundary function of an outer function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005034.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005035.png" /> of Hardy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005036.png" /> (cf. also [[Hardy classes|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005037.png" /> have been described in terms of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005038.png" /> (G. Szegő, [[#References|[a6]]], Chapt. 12). Here, the term outer means that the set of functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005040.png" /> is a polynomial, is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005041.png" />. The log-integrability hypothesis is automatically satisfied when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005042.png" /> is a trigonometric polynomial, and then the outer function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005043.png" /> is the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005044.png" /> 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|entire function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005045.png" /> of exponential type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005046.png" /> that is non-negative on the real axis and satisfies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005047.png" /></td> </tr></table>

can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005048.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005049.png" /> real, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005050.png" /> is an entire function of exponential type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005051.png" /> which has no zeros in the open upper half-plane (see [[#References|[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 [[#References|[a7]]], [[#References|[a8]]], [[#References|[a5]]], [[#References|[a9]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005052.png" /> be a trigonometric polynomial whose coefficients are operators on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005053.png" /> and which assumes non-negative selfadjoint values for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005054.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005055.png" /></td> </tr></table>

for some outer polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005056.png" /> whose coefficients are operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005057.png" />.

Here, the term outer is relative to the Hardy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005058.png" /> of functions with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005059.png" />: the meaning is that the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005061.png" /> is a polynomial with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005062.png" />, is dense in a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005063.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005064.png" /> for some subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005065.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005066.png" />.

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|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 [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R P. Boas,   "Entire functions" , Acad. Press  (1954)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Fejér,   "Über trigonometrische Polynome"  ''J. Reine Angew. Math.'' , '''146'''  (1916)  pp. 53–82</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Riesz,   B. Sz.-Nagy,   "Functional analysis" , F. Ungar  (1955)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Rosenblum,   J. Rovnyak,   "Hardy classes and operator theory" , Dover, reprint  (1997)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Yu.A. Rozanov,   "Spectral theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110050/f11005067.png" />-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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Szegő,   "Orthogonal polynomials" , ''Colloq. Publ.'' , '''23''' , Amer. Math. Soc.  (1975)  (Edition: Fourth)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Helson,   D. Lowdenslager,   "Prediction theory and Fourier series in several variables I"  ''Acta Math.'' , '''99'''  (1958)  pp. 165–202</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H. Helson,   D. Lowdenslager,   "Prediction theory and Fourier series in several variables II"  ''Acta Math.'' , '''106'''  (1961)  pp. 175–213</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  N. Wiener,   P. Masani,   "The prediction theory of multivariate stochastic processes I"  ''Acta Math.'' , '''98'''  (1957)  pp. 111–150</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  N. Wiener,   P. Masani,   "The prediction theory of multivariate stochastic processes II"  ''Acta Math.'' , '''99'''  (1958)  pp. 93–137</TD></TR></table>