REMARKS ON EXTREME EIGENVALUES OF TOEPLITZ MATRICES

Let f be a nonnegative integrable function on [-r,r], T.Cf) the Ca+l) X(n+l) Toeplitz matrix associated with f and k,. its smallest eigenvalue. It is shown that the convergence of )t,. to rain f(O) can be exrmnentiallv fast even when f does not satisfy the smoothness condition of Kae, Murdoeh and Szeg6 (1953). Also a lower bound for ),,. corresponding to a large class of functions which do not satisfy this smoothness condition is provided.


INTRODUCTION
Toeplitz matrices and operators arise in several areas of mathematics and its applications such as complex and harmonic analysis, probability theory and statistics, signal processing and information theory, numerical analysis, etc.Of particular interest in these areas are the determinant, and the distribution of the eigenvalues of finite sections of an infinite Toeplitz matrix.In this paper we are concerned with the estimates on the asymptotic behavior of the extreme eigenvalues of finite sections of Toeplitz matrices in the spirit of the works of Kac, Murdoch and Szeg5 [5], Widom [9] and Parter [7].While these uthors are concerned with the asymptotic behavior of extreme eigenvalues of finite sections of infinite Toeplitz matrices associated with continuous and continuously differentiable functions f(0), 0 [-r,r], see Condition A in Section 2, we study similar properties of Toeplitz matrices which are associated with functions which are not (necessarily) continuous nor differentiable.
Our restriction on f involves either frequency or order (multiplicity) of zeros of this function on I-Condition A imposes several severe restrictions on f which might be hard to verify or they may not be satisfied in some areas of application such as the prediction theory of stationary processes and signal processing, where f is viewed as the spectral density of a stationary stochastic process.As such f may have finitely many or countably many zeros and thus min f(0) is attained at several points so that Condition A is not satisfied.In view of this it is important to have some information about the rate of convergence of k,, to zero when Condition A is not satisfied.
The main results of the paper are in Scction 3. Thcorcm 3.2 providcs an upper bound for >,., in tcrms of outer capacity of the spectrum of f.This in turn is used to show that ),., can converge to zero exponentially fast even when thc Oondilfion A is not satisfied.Thcorcm 3.6 givcs a lowcr bound for kt., under a mild condition on the order of zeros of the function f. is called the nth finite section of the infinite Toeplitz matrix (c_i) associated with the function f.Let rn denote the essential infimum of the function f and Xl, the smallest eigenvalue of the matrix T,(f).It follows from a theorem of Szegq [3, p. 64], that as oo, Xl,, It is of interest to find the rate of this convergc.,.., So far this rate has been found under the following regularity (smoothness) condition on f [5, 6, 8].
Condition A. Let f(O) be real, continuous and periodic with period 2r.Let min f(O) --0--m and let --0 be the only value of 0(rood 2r) for which this minimum is attained.Moreover, let f(0) have continuous derivatives of order 2a in some neighborhood of --0 with f(2}(0) yg 0.
Under Condition A it is shown by Kac, Murdoch and Szego [5] and Parter [6] that X1, rn--0(1-) (2.1) In the following, without loss of generality, we assume that m -----0, i.e. f (a) _> 0. As a direct consequence of the well-known Weyl-Courant lemma cf.[6, p. 155], one can identify k, as the solution of a variational problem; This identification provides a very useful formula for finding bounds and consequently rates of convergences of ),,n to zero.
For a nonnegative function f, L(f) stands for the space of functions which are square integrable with d Le respect to the measure f on [-r,r] stands for the usual aebesgue space of functions on [-r,r].
3. BOUNDS AND RATES OF CONVERGENCE FOR )X 1,n" In this section rates of convergence of X, to zero is provided for some functions f which do not necessarily satisfy Condition A. This is done by showing that for any nonnegative Lebesgue integrable function f, X, is dominated by the well-studied quantity , where A --dct T,,(f).One may find the results of the first part A_ of this section a bit surprising.Since the rate of convergence of X,n to zero is much faster than (2.1) for some functions which do note satisfy Condition A. This, in particular, shows that smoothness of f is not necessarily required for fast rate of convergence of >,, to zero, cf. [5,6].
LEMMA 3.1.Let / be a nonncgative Lcbcsguc integrable function defined on [-rr,r I. Let Xt, denote the smallest eigenvalue of its associated Toeplitz matrix T(f).Then, PROOF.For each n, let P, denote the class o1' all a.n,dytic trigonometric polynonials of degree Icsb than or equN mn.We have from (2.2) that X,, ,,, f I, I: <_ f I I: for y P, with (0)=1.Next, we choose m be the unique polynomial in P, with (0)=1 which dO minimizes the ingraI f[ [=f over all Pn with (0) =1.It follows from a theorem of Szeg5 [3, p.38], that where {} is the sequence of orthogonormal polynomiNs obtained by orthogonormalizing the functions g-1,z,ze,..., in L2(f) and k is the leading coemcient of , i.e. k =.Thus, Since IIbll= _> 1, it follows that dO k,., < Thts lemma provides an upper bound for ),,,, in rtns of the ratio .The later, in prediction theory of a stionary smchtic process with spectrl density f, is identified the prediction error of a future vlue of the process bed on n immedia pt observations.As such the ymptic behavior of extreme eigenvducs of Toeplitz matrices T,(f) is relad e ymptic behavior or the fini prediction error of a stionary process with the spectrM density f.
For a nonnegativc function f the set E defined by E ={0;/(0) > 0} is referred the spectrum of the function f.For a set E in the complex plme r(E) sds for i Capity and r*(E) sds for our capity.For e definition of ese rms see [1, 2].
The following theorem is immedia consequence of Theorem of Babayan [1] and Lemma 3.1.
THEOREM 3.2.Let f mad kt,, bc zm in Lcmma 3.1 mad r'(E) deno the our capacity of i spectrum E.
Then, " lim S r "(E) Thus, in order that X,,, should decree m zero at let exponentiMly n , it is sufficient that the our capity of the spectrum of f be less th 1.As such the continuity d differentiability of f e not required for the exponentii ra of convergence of X,,, zero.The next two examples provide more informaon on the ra of convergence of X,,, zero for two specific functions.
I is imporg m no ghag in his example he essengiN infimum m =0 is agained a uneounly many poin d , in generN, hog ennuous on N or ag the endpin of N. Thus i doe no satisfy Cndiion A and the ra of convergence of X,, is much fr th (2.1).EXAMPLE 3.4.Let f(0) =cxp[(.2O-r).(O)]where (0) = ctn forsomc > 0 ltiscyshow that for this f 0 r lsin so that f(0) h a very high order contact xvith zero only at 8 =0.It follows from Rosenblatt [7, p. 810] md Lemma 3.1 th Thus, by choosing lm'gc enough one cm obtain very ft ra of convergence of kt., m zero even when f(0) is not differentile t =0.
In the rest of this section xve find a lower bound for kt,, corresponding functions f xvi f-L Agn no reference is me continuity or differentiability of the function f.Of course, ingrbility of fmposes restriction on the order of zeros of f.It is cy check (by using Holder's inequality) that for any L(f) dOl cf 111 ,,viU, c=f /-dO f I1 2.

Call for Papers
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