. Itroductio. Let T be the uit circle i the complex plae. For p<, let L p be the Baach space of all complex-valued Lebesgue measurable fuctios f o T f

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1 A Note o the Besov Space B y Raymod H. Cha Departmet of Mathematics The Chiese Uiversity ofhogkog ad Ma-Chug Yeug Departmet of Mathematics Uiversity of Califoria, Los Ageles Abstract. We cosider complex-valued fuctios f deed o the uit circle T that are cotiuous for all t T except at a poit t 0 where the left- ad right-had limits of f both exist. Usig matrix methods, we show that if f is i the Besov class B (T ), the f is cotiuous at t 0. I particular, we prove that if the left- ad right-had limits of f are ot equal at t 0, the P k=; jkjja k[f]j =, where a k [f] arethefourier coeciets of f. Key Words. Besov class, Toeplitz matrix, Circulat matrix, Hilbert matrix. AMS(MOS) Subject Classicatios. 6E30, 6B99 y Research supported i part by HKRGC grat o Typeset by AMS-TE

2 . Itroductio. Let T be the uit circle i the complex plae. For p<, let L p be the Baach space of all complex-valued Lebesgue measurable fuctios f o T for which the L p orm kfk p Z f(e i ) do p p ; is ite. For R, thesetofrealumbers, we dee the operator as ( f)(e i ) f ; e i(+) ; f ; e i 8 R: The for all atural umber, we let ; : For >0 ad p<, the Besov classb p B p = where is ay iteger such that >. is deed as Z o f L p : jj ;;p kfk p pd < ; Awell-kow theorem about the class B p states that if <p< ad >=p, the all fuctios i B p are cotiuous fuctios, see Bottcher ad Silberma [, p.]. I this paper, we will use matrix methods to discuss the case whe p = ad ==. Our mai result is the followig Theorem. If f B is cotiuous at every poit t T f;g ad both f(;+0) lim!0+ f; e i(;)

3 3 ad f(; ; 0) lim!0+ f; e i(;+) exist, the f(;+0)=f(; ; 0). As a immediate corollary, we also prove Theorem. Let f be ay arbitrary complex-valued fuctio deed o T. If f is cotiuous at every poit t T f;g ad both f(;+0) ad f(; ; 0) exist but f(;+0)6= f(; ; 0), the k=; where a k [f] are the Fourier coeciets of f. jkjja k [f]j = Before carryig out our proof, we eed several deitios ad lemmas.. Deitios ad Lemmas. Give f L,wedeeitsFourier coeciets a k [f] by a k [f] = Z ; f(e i )e ;ik d k =0 : Let A [f] deote the -by- Toeplitz matrix with the (j `)th etry give by a j;`[f]. If f is real-valued, the a ;k [f] = a k [f] ad hece A [f] is a Hermitia matrix. Let C [f] be the -by- circulat matrix i which the(j `)th etry is give by c j;`[f] where c k [f] = 8 >< >: ( ; k)a k [f]+ka k; [f] c +k [f] Clearly, C [f] will be a Hermitia matrix if f is real-valued. 0 k< 0 < ;k <:

4 A sequece of matrices fm g = is said to have clustered spectra if for ay >0, there exists a N > 0suchthatfor all, at most N eigevalues of M have absolute values exceedig. As examples, we cosider the followig Lemmas. Lemma. Let fm g = be a sequece of Hermitia matrices. where kk F deotes the Frobeius orm, the fm g has clustered spectra. If sup km k F < Proof. Sice the square of the Frobeius orm of a Hermitia matrix is equal to the sum of the square of its eigevalues, it follows that for ay give > 0, M has at most sup km kf eigevalues with absolute values greater tha. Lemma. Let f be a real-valued cotiuous fuctio o T. The the sequece of matrices [f] A [f] ; C [f] =0 has clustered spectra. Proof. See Cha ad Yeug [, Theorem ]. Lemma 3. If f is a real-valued fuctio i B =, the f [f]g has clustered spectra. Proof. We rst ote that the space B = admits a very simple descriptio, amely f B = () k=; (jkj +)ja k [f]j < () see for istace, Bottcher ad Silberma [, p.]. Sice the rst row of the Hermitia Toeplitz matrix [f] =A [f] ; C [f] is give by 0 ; a; [f] ; a ; [f] ; a; [f] ; a ; [f] ; ; a;+ [f] ; a [f]

5 5 we have ; k [f]k F = k= ; = k= ; k= ; = k= ( ; k)k a;k [f] ; a ;k [f] ( ; k)k ( ; k)k ; k ; kja k [f]j k= k=; ; ja;k [f]j + ja ;k [f]j kja k [f]j By Lemma, f [f]g has clustered spectra. ja ;k [f]j + ( ; k) k o ja k [f]j (jkj +)ja k [f]j < : Lemma. Let H bethe-by- Hilbert matrix, i.e. H = ; The for ay >0, the umber of eigevalues of H which exceed >0 is asymptotically : equal to log sech; : I other words, fh g does ot have clustered spectra. Proof. See Widom [3, p.3].

6 6 3. Proofs of Theorems. Proof of Theorem : It is eough to prove the theorem for real-valued fuctios. Thus let f be a real-valued fuctio i B =. Assume that f is cotiuous at every poit t T f;g with both f(; +0)= lim!0+ f(ei(;) ) ad f(; ; 0) = lim!0+ f(ei(;+) ) exist, but f(;+0)6= f(; ; 0). Dee g(e i )= for all ; < ad let = f(;+0); f(; ; 0) 6= 0: The f ; g is a cotiuous fuctio o T. By Lemmas 3 ad, both f [f]g ad f [f ; g]g have clustered spectra. Sice g = ; f ; (f ; g), [g] = [f] ; [f ; g] ad hece f [g]g has clustered spectra by Cauchy's iterlace theorem, see for istace Wilkiso [, p.0]. The Fourier coeciets a k [g] ofg are give by Z a k [g] = e ;ik d = ; 8 < 0 k =0 : (;) k k i k = : Therefore, for all m>0, therstrowofthem-by-m Hermitia Toeplitz matrix m [g] is give by 0 = m 0 ; a; [g] ; a m; [g] m m ; i ; (;)k+ i m ; m ; k i i : ; a; [g] ; a m; [g] m ; ; a;m+ [g] ; a [g] m

7 7 Let P m ad Q m deote the m-by-m diagoal matrices with (;) j+ i ad (;) m+j+ as their (j j)th etries respectively ad let m [g] be partitioed as Wm U m [g] = m W m U m where W m ad U m are m-by-m Toeplitz matrices. The Pm 0 P 0 Q m [g] m 0 Pm W m 0 Q = m Pm m Q m Um P m Pm W = m Pm J m H m P mu m Q Q m mw m Q m H m J m Q m W m Q m where J m = is the m-by-m ati-idetity matrix ad H m is the m-by-m Hilbert matrix. Let Pm W m = m Pm 0 0 Q m W m Q m ad The we have Y m = 0 H m J m J m H m 0 Pm 0 P 0 Q m [g] m 0 m 0 Q = m + Y m : () m Sice k m k F = kp m W m P m k F + kq m W m Q m k F m; m ; k =kw m k F = (m ; k) Z 0 k= ; t dt =log; ( ; t)

8 8 f m g has clustered spectra by Lemma. Recall that f m [g]g also has clustered spectra, therefore from () ad Cauchy's iterlace theorem, fy m g has clustered spectra. Let R m = p Im I m J m ;J m where I m is the m m idetity matrix. Clearly, R mr m = I m. Hece fr my m R m g has clustered spectra. However, R my m R m = Hm 0 0 ;H m This implies that fh m g has clustered spectra, a cotradictio to Lemma. : Proof of Theorem : Just use () ad Theorem. We ally ote that sice estimates of the form () oly hold for Besov space B p where p =ad ==, the matrix method used here will ot work for larger Besov spaces.. Ackowledgemet. We would like to thak Professors Olof Widlud ad J. Marti for their valuable commets ad help i the preparatio of this paper.

9 9 Refereces. [] A. Bottcher ad B. Silberma, Aalysis of Toeplitz Operators, Spriger-Verlag, Berli, 990. [] R. Cha ad M. Yeug, Circulat Precoditioers for Toeplitz Matrices with Positive Cotiuous Geeratig Fuctios, Math. Comp., 58 (99), pp. 33{0. [3] H. Widom, Hakel Matrices, Tras. Amer. Math. Soc., (966), pp. {35. [] J. Wilkiso, The Algebraic Eigevalue Problem, Claredo Press, Oxford, 965.