On the mean value of certain functions connected with the convergence of orthogonal series

Transcription

1 Analysis Mathematica, 4 (1978), On the mean value of certain functions connected with the convergence of orthogonal series B. S. KASIN Dedicated to Professor P. L. UVjanov on his 50th birthday In the study of the almost everywhere (a.e.) convergence of orthogonal series often arises the following question. Let Ф = Ф(и) = {<^(х)}" =1 be an orthonormal system of functions defined on the segment [0, 1]. Let us define the operator 1 S%: lg->~l 2 (Q, 1) in the following way. If у = {у^=1 б 1\, then (1) St(y) =f(x) = sup Let s(<p) denote the norm of the operator S%, i.e., (2) 5(Ф)= sup \\Si(y)h4o.i)- llyllm^l We would like to estimate the number,у(ф). The classical result of D.E. Mensov and H. Rademacher (cf. [3, p. 188]) says that for every Ф = Ф(п) 2 we have (3) s(^)^clnw. For every n^l Mensov (cf. [3, p. 192]) exhibited an example of a system Ф 0 =Ф 0 (п) such that (4) 5(Ф 0 )^с1пл. In the present paper we shall study the "mean values" of the norms,у(ф). To illuminate the precise meaning of the term "mean value" we consider the following set of orthonormal systems 2" = {Ф}. Received March 1, By 11 we denote the space R n with Euclidean norm. 2 In the sequel С, с, and В denote positive absolute constants.

2 28 В. S. Kasin If Ф б", then Ф = {(р 1 (х)}" ==1 and each function <р ( (х) is piecewise constant on the n subintervals of [0, 1] of equal length. There is a one-to-one correspondance between the systems <P Q n and the elements of the group O n of orthogonal matrices of order n. Namely, to the system Ф = {Ф^(Х)} corresponds the matrix А = {а^}^о п of the following form: (5),,.,==-L^i^/i) il*l,j*n). With the aid of this correspondance the Haar measure fx n defined on O n (cf. [6, p. 43]) may be carried over onto the set Q n. In the sequel the measure of a closed set Ее Q n shall be denoted also by ц п (Е). We have the following Theorem. There exist absolute constants С and y>0 such that for every пш\ and t^o we have (6) M* fi": *(*) s* t} ^ (Се-* 2 )". The statement of this theorem is included in [4] without proof. From this theorem we immediately obtain the corollaries below. Corollary 1. There exists an absolute constant В such that (б 7 )!л п {Феа п :8(Ф)^В}^е- п. Corollary 2. Let S(n) denote the set of all permutations of the numbers 1,2,...,TI. For G S(n). let Ф^ denote the rearrangement of the terms of Ф ( 6") according to a. Then [хлф^о 1 : max $(Ф а ) шс Yhin} < n" n. From the estimation (6') it follows that there are "very few" systems having the property discovered by Mensov (we emphasize that in (4) the system Ф 0 (п) may be chosen from Q n ). For the proof of Theorem 1 we need some lemmas. First we shall introduce some notations. The number of elements of any finite set E shall be denoted by \E\. Furthermore, if x R m, then (x)j (l^j^m) denotes the jth coordinate of the vector x, and, as usually, R\ ={x = {(х)$ =1 : (x)j ^ 0, 1 ^ j ^ n}; S» = {x R n : \\х\\ п = 1}. If L is a subspace in R n, dim L = q 9 then m L denotes the normed Lebesgue measure on the g-dimensional sphere S n DL (m L (S n f]l) = l). The measure m Rn (E) shall be denoted simply by m(e).

3 Convergence of orthogonal series 29 Lemma 1. For each rn^l one may choose a set of vectors Q m ={e} such that (i) GJ<C M ; (ii) if e Q m, then e R m + and e,-sl/3; (Hi) for any vector y S m C)R"l there is a vector e=e(y) Q m such that (e(y))js(y)j (ls/sm). Proof. Suppose that y S m f}r1. Let us set for 5=0,1,2,... * E s (y) = {j : 2 1 < уj ё 2-, 1 s ; = m}. It is easy to see that (7) a) b) ( i: г \E,(y)\ < 2* +», 2 /ST^-j) 1/2 = ]^1 2 " For each j.s m n.r+ let us define the vector e(y) R m in the following way: (2-*- 1 if j-сад, OSss [1 log 2 m] +1, (8) (e(y))j=\ (0 if ttl>e,(y), 0^ss[-log 2 mj. From the construction of e{y) it is obvious that (9) 0^(e(y))j^(y)j. Besides, in virtue of the definition of E s (y) and part b) of relation (7) we have l f l J ' Let ««= U e(y). y S m nr On account of (9) and (9') to conclude the proof of Lemma 1 it is enough to show that O w ^C m. It is obvious that for any j;<es m ni^ and s^o we have \E s (y)\^m. Consequently, the number of the different systems Я (у) of the form 1 AGO = { 3,(y),..., GOD does not exceed (w-f l) 2+(log2W)/2. From part a) of relation (7) it follows easily that the number of vectors e(y) of the system Q m with the same system k{y) does not exceed (O m П ст. l^$^(log 2 m)/2

4 30 В. S. Kasin Finally, for : the number \Q m \ we obtain the estimate l^s^(log 2w)/2 By using the elementary estimate C^^C q (m/q) q we obtain that a [(log 2 m)/2] ( m \ 22s [log 8 m] ( m \ -' sc " Ж Ш sc \g Ш 2q - This proves Lemma 1. { [log 2 m] / m \ 29 1 f[log 2 m] m 1 In Д (f j }= C m exp{ Д 2ЧпЩ g m\ 2 In [^C ffl exp{mc'}sc m. We shall use the following well-known fact (cf, for example, [5, p. 335]). On the sphere S n there exists a set = {<} such that \Q' n \^C n and for any y S n there exists a vector e / = e / (y) Q / n for which (10) Ik'OO-jlhsey. The estimate below is also well-known. Lemma 2. For t^0 9 l^n<, and x 0 S n we have the inequality f(t) = m{x^s n : \(x,x 0 )\ <* t}^ e~^\ For t>l the statement of Lemma 2 is obvious, as in this case f(t)=0. For Q^t^l the set {x S n :(x, x 0 )^t} is contained in the hemisphere of radius (1 1 2 ) 1/2 and consequently, ДО (i-./2)(n-i)/2 = [(2 / 2 ) 1 /' 2 ]* аи (,, - 1 >/ 2л ^ e~ f2n/4 (in obtaining the last inequality we used the estimation (1 xf^^e" 1, Lemma 2 has been proved. Lemma 2 immediately implies - 0<x<:l). Consequence. Let X be a subspace of R\ d\mx=m, (Mx 0 i? w. Then for t^q we have (11) m x {* w fu: (x,x 0 )l ^/} ^ exp{- mr 2 W ^2}. Lemma 3. ITiere are absolute constants С and a>0 л/сй that for every subspace XaR n 0>-3, diml^ 1+«/2), vector a={a f } *S r ", and number t^o we have the inequality (12) m L fj? 5"nL: max l^e, J ^ Л ^ Cfe-" 1 ".

5 Convergence of orthogonal series 31 Proof. It is enough to prove Lemma 3 for t^20jn w, since for t^20n~ 1/2 the statement of our lemma is obtained because of the choice of the constant С in (12). Without loss of generality we may suppose that a^o (1^Шп). n For $=0,1,2,...' let us split the sum 2ад i nt0 the "pieces" {P*}^1 in the following way: n Hi 1 n Р =2 1Уи Pl= 2 ЪУг + Яп^т* Р1 = (а П1 -ЯпОУт+ 2 а гугi=l i=l i=n 1 +l Analogously, the "piece" P* -1 is represented in the form ps 1 ps _j_ ps r v - r.2v't"' r 2v + l' A similar division is used in the proof of Erdos' theorem on the a.e. convergence of lacunary trigonometric series (cf. [1, p ]). It is not hard to see (cf. [1, p. 705, Lemma 2]) that the splitting {PUfJi 1, $=0, 1, 2,..., of the above form may be chosen in such a way that for each "piece" n P s v = 2 a \ v,s) yt (0^v<2") we have the estimation i=l (13) J[a/ V ' s) ] 2^2" s. 8 = 1. (13) implies that we can find an integer s 0 such that all "pieces" P* (0^v^2* 1) contain at most two terms. Then we have the inequality max 2^Уг so ^ 2 max \P S V\ + max а^,. 0^v<2 s=0 = v ' ля s 1^Шп Consequently, if for fixed vectors {a} and {y} and for s=0,l,... 9 s 0 inequalities (14) then Hence we have max \P S V 0^V<2 5 max VII r 2 i = l and max \a t y t \ ^, 5(s+l) 2 iiiiv ' "' - 3 we have the (15) m L \ytst\l: max 24j#' = 2 2 may^s'^l-.^^jt ^]^ + т ь {^ 5"П :шах в у, *4}.

6 .32 В, S. Kasin Due to (13), the consequence of Lemma 2 (cf. (11)), and the inequality diml^ ^ l+n/2 we obtain that <16> У^{у ЬП8':\Р'\^1^}^^{- 1^^У Besides, by using (11) we get that {16') m L \yzst\l: max \a,y t \ ё 4"} = Si Wt {,^nx: W>^} S iexp{-0}. Comparing inequalities (15), (16), and (16') we arrive at <17) mjye^nl: max 24-yJ = Л = io Г 2 s / 2 ra 1 " ( t 2 n] If г 2 и>-200, then it is easy to show that s o Г 2 s / 2 n 1 and, moreover, ^ех Н~^Л~^< ^ ехр -^4- j?2«exp{--~2^с'е-у* п (у > 0). By putting the last inequalities into (17) we obtain the estimation (12). Lemma 3 is proved. We are going to use the following property of the measure /л п on the group O n (cf. [6, p. 55]). Let usfixan arbitrary vector e 0 S n and denote by О й ~ 1 = {Л) the subgroup of the group O n whose elements are those matrices A for which Ae 0 =e 0. The symbol fi n^l9 as before, denotes Haar measure on О п ~ г. For e S" let us choose an arbitrary matrix A e from O n with A e e 0 = e. Then we have {18) ff(a) dfi n = J dm f f(a e A) dii n. x $n 0n-l for any measurable bounded function /04), A O n.

7 Convergence of orthogonal series 33 Proof of Theorem. We may suppose that л^10, as for ж 10 the statement of the theorem shall be satisfied if we choose the constant С in (6) large enough. Let a system Ф () п be given. Suppose that the matrix А(Ф)=А = {а (] } О п is defined by the equality (5). Then it is easy -to see that 2 \2у* а Л\ = sup 2 2 y + 2 \2УЫ\ # y={ yi }<=S*,{Nj}4 = l \i = l t i=[!t/2] + l Vi = l /J Consequently, it is enough to prove that for m=[n/2] and m=n [n/2] and for all t^o we have 2л 1/2 (20) M(t) = ti{aco»: '^ sup [J (i y ; a y j 2 ] 1/2 is -L} ^ (Ce-i**)"- Let the sequence {Д/}" =1 of numbers be fixed. By using the properties of the system of vectors Q' n (cf. (10)) we obtain that [ m / AT, \2i"ii/a Г m f N J /2 2 2w s sup 2\2(у-е'(у)Ш,^_ j=l \i = l /.J ye^lj^l 4=1 ^2-, 1/2 + supfi{f(e'0')) J a y } 8 ] 3,^S n Lj=i lj=i J J f и ч1/2 Taking into account that Z(^~^(7)) 2 =1/2 we arrive at 1 г ш г NJ i 2 ii/ 2 J J As ^ ^С И (cf. (10)), from the last inequality and from the definition of the function M{t) (cf. (20)) we obtain that (21) M(t) * {СУ sup JASO*: sup f 2 [2 *,<J1 =" 4=}. By virtue of Lemma 1 from inequality (21) it follows that (22) M(t) ^C n sup /*jv< ц \ A O n : sup zes n,e Q m I {#,} J? 2ъ<*! j i=l j=(e)j,l*j*m Let the vectors z^s", e Q m, and the number t^0 befixed.then (23) /iu 0":sup 1 3 Analysis Mathematica t 1 -=(e) y, lijsm = / {ty} V2 * nn o n J /=i = m } nx )dn,

8 34 В. S. Kasin where A = {a ij }eo n, l^j^rn, and I 1 if sup {0 otherwise. N j 2 z^ij i=l -Jf^> As \\e\\i^l/3 9 е О т (cf. Lemma 1), a multiple application of Lemma 3 and equality (18) show that f л m m ( а л (24) f IIxM)dfi n = ncap\- T t*n(ejn* on j=i j=i { z J ^ C m Qxpl-jt 2 n 2(e)j\ ^ C n exp(-^2) 5 where y>0 is an absolute constant. From (22) (24) we obtain that M(t)^ Cexp{-ynt 2 }. Thus inequality (20) is proved. As it was noted earlier, the statement of our theorem follows from (20). Remark. We estimated the mean value of the function я(ф) on the set Q n consisting of orthonormal systems. The estimation of the mean value of the function,у(ф) on the set of all (not necessarily orthogonal) systems such that Ф {(Pi(x)}" =lr> \<p.(x)\ = l 9 the function (р г (х) being constant on the intervals ((j l)/n, j/n), t^i, j=n, x [0, 1], may be carried out following the method of [2]. References [1] H. К. Бари, Тригонометрические ряды, Физматгиз (Москва, 1961). [2] G. BENNETT, V. GOODMAN, С. М. NEWMAN, Norms of random matrices, Pacific J. Math., 59 (1975), [3] S. KACZMARZ und H. STEINHAUS, Theorie der Orthogonalreihen (Warszawa Lwow, 1935) С. Качмаж и Г. Штейнгауз, Теория ортогональных рядов, Физматгиз (Москва, 1958). [4] Б. С. Кашин, О некоторых свойствах ортогональных систем сходимости, Труды матем. ин-та им. В. А. Стеклова АН СССР, 143 (1977), [5] L. FEJES TOTH, Lagerungen in der Ebene, aufder Kugel und im Raum, Springer (Berlin Gottingen Heidelberg, 1953) Л. Фейеш Т от, Расположения на плоскости, на сфере и в пространстве, Физматгиз (Москва, 1958). [6] A. WEIL, LHntegration dans les groupes topologiques et ses applications (Paris, 1940) А. Вей ль, Интегрирование в топологических группах и его применения, Иностранная литература (Москва, 1950).

9 Convergence of orthogonal series 35 О среднем значении некоторых функций, связанных со сходимостью ортогональных рядов Б. С. КАШИН В статье рассматриваются множества Q n, 1 ^ п < «*>, ортонормйрованных систем Ф = = i<pi ( *)}"= 1» состоящих из фушспжйэ постоянных на интервалах I» I» 1 ^j^n. На 6 й естественно переносится с группы ортогональных матриц порядка п мера Хаара. Изучается поведение # на Q n функции 8(Ф)= sup (/ sup (iyiviidfdxf*. Доказывается, что при / > 0 и п = 1, 2,-... /*{Ф Й П : л-(ф) ^ /} ^ (Се- у **) я. Б. С. КАШИН СССР, МОСКВА УЛ. ВАВИЛОВА 42 МАТЕМАТИЧЕСКИЙ ИНСТИТУТ ИМ. В. А. СТЕКЛОВА АН СССР 3*