estimate the diameters of the octahedron B, 1 <i?i<oo, in the metric of Let us specify some of the notation to be used below.

Transcription

1 SOME PROPERTIES OF MATRICES OF BOUNDED OPERATORS FROM SPACE i\ TO l\ B. S. Kashi n Izvestiya Akademii Nauk Armyanskoi SSR. Matematika, Vol. 15, No. 5, pp , 1980 UDC Introduction. Assume that l n n, 1 O<oo, n = l, Is a space of vectors x \ Vp with norm W" = 23 \x% \" I f o r i < p < r o and M,«-max x,, while B n is a unit ball in I. For every linear operator Г:/?"-> /? ш we set [7" / {p,?) = sup Г(у)((, т. If i n R we consider the basis!-z ; -)?_, with zj = (0,-, 1,-, 0), 1</<я, then obviously a one-to-one correspondence arises, such that every linear operator T corresponds to its own matrix T' = {tij], 1 < i < m, l<&] < n in basis u.i for which the j-th row coincides with vector T (zj) R m. For matrix T' we set, by definition, i.e., a matrix with n rows and m columns \T\ P, q) =\T\p,,j = sup v (1) where 1 <p> q < -f 1. Я q'i Note that the norm of matrix T" that we have introduced, чт\ р, q), coincides wlzi the (p, q 1 ) norm of bilinear 2 hi *> У/ defined in [1]. This paper, which consists of three sections, is an elaboration of the author's short paper [2] and contains the proofs of the assertions that are merely formulated in [2]. In 1 and 2, we investigate certain properties of matrices of operators ТгЩ-+1$ with norm Г )о>,2)<1 in particular, certain properties of orthonormalized matrices. In 3, we consider the properties of mxn matrices associated with bounds for the diameters, and estimate the diameters of the octahedron B, 1 <i?i<oo, in the metric of Let us specify some of the notation to be used below. 2<^<7<oc. We denote by E n m, n<m the set of all n-element combinations of numbers with 1 -C4*C* * "^C'VCm. For each matrix A [a,j}" ;W, we define matrix A* (a*?,,, setting I U, for 1981 by Allerton Press, Inc. # j a 'J> for i>j i,... l<^j ~~ 44

2 For given finite set G we denote by G the number of elements in it. Finally, if S n is the set of all permutations of the combination of numbers l,2,...,n and [kj< n, = з 5", then we denote by A Q the matrix oi. *,)",==1. i.e., the matrix obtained by permutation in order of a rows of matrix A. 1. For matrix A = [ai/l, 1 > / <m, KjKa with ИЬ. 2><1 and combination 2, we denote by A(2) a square matrix of order a fa*/}, / 2, 1 </< n. We will be interested in the behavior of the norms M'(2) i2,з> and \A C 2 )h. <,>. 1<9<2. 1</=у'< It is easy to see that matrix А п =\ац\, 1</<2л 1, 1</<п with в»/**! for л and o,-y==0 in the remaining cases possesses the property that 1) IIA,. 2,= 1; 2) for any combination ^^E" n _ v L4,< (2) <2,2) 1 At the same time, the following theorem is proved in [2]. Theorem A. For any e>0 there exists a constant p(e)>0. such that for any matrix А \ац\, 1 < im, 1</<л with <>(:) there exists a combination of integers m Q E% for which M(.')l2.2,<s. of e. In the proof of Theorem A in [2], no estimate was given for H 3 )as a function In this section, in particular, we will obtain such an estimate. Note that if for m x n matrix A with И Ь, 2) *C 1 we consider the average value of the norm \A (3) ( 2, with respect to all combinations Q: ЕЛА, q)^-±r 2 M C *, ), then it is obvious that for «7 = 2, E X (A, 2)-<l, but to obtain for every m x n matrix A wi t h И (2,2> < 1 a bound Е г (А, 2)<i<l, where у Is an absolute constant, it is necessary that n be much smaller than m. This follows from the fact that, as we can readily establish, for matrices А п,г=\а^\, 1</<[л 2 "'], 1 < j < n, e > 0, where aij 1 for 1</=У<л and aij 0 otherwise, we have WmE^An,... 2) =1. If fori?, 1<<7<2, for given m x n matrix A and combination 2 6 E n m we consider the quanity \A <8*s,4i> then, since Mj*<it,w * w H* for*6 ", it follows that Я И ( 2 M(2. </)-<л 1 "~" 2 '% 2 and therefore 2 j (A, q) < max И (2)fo, < л"«-"*иь. 2)- (2 ) Unlike the quantity E (A,2), f or E l {A, q), l<gr<2, even form>cn,* we can obtain a *Here and henceforth, K,C,C,denote positive absolute constants. 45

3 nontrivial bound (i.e., one that is better than fn"*-" 2 lafa.»7 < 1, ) On the basis of this, we can also prove a refinement of Theorem A (see Theorem 2 of this paper). We have the following. Theorem 1. For q, 1<<?<2 there exists a constant such that for any m x n matrix A with И1Ь, 2, < 1 and any #<1 we have the bound Theorem 1 yields the following Corollary 1. For any pair of numbers q> 1 < q < 2 and AT>0 there exists a constant C(q,K) for which, under the assumptions of Theorem 1, Ha) Н^:И ( 2 )l<2. >C(q, K)-1 ^(~)-n f < c m к-". Corollary 2. For r>l and 2), under the assumptions of Theorem 1, E r (A, q) a ((C» m )-> S И (S)^,)'" < C'(r,q).(\n ^V 2. n, - I / S Lemma 1. For any function /(x) L 2 (0, 1) with < 1, Wk*>ff. 1 s 9 <2 for any z<i we have the measure bound m = m {x (z (0, 1) : / >*) > - z') 2 ~ Q Proof. Clearly, У"<Ши = 'уа^х+f / * rfx< J/ /»rfx + z', (3) [0,!!/ О 2 where Xi- is the characteristic function of set E. (with indexes 2/2 - q and 2/q), we obtain Employing the Holder inequalil 1 I _2_ 2- I 2-я jv / * dx< * ' (J/V* ) 1, 2 < (m )~о 0 \ 2 2-y It follows from this last inequality and from (3) that?'<г'+(т )'. xj,«>yn Vq - l», Lemma 1 Is thus proved. Lemma 1'. If f or </, 1<<7<2 and y (0, 1) for vector х= х( # я we havewu<l, then \\i:\ Xi \>±yn-<j\>n(i) ,- 46

4 Indeed, consider function f(x) with / (x) = x t \ n for - < x <, l<i* n. Then n n i/k'<1, while 1 " ч 41 ( л < = 1 / ( Therefore, on the basis of Lemma 1, I j*m> рг«-,д } - l): /(x) >- -) > (A"jik. Lemma 2. For x = {JO } 6 with * *«<1. 1<»<S and (0, 1) we have the inequality Proof. Consider the set Q y {z'6[l, m] : \ Xi > ~ t/-n- 1/2 J. Since ui,m <l, we have In view of Lemma 1', we have the relation Ю,1<~ (4) С, с j Q 6 Е' я : IS t\qj > л (f) 2 " 2^ [-С;. (5 ) To bound \G' y \ for the given quadruple of numbers m, n, r, p with «>2n, г<я, / </>, m >/>-2we bound the number G (m, л, r, />)) of sets ^^ 'J, such that Qf){1, p]\ > r. We have min [p. я) C /л (л, r, p) = 2 C* Cj-4. Using the bound C*</^ ^» we find from the last equation that min ip. л),... v я с K/(m, л, r, p*<v V ^ у ^ у \ < (б) < К 1 ; р' (т - р)- г max? \ < С л в г,71" i<^«vs s (л s)"- s / In view of (4) and (5), G' does not exceed \G(m, n, r, p)\ with Therefore it follows from (6) that 2q 2tf ^- G >Ke Lc- n -(ij] «(7) Since Cj,>C, - "^ j, the right side in (7) does not exceed 47

5 In view of ( 5 ) and inequality ( 7 ), taking account of ( 8 ) and of the fact that sup у _ 2 (I ) J «i '-С,,, we find f ( 8 ) 1С,1 < (B;>- -ex P I n- (f) 24 ' 2 " in ^ ] c» ; 5;» C,Q. Lemma 2 is thus proved. Proof of Theorem 1. It is well known (see, e.g., [3]) that for n = 1,2,.. on sphere {x: H** 8 ***' there exists a combination of vectors A*= {*} with!д Л < C" such that for every vector x, x 7 n = 1, there exists a vector е л with \x el -L. *2 * 4 It is easy to see that for any vector 2 f E" ( 9 ) Therefore, 2 6E«n : U (ОЛЬ, > e, [d 6 ^ ( (j «y e, )' J > (10) consequently, 1/Л 1/0-1,2? \"«^ 1 - J / Л 1 1 / ( Ш Sinci we have for any vector / m / \2 \ 1/2 and, bounding the right side in (11) using Lemma 2, we obtain (see (11)): /(j,ko(i» f )-C;(i) (SO A,. ^-ВДф Theorem 1 is thus proved. Theorem?. There exists an absolute constant В > 0 such that for any m x n matrix A = (ои/ with ИЛа2)< 1 there exists a combination 2^ " Я for which H (2)1(2 21 < <*(-тг- The proof of Theorem 2 employs (in addition to Corollary 1) the following result of Grothendieck [ 4 ], which is formulated here in the particular case required by us. 48

6 Theorem В (Grothendieck). There exists an absolute constant С such that for any linear operator T:t* -» L 1 (0, 1) there exists a set <=(0, 1), m E > ~ with sup Г(,) и (,<С уе* л 2 Corollary В. For any matrix 5 = [Ьц)J_, 2 (1, there exists a combination 2, а с [1, 2л], 0 > n, such that Hfe/I ^8, 1, < ^ Д,, ( 1 2 ) To prove Corollary В we consider the operator T-. /" Z, 1 y = 'y : \^l n, into function f(z), where that carries vector /(*) = %m *«for liu < г < i,i < / < B, /=1 л л з Using Theorem B, let us find the set "=[0, 1], m > for which IT (o)lfm 4 < С sup ЦГ(»} (,. i,; we put yw n 2 It is easy to see that 2..* л and that expression (12) holds for Q. Proof of Theorem 2. Clearly, we can assume that m > 4n. Corollary 1 implies (for q = 1, к = 2) that there exists a constant С such that for matrix A' =- njtjj 1</<м, К/<2л, combination 2'^ ^with where а,, = а,у for 1 </ < л and <r y = 0 for />л, there exists a И i c ')ita, i) < (13) In" 2 я Now, applying Corollary В to matrix 4(2') K/<eWi8*, for which, in view of (13), IA (2') (2,1, < C t л 1 ' 2 In 1 '-, we obtain combination 2c=2', 2c ; с И (2)11(2. :><- 4^- Theorem 2 is thus proved. In" 2 я 2. The following problem, formulated long ago by A. N. Xolmogorov, is well known in the theory of orthogonal series: assume that l?*i*))*h> 1) is an orthonormalized system of functions. Does there exist a permutation of the natural series - = \K \n\, for which the system {?* (x))n^\ is a convergence system (i.e., every series converges almost everywhere)? The "finite-dimensional" version of this problem has the following form: does there exist an absolute constant С such that for any 49

7 orthonormalized combination Ф = k»\ ** * S# such that jtp (x))^l,, x (0, 1) there exists a permutation 1 ( I /VU) I ) ;;i.v^«. r>v m HJ! oe^hh <c? (14) In [ 5 ], Garsia expressed the hypothesis that a bound stronger than ( 1 4 ) obtains; specifically,, kn "J,, )6,.v -Р д л.» w,, я 2 <" **. ('j-(o.n<c. (15) Let us recall that, in view of the Mer.'shov-Rademacher lemma (see [ 6 ], p ), "nr any orthonormalized combination sup sup [в я! B$.V(.r,, 1.V(.«l< si 2 Л(л) a «t* ( *) i=(0, 1) <С1пЛ>. ( 1 6 ) Garsia himself proved the following result in [7]. Theorem С (Garsia). For any orthonormalized combination Ф= <<р я and any combination {a }(zb x there exists a permutation \кп} з = з (Ф, { ал }) for which лги),, SU P I 2 a^(4 <C. (17) Note that the proof of this result is based on the estimate for the given combination of numbers of the average The original reasoning of Garsia was complicated; in [5], he offered a simple proof of the bounds for S p ({b,,)} that is entirely analogous to the proof of the classical Kolmogorov inequality for the majorant of partial sums of a series in a system of independent functions (see [ 8 ], p. 6 8 ). The distribution function i 1 's /V _ behaves in many ways similarly to the distribution function of a sum of independ N j.-.t random variables. We can show, e.g., that for any combination b = 0 л=1 50

8 ~f(y, {о.}) < Cm,х -... i лг V 6 г я (х) >C t i/ [j у > О, where [r (*)) =i is a Rademacher system. Comparison of Kolmogorov's problem or of Garcia's assumption (14) and of Garcia's theorem (see (17)) reveals that bound (17) yields a solution of a simpler problem. Another possible simplification of Kolmogorov's problem is to obtain, for a given orthonormalized combination Ф {?«(x))jl, and given function /V(x),l < C/V(x)<yv, bounds for the quantity II,v <T > ii inf sup а» * (*) (18) ч*я!-»(?. Л'(г)) \a n\ В, 1 Ji/><0. li In 2, we obtain some results involving estimation of (18). We consider the model problem of estimating, for given matrix A = {at/}' j=i, the quantity inf i(a )"h. «) (the notation А д and A# was introduced at the beginning of this paper). Theorem 3. For any matrix A = {a,,},". with рл <г.2> < 1 and number q withl<<7<2 there exists a permutation of rows OQ such that Before proving Theorem 3, we should note that it follows from (16) (see also [9]) that for any n x n matrix A Ah t) <Cla n \Afa, 2). ( 1 9 ) Therefore for any n x n matrix A and ef [1, 2) И «Ь <»«^ 2) с ( I n л ). i, - V 2 Д 2 w < (20) 1 The typical example here of the Hilbert matrix H n = \п и \ with A//= ; for 1 <V» /<n. i j, andft, ; =0, 1 < i < n, for which \H^, 2)!<2* (see [1]), shows that estimate (20) cannot be improved. Lemma 3 (see, e.g., [8], p. 78, 8). Let [ft ( *)), =1 be a combination of independent functions specified on the segment [0, 1] and such that m (x :ft ix) =1} = X>0, m {x :/,(x) = 0 =1-л, 1</<S. Then 2/!(x) < j[ <r;0<-r.< T (/.)<1. Lemma 1 directly yields the following. Lemma 3'. Assume that {/<(*)}*,, -r^(0, 1)'is a combination of functions that assume only values 0 and 1, such that for every combination l?_, 1 </t<s with e ; = o or 1 51

9 where Х>0. Then т (*:/,(*)=e,, 1</ <*) п {*:Д + 1 (*) = 1))> >>т [*!/,»«,, 1 < / C/fc), / s - Proof of Theorem We denote by p the absolute constant p(l/2) (see Theorem A). For every matrix В={а,,}, 6 г </,/<А» with 1 ^4,<4 2 <n and $t>p~' we A. define its partition into four submatrices (5),, r = l, 2, 3, 4, setting A A (5)! =,a l; }, q + 1 < I, / * Ь 3 ; (B) 2 la,/}, q + 1 < i < 6 2. *i < ' < <7. (5) 3 = (my), f> x < /'< t/, (S) 4 = [aij], b x < / < q, q + 1< у < 6 г, where the number q = q(b) is determined from the expressions bt?«p (*t-ai+l), 6 2 -</ + i>p (6 г 6 г +1). 2. Let us set up the sequence of partitions Д 5 = д< (A), s= 0,1,-, s,, of the given matrix A into submatrices. The zero partition A Q consists of the matrix A = itself. If partition 'л, s>l is constructed and consists of submatrices [Я»- 1 )**-!, then to construct partition Д, those matrices of partition Л, which intersect the diagonal i = j of matrix A (and, by definition, they are all square) and are of order >l-f P -> can be divided into four parts (Z*-') r l<r<4 in accordance with the rule described in Ц, stops. We denote partition Д by Д. At some step with number s 0 <C In n the process in question s Let us specify the desired permutation!^///_i = 3 o= 5 oh) For this we first construct the combination of permutations - r, l<r<s 0, where a Q coincides with a g. To construct a^, using Theorem A as applied to matrix Wi)}, <i +1 < I < n, l</< with q = q(a) (see 1), we find a combination Qc[l, n],\q\=n-q such that fjoi/li q +1 < /<n, / < 23 (3> 2) < ~ and as rj, we take any permutation о for which э(2)=[<7+1> я]. If permutations a x, v««i-i s < s 0 have been constructed, then to construct а s we consider all submatrices A ^={at,,,_,</, } L + 1 < < L+\, 1 < i»<ft-i of the partition * a «l o f matrix m, y>l* t»n which intersect the diagonal i = j and are of order and, as in the case of a^, using Theorem A, combinations of integers -'.cfh + l, vn], Up<".,._,, such that M = /а H q (A; "') (««e par. 1), and I км, U ч ИГ') u l <' < fe+»> / s b.»'< j21) 52

10 Then we take a permutation a such that n 2 ) fl 3 ff'» + l. 'V+i]) = [^ -l, fc+d, l 0<.i,-i, (») =^[«7 ИГ 1 ) + 1, К P< (22) and we set j,e=}«e,-i' 4. We will show that * s, = 3 0= is the desired permutation; for this we bound the norm Под]. 1 <Л у < nj, 2 q). Assume that K^JW',^, is a combination of matrices constructed in accordance with the rule described in 1 for each matrix g, which intersects the principal diagonal and is of order > l -f sh, of the partition of Д of matrix {aia, (/) } 1<4 у<л. has the form Expression (23) and the bound Then it is easy to see that the matrix 4-1 ^ Д U U <#lb i-i i.i <2= (a 4 I, l< /<n, 1 < у, wherein, <, i</<. (23) IA&2,2> <. 1 imply that p and therefore IIQ-(A 0)!!I,2,2;<C, IQ-(A )-!,*. < Сл 1^. (24) Let us bound the norm i:qs<2. «> For this we first bound I у (&)J for given s, l<s < ln-l 1(2, 0) ^s 0 1. if for each matrix R s we denote by (where s) 1 or 3) the matrix of the partition д ен о <к<s 2, which contains matrix R, then, by definition of permutation a^, where p (is s) = % 1. *eiu. *-2 t К fyjto. 2. < irfh 2) < 2-"fih»), We can readily deduce from Lemma 3' that, for certain constants f, 0 <-;< 1, and c 0 >0, the sum of the orders of the matrices Obviously, where Ш 2,9) ( 2 5 } 2 rank (**)< m*. (26) 2, (?) I 1(2,?) & = и /г; ; & ==. и /г : 1*611. l*jl : Я (!* *) * c l'6 l. I* s l : /'(l*. j) > f*>' Using the following inequality to bound the normil/wtae): (27) 53

11 Z*,,, ) <(rankb)»«->/» fl r., 2), (28) as well as the inequality l#jb, 2) <1 (which follows from the fact that #Д 2. j)< < maxlajjjfl, 2) < И, 2, 2 )'< 1), and bound (26), we obtain RA..,)<«, "-" 2 T,,, "-,/2) ; 0< t <1. (29) To bound j/?j(2,o-) we employ inequalities (25) and (28), and find 3k < / l,"-, ' 2 /?:!(,, 2, < n 1 *- 112 max м,и.>. г) < ( 3 0 ) < n"~"* max ra (2, 2) <2 n "'-".2- ;;/' Iv-. s) >c,* It follows from (29), (30), and (27) that "Ьг) ^ ( 3 D and hence 1 _ Combining inequalities (32) and (24), we obtain the requisite bound f or И(Л Г )-Ц, 2, «> Theorem 3 is thus proved. In concluding this section, we will formula,e an assertion that can be prov using Corollary 2 and the partition of matrix!a» I; j into binary blocks. Assertion. For any matrix -4 = ao},%i with ИЬ.г><1 The assertion yields a better bound for the average (over all permutations of columns of A) for the quantity H tt i!(2. «> than does the bound С (In n) л"«- 1/2 which follows trivially from (19). 3. In this section we offer a bound for th«kolmogorov n-diameter d n (5, I ) of octahedron B in space % % the method of proof has some common features with that used in 1. The definition of the diameter, as well as a number of results regarding bounds for diameters of finite-dimensional sets, may be found in [10, 11]. is valid: It is well known ([12]; see also [10], p. 237) that the following equality «маг,/?)~(2^) ад (зз) It follows from (33) that for m>2n and 1<<7<2 54

12 1 У 2 <d n (ВТ, О <1. (34) Here we will obtain a bound that is uniform in n and m>'.ln for the diameter d {BT, С), 2< г/< то - T reasoning in this section is also applicable if m<2n, but we will com'ine ourselves to the case Л1>2л, which is of importance for applications to bou ids of diameters of functional classes. Theorem 4. For л = 1, 2, m>2n and 2<</<o we have the inequalities j min (1, m 1 '* n-" J ) < (B?, /-) < С (<j)min (I, m 1 -* я-"»). (35 ) Proof. a) Upper bound. For т^>я* я the upper bound is obvious, since in this case min (1, m 1^ л-"-') = 1, while the diameter d n {B'(\ / ) <1 for any numbers m, n, q. For m n Qn, using the obvious inequality H/"<<m" 4«and the bound for the diameter d n (B{', / ) that was obtained by the author in [13], in accordance with which, for m<m\ d n {B?, С (/.), we have d n (5;', /") < m 1 * d n {Bf, I'Z) <C(q)- m 1 '" в ' % b) Lower bound. For n = 1, the bound can be readily verified directly, artd therefore in what follows we will assume that п>1, т>2л. Lemma 4 ([14]; see also [10], p. 237). the inequality where for. I m 2 2,)>m - n, For any plane Lc R m, dim L = л we have The proof of the lemma is very simple; nonetheless, the lemma is frequently quite useful. In particular, it yields a lower bound f or 4 n (5, /?') (see (33)). Lemma 4'. For any combinations of vectors f«and [ЫЙ, the scalar product (et, ;,)=1, 1-%;/<2л, we have the inequality e «such that Vi/ tw- (36) ZJ II Е ' i v)r^n. Indeed, consider matrix A with 2n columns and n rows, where the i-th column of this matrix coincides with vector е.. Assuming that bound (36) is not true, we obtain that for subspace L<zR 2n, 2л stretched onto the row vectors of matrix A we have 2 p /» ' < C "» a n d this contradicts Lemma 4. Now assume that the number q, 2<<7<[oc is fixed. (n, m) we have the inequality </ (В, l ) >, then '' 4 If for pair of numbers 55

13 min (1, m x 4-V- ) ^ _1 < d n ( B m t [m^ i.e., in this case the lower bound in ( 3 5 ) is valid. If dn(b, I",'), we consider the plane L с R* t dim L = n, for which 4 sup щ (L, x) = max Pl (L, m) = d r. {BP, /"'). ( 3 7 ^ Taking in L the basis {y^}" v гл /?", we specify plane L by a matrix with n rows y- and m columns!&},*,. Expression ( 3 7 ) and the inequality d (B, Z )< imply that there exists a combination of vectors Ь f R a such that a) 2 l(i.,e>)l«<</«v^r. T). b) 1 <^ г ^ m Consequently, for combination of vectors?, = ii we have the bounds a) 2kb, /)! < (у)'<«(я\ С); 1 < i < m, ( 3 8 ) Consider the sum {h. e; )=1. It follows from ( 3 8 ) that 5-2 e y ) «. 1 i, r < m (j) <«(.9Г,./Г). ( 3 9 ) Now consider the sum Г=2 2, *,) «, where in sum T the outer summation is performed over all combinations term ej)\" appears T C'mZ 2 2 times in the sum, and therefore Q El\ т cs4-^<с2г5.т/^у..й(дг, о. (4o) It follows from (40) that we can find a combination 2 m such that Pi. 2j(1,-, e,)»< Г-(С^Г)- 1 --(уу.^.(5г, О О т -2 т (41) Since Сл?Г -(С«) 1 = 2п{2п-\)(т (т-\))"\ the right side in (4l) does not exceed 5 6

14 57 \з/ т-1 Now let us bound the left side in (41) from below. Ы,,> r4i- lli \x\,r, v>2, r = l, 2,-, Eq. (38b), and Lemma 4, we obtain 4 2 ^>(4п*)^- ю (2 (; e,) 2 )' /2 > (4 л') 1 '"-' V 2 = 4""-" 2 V'<- -J/2 ' ;e-' ' j Using the inequality (43) From (41) and (43), taking account of (42), we conclude that 1.3 / О ч и.in - 1/2 X(m-l)'"--> it " «*. 8 4 Theorem 4 is thus proved. REFERENCES 1. G. Hardy, D. Littlewood, and G. Polya, Inequalities [Russian translation], IL Press, Moscow, B. S. Kashin, "One property of bilinear forms," Soobshch. AN GruzSSR, L. F. Tot, Arrangements on a Plane, on a Sphere, and in Space [in Russian], Fizmatgiz, Moscow, A. Grothendieck, "Resume de la theorie metrique des produits tenzoriels topologiques," Bol. Soc. Mat., Sao Paulo, vol. 6, nos. 1-2, pp. 1-79, A. Garsia, Topics in Almost-Everywhere Convergence, Markham, Chicago, ' 6. S. Kaczmarz and G. Steinhaus, Theory of Orthogonal Series [Russian translation], Fizmatgiz, Moscow, A. Garsia, "Rearrangements for Fourier series," Ann. of Math., vol. 79, no. 3, PP , V. V. Petrov, Sums of Independent Random Variables [in Russian], Nauka Press, Moscow, D. E. Men'shov, "Sur les series des fonctions orthogonales III," Fung. Math., vol. 10, pp , V. M. Tikhomirov, Some Problems of Approximation Theory [in Russian], Izd. MGU, Moscow, B. S. Kashin, "Diameters of some finite-dimensional sets and classes of smooth functions," Izv. AN SSSR. Ser. matem., vol. 4l, no. 2, pp , S. B. Stechkin, "Best approximations of specified classes of functions by arbitrary polynomials," UMN, vol. 9, no. 1, pp **, B. S. Kashin, "Diameters of octahedra," UMN, vol. 30, no. 4, pp , A. N. Kolmogorov, A. A. Petrov, and Yu. M. Smirnov, "One Gauss formula from the theory of the least-squares method," Izv. AN SSSR. Ser. matem., vol. 14, pp , October 1979 Steklov Mathematics Institute, AS USSR