DOI 10.1007/s11253-016-1162-0 Ukrainian Mathematical Journal, Vol. 67, No. 9, February, 2016 (Ukrainian Original Vol. 67, No. 9, September, 2015)

MULTIPLE HAAR BASIS AND ITS PROPERTIES V. S. Romanyuk

UDC 517.51

In the Lebesgue spaces Lp ([0, 1]d ), 1  p  1, for d ≥ 2, we define a multiple basis system of functions Hd = (hn )1 n=1 . This system has the main properties of the well-known one-dimensional Haar basis H. In particular, it is shown that the system Hd is a Schauder basis in the spaces Lp ([0, 1]d ), 1  p < 1.

Introduction �1 � � � In 1909, Haar [1] constructed a complete system of functions hn (x) n=0 , x 2 [0, 1], in the space L [0, 1] orthonormal on the segment [0, 1]. The Fourier series for continuous functions expanded in this system are uniformly convergent on [0, 1]. This system has a sufficiently simple structure and consists of piecewise continuous functions on the intervals of binary partition of the segment [0, 1]. We now recall the definition of this system in terms of notation convenient for our subsequent presentation. By Dj , j = 1, 2, . . . , we denote a set of binary intervals of the j th level on the segment I := [0, 1] : � Dj = Ijs : s = 0, 1, . . . , 2j−1 − 1 ,

� � where Ijs = s2−j+1 , (s + 1)2−j+1 . We also set I00 := I and D0 = {I00 }. We define the Haar functions by setting HI00 (t) = 1,

t 2 I,

and, for j = 1, 2, . . . , s = 0, 1, . . . , 2j−1 − 1,

HIjs (t) =

8 > > |Ijs |−1/2 , > > > > > > <

−|I s |−1/2 , > > j > > > > > > :

0,

t2 t2

✓

s2−j+1 ,

✓✓

1 s+ 2

✓

◆

1 s+ 2

◆

2−j+1

2−j+1 , (s

+

◆

,

1)2−j+1

◆

,

t 2 I \ Ijs ,

where |Ijs | = 2−j+1 is the length of the interval Ijs and Ijs is its closure.

At all interior (with respect to the segment I) points of discontinuity, the functions HIjs (t) are assumed to be equal to the half sum of their left-hand and right-hand limits and, at the endpoints of the segment [0, 1], the functions are regarded equal to their limit values inside the segment. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine.

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 9, pp. 1253–1264, September, 2015. Original article submitted March 27, 2015. 0041-5995/16/6709–1411

c 2016 �

Springer Science+Business Media New York

1411

V. S. ROMANYUK

1412

The system H = {HI00 }

S

is called the basis Haar system.

{HIjs } j=1,2,...

s=0,1,...,2j−1 −1

We order the system H as follows: We set

h0 (t) = 1,

t 2 I00 ,

and, for 0  s < 2j−1 , j = 1, 2, . . . , h2j−1 +s (t) = hsj (t) = HIjs (t). By H we denote the obtained sequence hn , n = 0, 1, . . . . space Lq ([0, 1]), In 1928, Schauder [2] showed that the system H = (hn )1 n=0 is a basis in �the Lebesgue � 1  q < 1. Despite the fact that the system H cannot be a basis in the space C [0, 1] of functions continuous on the segment [0, 1], in 1910, Faber showed that each function continuous on [0, 1] can be uniquely represented in the form of a series in the system of functions �1 ⇢ Z x hn (t)dt 1, 0

n=0

� � continuously convergent to this function, i.e., this system of functions is already a basis in the space C [0, 1] . Series in the Haar system H were systematically studied in [3], where, in particular, the problems of convergence of Fourier series of functions from some classes in the system H were investigated and the estimates for the Fourier coefficients in the system H were established for elements of the space of functions measurable on [0, 1]. In 1972, the fundamental results characterizing � � � the� approximating properties of the system H for functions from the spaces Lq [0, 1] , 1  q < 1, and C [0, 1] , namely, the corresponding direct and inverse theorems, were presented by Golubov in [4] . In the present paper, on the basis of the system H, we determine one of the multiple Haar systems Hd0 of functions defined in the cube [0, 1]d of the Euclidean space Rd , d ≥ 2. The structure of this system somewhat differs from the structure of the classical tensor Haar system Hd (see the definition in Sec. 1). However, it can be shown that this system has important properties typical of the one-dimensional basis � Haar� system H. In particular, we show that the properly ordered system Hd0 is a Schauder basis in the spaces Lp [0, 1] , 1  p < 1. used for the solution of some problems of Then the established properties of the system Hd0 are� efficiently � d approximation of the classes of functions in the spaces Lq [0, 1] . In the present paper, we use the conventional notation N, R, R+ , Z, and Z+ for the sets of natural, real, real nonnegative, integer, and integer nonnegative numbers, respectively. Yd A, d 2 N, we denote the Cartesian product of d sets A, where A is one of the sets N, R, By Ad = i=1 d N R+ , Z, Z+ or the segment [a, b] ⇢ R. By M(i) we denote the tensor product of some sets M(i), i = 1, d, i=1

and in particular, of functional sets, ]A denotes the number of points of a finite set A ⇢ Zd , cardA is the number of elements of a finite set A, |A| or vol A is the volume (Lebesgue measure) of the set A ⇢ Rd , and suppf (x) denotes the support of a function f, i.e., the set of interior points of the set A such that f (x) 6= 0, x 2 A. For the expressions a and b defined by a certain collection of parameters, the notation a ⇣ b means that there exists positive quantities c1 and c2 independent of one essential parameter such that c1 b  a  c2 b. If a  c2 b (c1 b  a), then we write a ⌧ b (a � b). By C(p), C1 (d, p), etc., we denote the quantities that (possibly) depend only on the parameters in parentheses and are positive for all admissible values of these parameters; C, C1 , and C2 are absolute positive constants not necessarily identical in different places of the text.

M ULTIPLE H AAR BASIS AND I TS P ROPERTIES

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In addition, we also introduce a series of definitions and notation used in the paper. By Lq (⌦), 1  q  1, we denote a space of functions ' : ⌦ ! R measurable on the measured space ⌦ ⇢ Rd with finite norm 0 11/q Z k'kLq (⌦) = @ |'(x)|q dxA , 1  q < 1, ⌦

k'kL1 (⌦) = ess sup |'(x)|. x2⌦

For ⌦ = Id , we sometimes write Lq instead of Lq (Id ) and k · kq instead of k · kLq (Id ) for 1  q  1. For 1  p  1, we define the modulus of continuity of a function ' 2 Lp by the equality !(', t)p :=

sup 0⌧i
k∆⌧ 'kLp (Id⌧ ) ,

where ⌧ = (⌧1 , . . . , ⌧d ) 2 Id ,

Id⌧ :=

d Y i=1

[0, 1 − ⌧i ]

and ∆⌧ (', x) := '(x + ⌧ ) − '(x) for x, x + ⌧ 2 Id . Finally, by the same letter (but in different fonts), we denote various systems of Haar functions: H is a system of Haar functions of one variable; H is a Haar basis in Lp (I), 1  p < 1 (an ordered sequence of functions of the system H); Hd is a tensor Haar system of functions of d variables, d 2 N; Hd0 is a basis Haar system with “interval” indexation of functions of d variables; Hd0 is a basis Haar system with vector indexation of functions of d variables; Hd is a Haar basis in Lp (Id ), 1  p < 1, d 2 N (an ordered sequence of functions of the system Hd0 ). 1. Definition of the Functional Systems Hd0 , Hd0 , Hd , and Hd We first introduce a multiple basis Haar system Hd0 of functions defined in a unit cube Id , d ≥ 2. By Qj :=

d O

Dj ,

j = 1, 2, . . . ,

i=1

we denote the set of cubes I of a binary partition of the cube Id of volume |I| = 2(−j+1)d , i.e., ( ) d Y Ijli : l = (l1 , . . . , ld ), 0  li < 2j−1 , i = 1, d . Qj = Ijl = i=1

By Q :=

S1

j=1 Qj

we denote the set of all cubes of the binary partition of Id . We set Hd0 := {HId } [ {HI }I2Q ,

V. S. ROMANYUK

1414

where the function x 2 Id ,

HId (x) = 1, and, for j 2 N and I 2 Qj

⇣

i.e., I =

Yd

i=1

⌘ Ijsi ,

HI (x1 , . . . , xd ) =

Y

i2E

HI si (xi ) ⇥ j

Y

i2T\E

|HI si (xi )|. j

(1)

Here, E is an arbitrary nonempty subset of the set T := {1, 2, . . . , d} including the case where E = T; in this Y case, the factor is replaced by unity. i2T\E

Note that the collection of all subsets E with given number card E 6= d and the set E = T by relation (1) is determined by 2d − 1 functions with supports on a fixed cube I 2 Qj and, hence, on each cube Ijl =

d Y

Ijli ,

0  lj < 2j−1 ,

l = (l1 , . . . , ld ),

i=1

i = 1, d.

By H(j, ¯l ) we denote the corresponding sets of these functions. We now represent the system Hd0 , d ≥ 2, in other way based on the one-dimensional Haar basis H and using a vector enumeration of functions of this system. To this end, we decompose the set Zd+ into the nonintersecting subsets Z0,d := Y0,d and Zj,d :=Yj,d \ Yj−1,d , j = 1, 2, . . . , where � Y0,d = {0} = (0, 0, . . . , 0) 2 Zd+ ,

� Yj,d = k¯ = (k1 , . . . , kd ) 2 Zd+ : 0  ki < 2j , i = 1, d ,

It is clear that Zd+ =

S1

j=0 Zj,d .

j = 1, 2, . . . .

We also note that

]Yj,d = 2jd

and

]Zj,d = (2d − 1)2(j−1)d ⇣ 2jd .

Hence, we define a system of functions of d variables Hd0 = {hk¯ }k2Z ¯ d := +

j=0

setting h0 =

1 [

d O

{hk¯ }k2Z ¯ j,d

h0

i=1

and, for k¯ 2 Zj,d , j = 1, 2, . . . , hk¯ =

O i2E

hki ⌦

O

i2T\E

|h2j−1 +ki |,

where E = {i 2 T : 2j−1  ki < 2j }, moreover, if E = T, then we set hk¯ =

N

i2T hki .

M ULTIPLE H AAR BASIS AND I TS P ROPERTIES

1415

d It is clear that Hd0 = Hd0 , i.e., the sets {hk¯ }k2Z ¯ d and H0 coincide. Moreover, there exists a one-to-one +

correspondence between indexation of functions of the set Hd0 by binary cubes from Qj and indexation of functions of the set Hd0 by vectors from Zj,d such that {hI }I2Qj = {hk¯ }k2Z ¯ j,d ,

j = 1, 2, . . . .

By Zj,d (l) we denote a set of indices k¯ 2 Zj,d of functions hk¯ 2 H(j, l). We arrange vectors k¯ = (k1 , . . . , kd ) of the set Zd+ in the form of the sequence k¯(1) , k¯(2) , . . . , k¯(m) , . . . such that k¯(1) = (0, 0, . . . , 0) 2 Zd+ and, for i = 2, 3, . . . , o n o n (i) (i+1) : j = 1, d . max k¯j : j = 1, d  max k¯j

�1 � By Hd we denote the sequence hk¯(i) i=1 of functions of the system Hd0 corresponding to this arrangement. In this case, if, for some number i, the inequality ]Zj−1,d < i  ]Zj,d with j 2 N is true, then k¯(i) = k¯ for some k¯ 2 Zj,d , and if k¯ 2 Yn,d , then, for some 1  i  ]Yn,d , we have k¯ = k¯(i) and k¯(i) 2 Yn,d . Enumerating functions of the system Hd according to the correspondence k¯(i) ! i, we write Hd = (hi )1 i=1 . In conclusion of this section, we note that the multiple system of Haar functions Hd mentioned in the introduction is defined as a tensor product of basis Haar systems of functions of one variable with the corresponding indexation of functions by parallelepipeds of the set Dd of binary partition of the cube Id : d

H =

d O i=1

H = {HI }I2Dd .

Hence, for given j = (j1 , . . . , jd ) 2 Zd+

and

s = (s1 , . . . , sd ),

sk = 0, . . . , 2jk −1 − 1,

k = 1, d,

and I=

d Y

Ijskk ,

Ijskk 2 Djk ,

k=1

we set

HI (x1 , . . . , xd ) :=

d Y

k=1

k = 1, d,

HI sk (xk ). jk

Note that the recent works by Temlyakov and other authors deal with the investigation of the properties of the system Hd , d ≥ 2, mainly in connection with the problems of nonlinear approximation of functions, and the pioneering Temlyakov’s work is [5]. It is worth noting that, for d = 1, the systems H1 and H10 := H a priori coincide. 2. Representation of Partial Fourier–Haar Sums d d By the definition of the orthonormal system Hd0 = {hk¯ }k2Z ¯ d in L2 (I ), any function f 2 L1 (I ) can be + decomposed in the Fourier–Haar series

f (x) ⇠

X

¯ d k2Z +

(f, hk¯ )hk¯ (x),

V. S. ROMANYUK

1416

where (f, hk¯ ) =

Z

f (x)hk¯ (x)dx,

Id

k¯ 2 Zd+ ,

are the Fourier–Haar coefficients of the function f. By Pn we denote the operator Pn : L1 ! Vn of orthogonal projection of the space L1 (Id ) onto the subspace 9 8 = < X � ck¯ hk¯ , ck¯ 2 R , Vn := span hk¯ , k¯ 2 Yn,d = u : u = ; : ¯ k2Yn,d

i.e.,

Pn f (x) =

X

(f, hk¯ ) hk¯ (x),

¯ n,d k2Y

f 2 L1 (Id ).

We call the functions Pn f (x), x 2 Id , polyhedral (cubic) Fourier–Haar sums of the function f. Proposition 1. For any function f 2 L1 (Id ) and n 2 Z+ , the following representation is true: 1 Pn f (x) = |I|

Z

f (y)dy,

I

x 2 I,

(2)

where I 2 Qn+1 , i.e., I is a binary cube, I ⇢ Id , vol I = 2−nd . Proof. For the case d = 1, the proof of Proposition 1 can be found in [6, pp. 78–80] (Chap. 3, Sec. 1). For d > 1, the statement is proved similarly. To do this, it suffices to note that, for each n 2 Z+ , f 2 L1 (Id ), and I 2 Qn+1 , the functions Z 1 f (x)dx, x 2 I, mn (f ; x) := |I| I

mn (f ; x) = 0,

x 2 Id \I,

for any I 2 Q and x 2 I satisfy the equality

as in the case d = 1.

� � Pn f (x) = Pn mn (f ; ·) (x) = mn (f ; x)

Remark. Representation (2) does not trace the values of the function Pn f (x) on the “grid” of binary partition of the cube Id . Nevertheless, these values of the function Pn f (x) are insignificant for the analyzed properties of this function. 3. One Property of the System Hd0 We now formulate and prove an important auxiliary statement concerning the estimation of the Lp -norm of elements of the space Wj := span{hk¯ ; k¯ 2 Zj,d }, j 2 N.

M ULTIPLE H AAR BASIS AND I TS P ROPERTIES

1417

Lemma 1. For any system of real numbers {ak¯ }k2Z ¯ j,d , j = 1, 2, . . . , the following relations are true: � � 11 0 � � p � � d d X � X � −j p − 2 p � � @ ak¯ hk¯ � ⇣ 2 |ak¯ | A , � �k2Z � ¯ ¯ k2Z j,d

and

1  p < 1,

(3)

j,d

p

� � � � X � � � ak¯ hk¯ � � � � �k2Z ¯ j,d

jd

(4)

⇣ 2 2 max |ak¯ |. ¯ j,d k2Z

1

Proof. We first deduce (3) with sign ⌧. Fixing j, we split the set of indices Zj,d into disjoint subsets Zj,d (l) according to the procedure established in Sec. 1. Thus, Zj,d (l

(1)

) \ Zj,d (l

(2)

for l

)=?

(1)

6= l

(2)

and Zj,d =

[

Zj,d (l).

l2Yj−1,d

Moreover, {hk¯ }k2Z ¯ j,d =

[

H(j; l).

l2Yj−1,d

On the other hand, taking into account that card H(j; l) = 2d − 1 for any l 2 Yj−1,d , we split the set (i) d d {hk¯ }k2Z ¯ j,d into 2 − 1 nonintersecting subsets Hj , i = 1, 2, . . . , 2 − 1, such that each function hk ¯ 2 H(j; l), (i)

l 2 Yj−1,d , is associated with exactly one of the sets Hj and, hence, {hk¯ }k2Z ¯ j,d =

d −1 2[

i=1

(i)

Hj .

(i)

\ supphk(2) =? It is clear that card Hj = ]Yj−1,d = 2(j−1)d for any i = 1, . . . , 2d − 1 and supphk(1) ¯ ¯ (i) d ¯ ¯ if hk(1) 2 Hj (for some i = 1, . . . , 2 − 1) and k(1) 6= k(2). ¯ , hk(2) ¯ (i) (i) � (i) By Zj,d we denote a set of (vector) indices k¯ such that the function hk¯ belongs to the set Hj thus, ]Zj,d = � 2(j−1)d . We first note that direct calculations yield the following for j = 1, 2, . . . and l 2 Yj−1,d : khk¯ kp = |Ijl |1/p−1/2 = 2−jd(1/p−1/2) ,

k¯ 2 Zj,d ,

1  p  1.

Hence, by using the inequality (a + b)p  2p−1 (ap + bp ),

a, b > 0,

1  p < 1,

(5)

V. S. ROMANYUK

1418

we obtain �p �p � � �d � � � 2 −1 � � X X � � X � � =� � a h a h � ¯ ¯ ¯ ¯ k k� k k� � � � � � k2Z (i) ¯ j,d � i=1 k2Z � ¯ p j,d

p

� �p � d −1 d −1 Z Z �� 2X 2X � X � � = � ak¯ hk¯ � dx  C(p, d) � � (i) i=1 i=1 d � � ¯ k2Z Id I

� �p � � � X � � � ak¯ hk¯ � dx � � � (i) � k2Z � ¯

j,d

= C(p, d)

= C(p, d)

� �p � � d −1 2X � X X � � ak¯ hk¯ � = C(p, d) |ak¯ |p khk¯ kpp � � � (i) (i) i=1 k2Z � k2Z � ¯ ¯

d −1 2X �

i=1

d −1 2X

j,d

j,d

j,d

p

2−jd(1/p−1/2)p

i=1

X

(i) ¯ k2Z j,d

|ak¯ |p = C(p, d)2−jd(1/p−1/2)p

X

¯ j,d k2Z

|ak¯ |p ,

which yields the upper bound in relation (3). The lower bound in relation (3) for p = 2 is a trivial corollary of orthonormality of the system {hk¯ }k2Z ¯ d

+

in L2 (Id ). Moreover, the following equality is true: � �2 � � X � X � � ak¯ hk¯ � |ak¯ |2 . � � = � k2Z � ¯ ¯ k2Z j,d

(6)

j,d

2

We prove the lower bound in (3) for any 1  p < 1. � � according to Sec. 1, it is an ordered sequence of functions from Hd0 = {hk¯ }k2Z is Since Hd = (hi )1 d ¯ i=1 +

a basis in the spaces Lp (Id ), 1  p < 1 (see Theorem 1), in view of the statement in [7], there exists 0 < ↵ < 1 such that, for any collection of real numbers {ck }n+m k=1 , n, m = 1, 2, . . . , the following inequality is true: � � � � n+m n � � �X �X � � � � c k hk � ≥ ↵ � c k hk � � � � � � k=1

k=1

p

(7) p

(for p = 2, it is obvious that ↵ = 1). In turn, by analogy with [4, pp. 261, 262] (Sec. 2) for d = 1, relation (7) yields the inequality � � � n+m ◆ � n+m ✓ � � �X 1 −1 � � � X � � c h ≥ 1 + c h (8) � � k k� k k� . � � � � ↵ k=1

k=n+1

p

p

Indeed, by virtue of inequality (7) for 1  p < 1,

� � � � � � � � n+m � n � n+m n+m �n+m � � � � � � X �X � X 1� �X �X � � � � � � � � c h ≥ c h − c h ≥ c h − c h � � � � � k k� k k� k k� k k� k k� , � � � � � � � � � ↵� k=1

which yields (8).

p

k=n+1

p

k=1

p

k=n+1

p

k=1

p

M ULTIPLE H AAR BASIS AND I TS P ROPERTIES

1419

In particular, it follows from inequalities (7) and (8) that, for some constant γ > 0, � � � � m M �X � � �X � � � � c k hk �  γ � c k hk � � � � � � k=n

k=l

p

(9) p

for any natural n, m, l, and M connected by the relation l  n < m  M. By virtue of multiindex (vector) enumeration of functions of the basis Hd , i.e., the system Hd0 , on the basis of (9), we can write � � � � � � � � � X � X � � � � � ak¯ hk¯ �  � ak¯ hk¯ � (10) max � � . � 1i2d −1 � � � k2Z (i) ¯ j,d � k2Z � ¯ p j,d

p

In the proof of the upper bound in Lemma 1, it was shown that, for any i, 1  i  2d − 1, � � 11 0 � � p � � � X � 1 1 −jd p − 2 B X � C � ak¯ hk¯ � = 2 |ak¯ |p A . � @ � � (i) (i) � k2Z � ¯ ¯ k2Z j,d

j,d

p

By using this equality and inequality (10), we obtain

� �p � �p � d −1 � � � �1 1� 2X � � X X � � 1 −jd p − 2 p X � � � � ak¯ hk¯ � ≥ d ak¯ hk¯ � = C(d, p)2 |ak¯ |p , � � p � � (2 − 1)γ � k2Z � (i) ¯ j,d ¯ j,d i=1 � k2Z � ¯ k2Z p j,d

p

i.e., the required lower bound in relation (3). To complete the proof of Lemma 1, it suffices to note that relation (4) in this lemma is a direct corollary of equality (5). Note that, in the case 1 < p < 2, a lower bound in relation (3) can be obtained on the basis of other arguments by using only the H¨older inequality kf k22  kf kp · kf kp0 ,

1 1 + 0 = 1, p p

f 2 Lp (Id ),

equality (5), and the upper bound in (3) for the exponent of summability p0 . Indeed, for f = � � � � � X � kf k22 � ak¯ hk¯ � � � ≥ kf k 0 � p � k2Z � ¯ j,d

p

�

d d −j p0 − 2 2

X

¯ j,d k2Z

� ✓X

|ak¯ |2

¯ j,d k2Z

|ak¯ |

p0

X

¯ j,d k2Z

ak¯ hk¯ we get

◆1/p0 =: P.

Hence, if 1 < p < 2, then 2 < p0 < 1 and, according to the known inequalities N X s=1

|bs |γ

!1/γ



N X s=1

|bs |µ

!1/µ

,

(11)

V. S. ROMANYUK

1420 N 1 X |bs |γ N s=1

!1/γ

≥

N 1 X |bs |µ N s=1

!1/µ

,

(12)

where b = {bs }N s=1 is an arbitrary system of real numbers and 1  µ < γ < 1, we have X

¯ j,d k2Z

and 0 @

0

|ak¯ |2 ≥ @ X

¯ j,d k2Z

12/p

X

|ak¯ |p A

¯ j,d k2Z

p0

11/p0

|ak¯ | A

In this case, the following estimate is true:

0

@

· 2−jd(1/p−1/2)·2

X

¯ j,d k2Z

11/p

|ak¯ |p A

.

12/p,0 0 11/p 1 X B jd(1/2−1/p0 ) @ X C P ≥ 2−2jd(1/p−1/2)@ |ak¯ |p A |ak¯ |p A A @2 0

¯ j,d k2Z

0

≥ C(d, p)2−jd(1/p−1/2) @

¯ j,d k2Z

X

¯ j,d k2Z

11/p

|ak¯ |p A

.

4. Some Properties of the Operator Pn and Elements of the Subspace Vn Lemma 2. For any function f 2 Lp (Id ), 1  p  1, and n 2 Z+ , the following inequalities are true: kPn f kp  C1 (d, p)kf kp ,

(13)

kf − Pn f kp  C2 (d, p)!(f ; 2−n )p ,

(14)

and, for f 2 Vn , � �1/p kf kp . !(f ; δ)p  C(d, p) min{δ2n ; 1}

(15)

Proof. To prove inequality (14), we use the scheme of the proof of this inequality in the case d = 1 (see [6, pp. 81–83], Theorem 2 for p = 1, Theorem 3 for 1  p < 1) and Proposition 1. Details of the proof are omitted. Inequality (13) is a direct corollary of inequality (14). Indeed, the obvious inequality !(f ; δ)p  2kf kp , which is true for any f 2 Lp (Id ) and 0  δ  1, yields kPn f kp = kf − (f − Pn f )kp  kf kp + kf − Pn f kp � �  kf kp + C2 (d, p)!(f ; 2−n )p  1 + 2C2 (d, p) kf kp = C1 (d, p)kf kp .

M ULTIPLE H AAR BASIS AND I TS P ROPERTIES

1421

Proceeding to the proof of (15), we recall the definition of !(f ; t)p . If f 2 Lp (Id ), 1  p  1, and λ = (λ1 , . . . , λd ) 2 Rd+ , then !(f ; t)p :=

k∆λ (f ; ·)kL(Id ) ,

sup

λ

0λi
where Idλ

=

d Y i=1

[0; 1 − λi ]

and ∆λ (f ; x) = f (x + λ) − f (x)

for x, x + λ 2 Id . As already indicated, the inequality !(f ; δ)p  Ckf kp with constant C = 2 is uniformly true for all 0  δ  1 for any function f 2 Lp (Id ) and, in particular, for f 2 Vn . It can be defined more exactly for f 2 Vn and 0 < δ < 2−n for n = 1, 2, . . . . Indeed, let j 2 N and let δ, 0 < δ < 2−j , and 0  λi < δ, i = 1, d, be given. Consider the difference k¯ 2 Zj,d ,

∆λ (hk¯ ; x) = hk¯ (x + λ) − hk¯ (x),

x 2 Idλ . ⇤

With regard for supp hk¯ 2 Qj for k¯ 2 Zj,d or, more exactly, supp hk¯ = Ijs for some s⇤ = (s⇤1 , . . . , s⇤d ), ⇤ 0  s⇤i < 2j−1 , i = 1, d, and |Ijs | = 2(−j+1)d , it is easy to note that, in view of the definition of the function hk¯ , ⇤ k¯ 2 Zj,d , the difference ∆λ (hk¯ ; x) is different from zero only on some set ⌃(λ) ⇢ Idλ \Ijs of volume |⌃(λ)|  Cδ d , C > 0. By virtue of the inequality |a ± b|p  2p−1 (|a|p + |b|p ),

a, b 2 R,

1  p < 1,

we get |∆λ (hk¯ ; x)|p  2p |hk¯ (x)|p ,

(16)

x 2 ⌃(λ),

and we arrive at the relation � � �∆λ (h¯ ·)�p d = k Lp (I ) λ

=

Z

|hk¯ (x + λ) − hk¯ (x)|p dx

Idλ

Z

|hk¯ (x + λ) − hk¯ (x)|p dx < |⌃(λ)| max |∆λ (hk¯ ; x)|p x2⌃(λ)

⌃(λ)

d p

 Cδ 2

p (2jd ) 2

d

jd

= C(p)δ · 2

·

✓

�

1 1 −jd p − 2 2

We also mention the obvious equality k∆λ (h0 ; ·)kLp (Id ) = 0. λ ✓ X Xn ck¯ hk¯ (x) = Moreover, for f 2 Vn i.e., f (x) = ¯ k2Yn,d

j=0

X

�◆p

 C3 (d, p)δ · 2j khk¯ kpp .

c¯ hk¯ (x), ¯ j,d k k2Z

ck¯ 2 R

◆

and 0 < δ <

2−n , n 2 N, by using the same reasoning as in the proof of Lemma 1 and relations (16) and (9), we obtain

V. S. ROMANYUK

1422

� � � n � X X � � �  ck¯ ∆λ (hk¯ ; ·)� � � � ¯ j=0 � k2Z

k∆λ (f ; ·)kLp (Id ) λ

j,d

 C4 (d, p)

n X j=1

Lp (Idλ )

−jd( p1 − 12 )

(δ2j )1/p · 2

0 @

� � � � X � � j 1/p �  C5 (d, p) (δ2 ) � ck¯ hk¯ � � � k2Z � ¯ j=1 n X

j,d

 C7 (d, p)

n X j=1

n,d

¯ j,d k2Z

11/p

|ck¯ |p A

p

� � � � X � � j 1/p �  C6 (d, p) (δ2 ) � ck¯ hk¯ � � � k2Y � ¯ j=1 n X

X

p

(δ2j )1/p · kf kp  C(d, p)(δ2n )1/p kf kp .

(17)

Taking into account the definition of !(f ; δ)p and relation (17), we arrive at inequality (15). Lemma 2 is proved. Inequalities (13) and (14) can be generalized to the case of operators more general than Pn . For any set ⌦ ⇢ Zd+ such that Yn,d ⇢ ⌦ ⇢ Yn+1,d , we define operators Pn⌦ , n 2 Z+ , acting by the formula X

Pn⌦ f (x) =

(f, hk¯ )hk¯ (x),

¯ k2⌦

x 2 Id .

Denote S := where Zn+1,d = Yn+1,d \ Yn,d . Then

[

Pn⌦ f (x) =

¯ k2⌦\Z n+1,d

supp hk¯ ,

8
x 2 S,

: Pn f (x),

x 2 Id \ S.

In view of the properties of the modulus of continuity of !(f, t)p , f 2 Lp (Id ), 1  p  1, we get the following relations from (14): kf −

Pn⌦ f kpp

=

Z

|f (x) − Pn⌦ f (x)|p dx

Z

|f (x) − Pn+1 f (x)|p dx +

Id



S

Z

Id \S

|f (x) − Pn f (x)|p dx

M ULTIPLE H AAR BASIS AND I TS P ROPERTIES



Z

Id

1423

p

|f (x) − Pn+1 f (x)| dx +

Z

Id

|f (x) − Pn f (x)|p dx

� �  C8 (d, p) ! p (f, 2−n−1 )p + ! p (f, 2−n )p  C9 (d, p)! p (f, 2−n )p

for Yn,d ⇢ ⌦ ⇢ Yn+1,d , n 2 N, i.e.,

kf − Pn⌦ f kp  C10 (d, p)!(f, 2−n )p .

(18)

As a consequence of (18), we obtain the inequality kPn⌦ f kp  C11 (d, p)kf kp

(19)

more general than (13). 5. On the Basis Property of the System Hd0 in the Space Lp (Id ), 1  p < 1 By using the results obtained in the previous sections, we can now state and prove the main assertion concerning the system Hd0 . d = (h )1 be a sequence of functions from Hd ordered Consider the system Hd0 = {hk¯ }k2Z d . Let H ¯ i i=1 0 + according to Sec. 1. We define the operators Rj : L1 (Id ) −! Wj , where Wj := span{hk¯ ; k¯ 2 Zj,d }, j = 0, 1, . . . , by the formula Rj f (x) =

X

(f ; hk¯ )hk¯ (x),

¯ j,d k2Z

f 2 L1 (Id ).

Theorem 1. For any function f 2 Lp (Id ), 1  p  1, the Fourier–Haar decomposition f (x) =

1 X

Rj f (x)

(20)

j=0

d convergent in the space Lp (Id ) is true. The sequence Hd = (hi )1 i=1 is a Schauder basis in Lp (I ), 1  p < 1.

Proof. To prove the second part of the theorem, it suffices to verify the criterion of the basis property of a given sequence of elements of the Banach space for Hd (see [6, p. 19], Theorem 6). First, according to inequality (14) [more exactly, inequality(18)], system Hd orthonormal in L2 (Id ) is complete in Lp (Id ), 1  p < 1. Second, the system Hd is minimal in these spaces because an analog of Proposition 4 in [6, p. 16] for d > 1 is true for this system. Finally, in view of the fact that inequality (19) is true for system Hd , according to the criterion of the basis property, we conclude that Hd is a basis in Lp (Id ). Hence, representation (20) is true for 1  p < 1. For p = 1 (and 1  p < 1), representation (20) directly follows from inequality (14). In conclusion, we note that the present paper contains some results from the previous publication [8].

V. S. ROMANYUK

1424

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

A. Haar, “Zur Theorie der ortohogonalen Funktionensysteme,” Math. Ann., 69, 331–371 (1910). I. S. Schauder, “Eine Eigenschaft des Haarschen Orthogonalsystems,” Math. Z., 28, 317–320 (1928). P. L. Ul’yanov, “On the series in the Haar system,” Mat. Sb., 63, No. 3, 357–391 (1964). B. I. Golubov, “Best approximations of functions in the metric of Lq by Haar and Walsh polynomials,” Mat. Sb., 87, No. 2, 254–274 (1972). V. N. Temlyakov, “The best m-term approximation and greedy algorithms,” Adv. Comput. Math., 8, No. 3, 249–265 (1998). B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow (1984). M. M. Grinblyum, “Some theorems on basis in a space of type B,” Dokl. Akad. Nauk SSSR, 31, 428–432 (1941). V. S. Romanyuk, Basis Haar System of Functions of Many Variables and Its Approximate Properties on the Besov Classes and Their Analogs [in Russian], Preprint No. 2012.2, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2012).