on shape-preserving probabilistic wavelet approximators

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On shape-preserving probabilistic waveletapproximatorsLubomir Dechevsky a & Spiridon Penev ba Institute of Mathematics & Informatics , Technical University , Sofia, Bulgaria, 1156b School of Mathernatics Department of Statistics , The University of New South Wales ,Sydney, 2052, AustraliaPublished online: 03 Apr 2007.

To cite this article: Lubomir Dechevsky & Spiridon Penev (1997) On shape-preserving probabilistic wavelet approximators,Stochastic Analysis and Applications, 15:2, 187-215, DOI: 10.1080/07362999708809471

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STOCHASTIC ANALYSIS AND APPLICATIONS, 15(2), 187-2 15 (1997)

ON SHAPE-PRESERVING PROBABILISTIC WAVELET

APPROXIMATORS

Lubomir Dechevsky

Institute of Mathematics R- Informatics Technical Cniversity, Sofia

1156 Sofia, Bulgaria

Spiridon Penev

School of Mathematics Department of Statistics

The University of Kew South \Vales Sydney 2052. Australia

ABSTRACT \Ve introduce a general class of shape-preserving wavelet approsimating opera-

tors (approximators) which transform cumulative distribution functions and densities into functions of the same type. Our operators can be considered as a generalization of the operators introduced by Anastassiou and Yu [I]. Further, we extend the consideration by studying the approximation properties for the whole variety of L,-norms, 0 < p 5 m. In [l] the case p = o is discussed. Using the properties of integral moduli of smoothness, we obtain various approximation rates under no (or minimal) additional assumptions on the functions to be approximated. These assumptions are in terms of the function or its Riesz potential belonging to certain ho~nogeneous Besov. Triebel-Lizorkin. Sobolev spaces, the pace BT., of functions ~v i th bounded Wiener-Young p-variation, etc.

0. PRELIMISARIES

Copyright G 1997 by Marcel Dekker. Inc.

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188 DECHEVSKYANDPENEV

is well-known that L, is a quasi-Banach space for 0 < p < m ([2], [IS]) with em-

bedding constant in the quasi-triangle inequality c, = max{l,2$- ') : 1 1 f f glL,II < c,((l f iL,JJ -+ JlglL,ll). and, in particular. L p is a Banach space for 1 5 p 5 x?. Fur-

ther we denote by L1,i,, = { f : f l c € L1(C) for every compact C C R) xvhere f l c is the restriction o f f on C.

The spaces of lo call^) absolutely continuous functions are defined. as usual.

by .ACloc == {f E Ll,i ,c : 3 f ' ( the derivative of f ) Lebesgue almost o l e r \nhe re and

""j is the r/-th j' E I,l,i,l). AC = .AC(R) = {f E ACI,>, : f' E L1). f (" ' = - dxu

derivative of a univariate function. For a 2 O.[a] is the integer part of n. For

f E Ll,i,,.p E N, 0 < h < x. the Steklov's function (Steklov-means) j,,,,, is defined

by (see [12].[13]):

t .>c.

V f denotes the usual variation of a function f in j-x,fx). \ lk shall also use -

b

the general notation V f for the variation in [a. b ) . -x < (1 5 6 < x. E'os a

+ x

1 5 p < x? V f = I/p f is the IViener-hung variation of f . Denote ljy 131; = -= P

the space of functions with a bounded LYiener-Young variation 1;. Note tha t B l ;

is the space of functions xvith bounded variation in the usual sense.

It is easily seen tha t BV;!" BL;lq for 1 5 p 5 q 5 x. The space BT$'" is a

seminornied space, whileas BVp is a quasi-seminormed abelian group with imbedding

constant in the quasi-triangle inequality c, = Zp-l, 1 < p < x. Here and in the se-

quel "A (I B", when applied to quasi-seminormed spaces .A and B , has the meaning

of continuous imbedding (see, e.g. [2],[15]). For such A and B . A n B is the inter-

section of the two spaces with quasi-seminorm 11 f Id fi BII = n~ax{ll fi.-lii. l i flB11).

and A + B is the sum of the two spaces with quasi-seminorm

inf l l f I A + = f = f o + f , . f 0 E A . f l tB (IlfolAii + IlfllBll)

(see [2], [15]). For t E (0 ,cc) Peetre's Ii-functional (e.g., [I l l) is defined by Dow

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PROBABILISTIC WAVELET APPROXIMATORS 189

The homogeneous Sobolev spaces are defined by ~@F(R) = {f E AClOc : 1 1 f l ~ ; l l =

Il.f('")lLPll < x) , 1 5 p 5 x, p E N. Their inhomogeneous analogs are defined by

\Ve introduce the homogeneous Besov and Triebel-Lizorkin spaces B;, and

F;,, s E R, 0 < p , q 5 x respectively, as well as their inhomogeneous analogs

B,&, F;,. Our notations are the same as in [2]. [15]. [lG]. For equivalent definitions

and properties (imbeddings, dual spaces, interpolation) we refer also to these. as

well as [ll]. [8]. Here we only make few useful notes for the sake of the reader's

orientation which cannot be explicitly found in the references but are collections

or implications of known facts to be found in these references. If the homogeneous . .

spaces B;,, FpSq are defined using Tl'iener-Paley's theory (via Peetre's function (see

121. [lj]) or Calderon's function (see [6]) as a basis for their atomic decomposition).

then a factorization is carried out modulo polynomials orthogonal to the concretely

chosen Calderon's function (with respective modification for Peetre's function). The

factor-spaces obtained are independent of the concrete choice of Calderon's or Pee-

tre's function and are quasi-Banach spaces (Banach spaces for 1 < minip. q ) < x).

It is often collvenient t o consider the elements of these spaces as functions rather

than as equivalent classes of functions and, for a fixed choice of Calderon's (Pee- . .

tre's) function. consider B;,. F;, to be (quasi) seminorrned. (Recall that a similar

situation arises with L,-spaces).

For s > mas{;. l } - 1. respectively s > max{ l P ' 1 . 1 ) - 9 1, the Besov. respectively

Triebel-Lizorkin spaces admit equivalent (quasi-)norms via finite differences and

functional moduli of smoothness. The above restrictions on s are essential and

are related to the fact tha t for these ranges of parameters the Besov and Triebel-

Lizorkin spaces are contained in Ll,i ,c. Outside these ranges the Besov and Triebel-

Lizorkin spaces contain generalized functions which are not regular. Other important

ranges of parameters are .c > b. respectively s > max{b, a ) . Then each element

(equivalence class) of the Besov, respectively Triebel-Lizorkin space is (contains) a

continuous function.

The inhomogeneous versions B;,, FpSq are quasi- Banach spaces (Banach spaces

for 1 < m i n { ~ , ~ ) < m). It is important tha t for s > maxi; , 1) - 1 : B,S, =

L , n B;,, F h ~9 LP n F,",. For p = q F;, = Bi9, F;q = B& (with equicalent quasi- norms). If p # q. s E R then the Besov and Triebel-Lizorkin spaces are essentially

diverse (see e.g. [ l j ] ) .

;In enormous variety of well-known function spaces can be identified as 110-

mogenous or inhomogeneous Besot or 'Triebel- Li~orkin spaces for specific values of

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190 DECHEVSKY AND PENEV

p,q and s . For an orientation we refer t o [6] and [ E l . Two (quasi-semi)normed

spaces are meant here to be identical if they are isomorphic (or. even simpler. coin-

cide as sets and have equivalent (quasi-semi)norms). Here we mention fea relevant

examples only:

11 E N. For p = 1: x' W F and are not Triebel-Lizorkin. or Besov spaces. The

same refers to L p . 0 < p 1 or p = x. For the definition ant1 relevant properties of

the Riesz potential I S f . s E R, we refer to [ 2 ] . \Ye note that it exists in any case for

f that may be of interest t o us. The following property will be of interest t o us (see

[2]. [IS]): IIIS f IL,II is an equivalent norm in F ; ~ for 1 < p < x. s 6 R. hloreolrer.

if o < q := cx: s , g E R then 1 1 1 " j ~ ; ~ ( l is an equivalent (quasi-)norm on F;:' for

0 0, LL E -Y. Denote by ;,(f, h ) , . 1 5 p 5 x the iu t eg~a l p-modulus of

smoothncw of f . The latter is defined b j :

l e e A ( = f + t - f ) : A ( 2 ) = A A:-' f ( z ) ) . 1, = 2.3.

The moduli of smoothness (which for 11 = .x, are usually referred to as moduli of

continuity). are a basic error estimation tool in approximation theory. For some more

details see [12], [ 1 3 ] , [14]. Their properties related to the spaces BT.',. 1.i7, B,",: F;", etc. , will be used essentially later in the text. Here we only note that (see. e.g..

[ lo]? [14:1) for 1 I p < m xjw,(f,t), = ot( l ) , t -+ O+ if and only if f E L,. For

p = m :~,(f , t) , = o t ( l ) , t + O+ if and only if f is continuous. Besides (see [7]:

[12]. [13]), for 1 I p 5 m, Steklov's function f,,h is related to f by:

These bounds are useful for a variety of purposes. In particular, they allow a

very convenient and precise error estimation technique to be developed which, in

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particular, can be used in proving that the moduli of smoothness are equivalent t o

appropriate A'-functionals ( [ 7 ] ) .

Formally, the above definition of the integral moduli of smoothness for 1 5 p 5 rn can be also extended for the case 0 < p < 1 (see, e.g., [12]). However, for

0 < p < 1, the moduli so defined have much worse properties as an error estimation

tool. In particular, the inequalities about Steklov's function f,,), and its derivative

fir2 are generally not true if 0 max (i. 1) - 1. na logous ly

l/q L + [ ( ( ) ) is an equivalent (quasi- )norm in B;q if s >

max {i, 1) - 1 (see [ l j ] ) . (Readers who use the source 1121 should take our remark

in consideration).

In view of the last observations for the case 0 < p < 1. we consider the following

definition of integral moduli which holds for 0 < p 5 x , p E N:

This definition. while preserving the useful properties of the previous definition

for 1 _< p 5 w. has an essential adbantage. To begin with. for 1 5 p 5 cc it is

equitalent t o the old one (see the proof of Theorem 2.1.1. ii) belou). Next. similar

to the prelious definition (see [15]), for ~ , ( f , 6 ) ~ it holds in case 0 < p = q < 1 I?

1. - 1 < s < p tha t [ s f p ) ] is an equivalent quasinorm in

F& = B,S, (which equally implies &,( f . S)? 5 c(p.p, S ) S ' f l ~ ; ~ j l for all f E B;,).

Finally, for this definition a version of the inequalities involving Steklov's func-

tion and its derivatives continues to hold true now also for the case 0 < p < 1

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(compare with (3.25) and (3.25') in the sequel of the present paper.

We assume that the reader is familiar with the basic notions of wavelets theory.

For references on unconditional bases, atomic decompositions and wavelets. we refer

1. INTRODUCTION

This is the first paper of a sequence of several papers whose objective is as

follows:

1. To introduce a general class of shape-preserving wavelet approximating oper-

ators (approximators) which transform cumulative distribution functions and

densities into functions of the same type.

2. To analyze the approximation properties of these operators under minimal as-

sumptions about the regularity of the cumulative distribution function (cdf)/density.

3. To introduce, as data-dependent versions of these approximators, shape-preserving

wavelet estimators with an a priori prescribed smoothness properties.

4. To evaluate the risk of the estimators.

5. To analyze the asymptotic optimality of the estimator class considered

In the pioneering paper [l] the following operator is considered (modified nota-

tions):

where F is a cdf and pk, is defined via p as in (2.1.1). It is shown in [I] that , if 3 is

supported in [ -a ,a ] and satisfies (2.1.2-5), then the class of cdf's is invariant under

the action of Tk . It is proved further that for continuous cdf F

Moreover, if, additionally, p satisfies the condition

then the order of approximation increases from 1 to 2 and

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IITk(F) .- FILII I w ( F , 21-ka)m (1.3)

A model example of an admissible q satisfying (1.2) is also given there.

I t is remarkable that the explicit construction of the monotone operators Tk

produces orthogonal wavelets only in very restricted cases of the choice of +. In

particular, we observe that , because of the monotonicity, T k ( F ) cannot simultane-

ously be an orthogonal wavelet expansion and a continuous function. In other words,

it is possible to combine monotonicity of the wavelet approximation of a cdf with

certain regularity requirements about the image cdf (e.g., continuity or smoothness

of a certain order) only if the orthogonality of the wavelet is lost (and replaced by

almost-orthogonality).

In the present paper we develop the idea of using almost- orthogonal wavelets

further by considering (see (2.1.1))

where 9 and y do not necessarily coincide. We show that , if p satisfies (2.1.2-5) and

y satisfies (2.1.2) and (2.1.6), t h m Ah is cdf-preserving . Kote that the conditions

(2.1.2,6) are weaker than (2.1.2-5) because, as shown in [I], (2.1.3) implies (2.1.6).

In general, Ak is non-orthogonal, although (1.4) includes some orthogonal wavelet

operators, too. We extend the consideration to estimating ( (Ak(F) - FIL,(( , 0 < p 5 m. noting that all cases 0 < p 5 m are equally important for stochastic applications.

Our estimates for IIAk(F) - FIL,j are both an extension and an improvement of

(1.1) and (1.3)

IlAk(F) - FIL,II i cl(a)lla,,,l ~ , ( l w . l ( ~ , 2 ' - ~ a ) , t c2(a,p. y. $ ) 0 4 ( ~ . 2 ~ - ~ a ) , ,

(1 .5) for 1 5 p 5 m. and for 0 < p < ffi the estimate is the same, with an inessential

modification. Here

Note that the condition a,,, = 0 Lebesgue a.e. on R is an essentially weaker t 2c

condition than 3 = zi.. J ~ ( C - J ) = ( (which implies (see [I]). r g ( r ) d r = j=-w

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1 94 DECHEVSKYANDPENEV

The rich variety of properties of the moduli of smoothness implies various ap-

proximation rates under no (or minimal) additional assumptions about F which are formulated in corollaries 2.1.1-15. For example, under no additional assumpt,ions

(i.e., F being a cdf is the only assumption) we obtain that

-4dditional assumptions on F are in terms of F or its Riesz potential (derivative,

fractional derivative) belonging to certain well-known function spaces: the homoge-

nous Besov or Triebel-Lizorkin spaces, the Sobolev spaces (including the limit cases

p = l , ~ , when Sobolev spaces are not in the Triebel-Lizorkin scale), the spaces

BV, of functions with bounded Wiener-Young p-variation, etc.

As far as densities f are concerned, we show that , if both p and II, satisfy (2.1.2)

and (2.2.1), then Ak preserves simultaneously both cdf's and densities.

The estimation of / \Ak( f ) - flLpjj is carried out along the same lines as the one

of J IAk(F) - FIL,II. In particular, (1.5) holds true, with F replaced by f .

The corollaries (under additional assumptions) about f are similar to the ones

about a cdf. However, if no additional assumptions are made about f . then there

are essential differences in the rates of convergence (due to the different "natural"

properties of a cdf F and a density f - more precisely F E BVl, respectively f E L1).

Thus the analog of (1.6) is only (IAk(f) - flLlli = ok( l ) , k -i +w.

2. &LAIN RESULTS

2.1 Approximation of the CDF

Let F be a cumulative distribution function (cdf). Consider the wavelet oper-

ator "at the k-th resolution level"

+ m

< f , >= lw F(t)ykj( t )dt , where ykj(x) = 2k /2p(2kz - j) :

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PROBABILISTIC WAVELET APPROXIMATORS

+ 00 V p < m , p - right continuous - 05

There exists b E ( -a ,a ) such that cp is non- decreasing on (-m,b] and non-

increasing on [b, tee,) (2.1.5)

$ satisfies (2.1.2) ; $ E L1(R) and $(x)dx = 1 (2.1.6)

The operators A k ( F ) be the main objects of our investigation. They will be used

for approximating the cdf F. Operators of the same type when applied to a density

f , will be used later in Section (2.2), for density approximation. The function II, will

have to satisfy some additional conditions then.

The construction of the operator Ak(E) is inspired by the pioneering work of

Anastassiou and Yu [I] who consider the partial case 9 = $. Our generalization

makes it possible t o distinguish the essential differences of the roles of p and y in

the approximation process. In particular, $ may be selected in a much more general

class of functions than q. Note that (2.1.2,3) imply that p E L, , iipiLmll I: 1 and

that (see [I] p. 255) (2.1.3) implies v(x)dx = 1, but the opposite implication

is not true.

Denote E k ( F ) = A k ( F ) - F.

Our main result on estimating the rate of approximation of F by A k ( F ) is given

in the following

Theorem 2.1.1 Let i l k , ~ , y satisfy (2.1.1-6). Denote

i) For 1 5 p 5 cc, there exist constants cl > O , q > 0 such that

where cl = c l l ( 2 a j . c1 = c; . ua . 2 ' l p . a l / p , c ; , c ~ being absolute positive constants.

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p' E 11, co] is such that $ + $ = 1

ii) For 0 0 such that

where c3 = a-p .2-p(l+ 22p), q = va . ap. 2(1 f 23p), cs = 2aP(2P-1 + v,(l + 2 3 ~ ) ) .

Let us note that in view of the importance in applications and especially in view

of the future statistical applications (see Dechevsky and Penev [5]) the quasi-Banach

case 0 < p < 1 is also included in Theorem 2.1.1.

A set of important corollaries can be derived from the main result.

Corollarv 2.1 . I . Under the conditions of Theorem 2.1.1, let o,,+(C) = 0 V ( E W. Then

ii) For 0 < p < 1

i.e. the rate of approximation is evaluated from above by the second and third

integral modulus of smoothness. Therefore, one is interested in finding couples of

functions p and $ for which a,,$(x) 0 holds.

Remark 2.1.1. In [I] convergence of second order of the approximation was shown

under the assumption

Our Theorem 2.1.1. shows that (2.1.7) can be replaced by the essentially weaker

requirement

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Note that (2.1.8) is an essentially less restrictive condition than (2 .1 .7) even in the

special case y - Q. Indeed. (see [I], p. 264) the property (2.1.7) implies that + n

L r p ( r ) d r = r p ( r ) d r = 0 and (2.1.8) will hold for 9 5 Q. On the other 1: hand there exist y's for which (2.1.8) is true but not (2.1.7).

Examples illustrating this fact are given below.

Example 2.1 . I .

i) Case p ( x ) E $(x). Let us define

~ $ 1 - c o . x E ( - 1 t c o , c o l t co - a: , 2 E (co, 1 t co]

, elsewhere

where co # 0 with 0 < / co J < i. Now it is easily seen that 9 satisfies all requirements t o o

(2.1.2-5), ( - p ( j - () = co # 0 holds but , yet. (2.1.8) holds true and there is j=-CC

a second order convergence of the approximation.

i i ) Case p ( x ) + f:[xj. Define p as in i) and let

) = ( 1 2 E ( - 1 + co. 1 t co) 0 elsewhere

Then (2.1.8) is again fulfilled, and there is a second order convergence of the ap-

proximation.

Rcmark 2 . 1 . 2 The problem of explicitly finding all appropriate p and q) for which

(2.1.8) holds can be approached from the Fourier-transform side. This \\ill not be

discussed In details here.

R c r n c ~ ~ l : 2.1 .i \Ye note another case of G satisfying (2.1.6) which has interesting

extremal propeltles with respect t o the embedding constants in 'Theoren1 2.1 1 and

colollary 2.1.1. If A c [--a, a] is measurable, with Lebesgue measule /A1 = 1 ( a 2 $)

and cl = y,, the indicator function of the interl~al A. then apparently i.lL,iJl = + x

L , ! L * I I = lm V ( T ) ~ T = 1 for every p : 1 < p _< r: (even 0 < p 5 x' if we assume

the natural convention p' = x for 0 < p _< 1 j. Moreover, the inf { ~ I L ' L , ~ 1 : Q

satisfies (2.1.2.6) ) = 1, 0 < p 5 x is attained exactly for 9 = 1,. Since this

nlinirliizes the embedding constant, we have a hint for choosing suitable functions

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$(z). Note also that choosing a discontinuous ~ ( x ) does not effect the smoothness

properties of l ik (F) , as long as we can take y # ~. Besides, example 2.1.1. ii) can be

easily modified to display a +(x) = x, (2) which, in combination with appropriate

(smooth or non- smooth) p(x) satisfies 2.1.8.

The following assertion holds:

Lemma 2.1.1. The operators Ak(F) are shape-preserving for cdf, i.e., if F is a cdf,

then Ak(F) is also a cdf.

Remark 2.1.1. It is worth noting that Lemma 2.1.1 is the only assertion which makes

use of the condition (2.1.5) for ~ ( x ) . We would also like to point out that our lemma

2.1.1. shows that the condition about continuity of the cdf F which is imposed in

[I] is in fact not necessary for the operators Ak(F) to be shape- preserving.

There follow important corollaries of Theorem 2.1.1 and of Corollary 2.1.1 which

are obtained by utilizing the properties of the moduli of smoothness. It is convenient

to divide these corollaries into two classes depending on whether IIu,,$IL,H = 0 or

not. In all corollaries F is understood to be a cdf. In corollaries 2.1.2-7 we assume

ll~v,*lLmll > 0. Corollaru 2.1.2. There exists c > 0 independent of F , such that ( ( E k ( F ) / L 1 / / < ~ . 2 - ~ .

Corollaru 2.1.3. Let 1 5 p < co. Then, there exists c > 0, independent of F, such

that

Corollaru 2.1.1. If F E .4C(R), i.e. F is absolutely continuous, and, additionally,

f = F' E L, , 1 0, independent of f . such that

Corollarz, 2.1.5. Let F E B,", , 1 5 p < co , 0 < s < 1. Then, there exists c > 0,

independent of F, such that

Corollarv 2.1.6. Let F E gip , 0,

independent of F , such that

Remark 2.1.5. Since every cdf F E ~ ; k . 1 _< p 5 co, with no loss of generality we --

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may assume tha t instead of s : 0 < s < 1, only s : 0 < < s < 1 is required. This makes sense for 1 < p < m.

R ~ r n a r k 2.1.6. The imbedding B ; , ~ C B,",,~ , 0 < q1 5 43 5 X , shows tha t the

estimate in Corollary 2.1.5 is the best possible in the Besov scale. i.e. replacing

11. !B,S,!( by 11. ~ B ; J for some other q E ( 0 , x ) is true but would only make (2.1.10)

rougher. For the same reason estimates in terms of the inhomo~eneous Besob spaces

B,S, = L , n B,", C B,", , 1 0 are rougher. The same applies for the

case 0 - 1.

Corollarv 2.1.7. Let 1 < p 5 K , < s < 1 and assume tha t F E B,", . Then, for every q : p 5 q < x. there exists c independent of F. such tha t

Yext. in corollaries 2.1.8-15 u e assume lia,,,lL,li = 0 .

In this case corollaries 2.1.2-7 hold true with embedding constants which are

smaller than if llu,,,,lL,ll = 0 was not fulfilled

Cbrollary 2.1.8. In Corollary 2.1.5 s may be taken 0 < s < 2 (or, with no loss

of generality for cdf. $ < s < 2 ) . Moreover. for 1 5 s < 2, Inequality (2.1.10)

is equivalent t o the statement: there exists c > 0. independent of F. such tha t

! j E k ( F ) I L p ! ! 5 c.2-"!! f I B & ~ /I, where f = F' is the density.

Corollnru 1.1.9. In Corollary 2.1.6 one can take $ 0. independent of F , such that

~ h c r e f = 1:' is t,lle density.

C'orollnr.71 2 .1 .1 1. Let F E ~i~i. 1 < 11 < X. Then, for every q : p 5 q 5 x there

cxists r. > 0. independent of 6'. such that I E I ; ( F ) i L , l < c . 2 - " ' - ~ + ~ ' ~ 1 ~ i ~ ~ / l

(-'omllaru 2.1.12. Let I E 1Cr(IW). with density f = P'. Thcn. for cvcry 11 : I 0 such

< C . 2 - k 4 (f, 21-A[ l )y -

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Corollaru 2.1.11. For 1 0, independent of F, such that

Corollaru 2.1.15. If F E AC(R), f = F' is the density and f E BVl, then there +03

exists c > 0, independent of F, such that ( I E k ( F ) ( L l j ( 5 ~ . 2 - ' ~ V f -co 1

2.2 Densitv Approximation

Let F be an absolutely continuous cdf, with a density f(x) = F 1 ( x ) . Consider

A k ( f ) defined as in (2.1.1). with F replaced by f . Function p is now supposed to

satisfy (2.1.2, 3) but with (2.1.3) loosened:

p(r - j ) E I Lebesgue a.e. x E R (2.2.1) +-.a

The function $ ( x ) is now supposed to have the same properties as p, i.e. (2.1.2)

and (2.2.1). These conditions imply that p, t,h E Ll n L,. Besides (see [ I ] ) , (2.2.1) +m

also implies l_ Theorem 2.2.1. Theorem 2.1.1 and Corollary 2.1.1 continue to hold true if the cdf -- F be replaced by a density function f . Corollaries 2.1.2-15 continue to hold true

with f instead of F, if all conditions of the type 5 s be replaced by 0 < s.

Lemma 2.2.1. Let p,$ satisfy (2.1.2) , (2.2.1) . Then A k ( f ) is a density. i f f is such. -- i.e., the operator Ak is shape-preserving for densities.

3. PROOFS

Proof of Theorem 2.1.1

i ) First consider the case 1 _< p _< co +oo

We consider the homogeneous Sobolev space I&':. Since F is a cdf, i.e. V F =

1 < co, it follows that F E L1 + I@: for every p E N. This means that

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where 1 1 . IL, + W:)) is any of the equivalent norms

or w.,( f , h),-the integral p-modulus of smoothness of f, h > 0. It is well known

that for F E L, f there exists the Steklov function Fu,hrp E -V, h > 0 (see

Petrushev and Popov [12, Section 31 and the Preliminaries for the explicit definition

of F,J,). This function has derivatives F::; , u = 1 , 2 . . . . , p and satisfies follouing

relations:

where c,,, are positive constants. There exist explicit estimates from above for

those constants (see [12], [13]). For a given cdf F we shall concentrate on the

Steklov functions Fk := F u , h for p = 2. h = 21-ka. Let us note that Fk may be. or

may be not a cdf. By llinkowski's inequality:

IIEk(F)ILpII I lIEk(F - J-~)lLpll + IIEkJFk)ILpII (3.3)

U e shall estimate from above each of the summands in the right hand side JRHS)

of (3.3). ITsing (2.1.3). we have

+cs

Hence. for any function f J x ) n e can write: f ( x ) = 2-"2pi,,[x) f ( n ). whence

+ s Since L ' ( T ) ~ T = I , one can ~vr i te 2-$ =< 1 , ~ ' ~ ~ > and a substitution in (3 .4 ) .i,, leads to

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Application of the above equality for f(x) := F(x)-Fk(x) leads to following estimate

from above of the first summand in (3.3):

Note that we used the fact that the non-zero summands in the infinite sum are only those for which the summation index j (2% - a, 2% + a). Since these are a finite

number of summands, we can utilize for 1 5 p < m, p : $ f $ = 1 and for arbitrary

numbers a,, i = 1 ,2 , . . . , 12 the inequality

LVe can easily see that

Hence the RHS in (3 .6) can be further bounded from above by

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Further application of Holder's inequality and change of variables r = 2k t - j yields

Hence

Sote . however, tha t Ilpks ILmii = 2"' IlpiLmll.

.inother fact easy to observe is that

- Hence. if we introduce the constant t := V,P' . 2 k l p l w l ~ p , / . llplLmil, we can further

evaluate the RHS of (3.8) from above and by Fubini's theorem arrive a t

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Our goal is to evaluate the bound from above in terms of the second integral modulus

of smoothness. Using the property

w i f , h ) p I2llflL,ll . h > 0

of the integral modulus and property (3.1) with p = 2. we obtain from (3 .9) for

f = F - Fk that

Now we start with the evaluation of the second summand in the RHS of (3.3). For

f = F k r similar to the estimate of IIEk(F - F k ) l L p J J , we have

(3.11)

Under the assumption that w z ( F , 21-ka)p < oo, it follows that f = Fk E L p , w;, 147;.

Hence f admits a local Taylor expansion of second order with an integral remainder,

i.e.

f ( t ) == f (x) + ( t - x ) f l ( x ) + ( t - x)' ( 1 - 0 ) f "(x + O(t - x))dO , x, t E R d' Hence

2-k(j+a) 2-" ( 2 - j ) )

2 - k ( 3 - a )

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(3.12) In the expression in the large brackets we change the variables by introducing T =

2k t - j . Utilizing $(r)d~ = 1 and the property (2.1.3) for y, this expression 1: can be easily simplified. For the second expression in (3.12) we use the fact that

Zkx - a < < < 2kx f a imposes that 1x - ti 5 21-ka in the range of integration of

t . This helps to further estimate this expression from above. Finally we obtain by

Fubini's theorem

J - a

Applying Minkowski's inequality in (3.11) and using (3.13) leads t o

But on applying (3.2) with v = 1,h = 21-k. a , we get:

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DECHEVSKYANDPENEV

ci > 0 being an absolute constant.

We note on passing that apparently

(3.16)

Further, on applying generalized Minkowski's inequality, we have

(3.17)

Repeating the same steps as in evaluating IIEk(F - Fk)ILplI after (3.6), the RHS of

(3.17) can be bounded from above by

By changing variables t into r = t - z and analogously to the estimation of the RHS

in (3.8) one can estimate (3.18) from above by

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Putting c* into (3.19), utilizing again inequality (3.2), the fact that f" = F[ and

$ + b = l , w e g e t

where i;; is an absolute constant.

From (3.10), (3.15) and (3.20) we get with ck = 2 + $ the final evaluation

ii) Kow we shall consider the case 0 < p < 1. The definition of the integral

p-modulus of smoothness given in the preliminary section for 0 < p < 1 can be

shown to be more general and to work equally well also for the case 1 5 p 5 co. To see this, let us consider

also for p E [I , m] and compare it to the usual definition in case p E [ I . m]:

On the one hand, L ~ , ( f , 6 ) ~ 5 u ~ , ( f , S ) ~ is obvious (take k = 1. X = 0. bl = p).

In the opposite direction, it is easily seen that w.,(f,6), 5 c;llflL,jl V f E L, and

j , ( f ,&) , 5 c;6,ll f l ~ p " l l V f E ~kp" hold wherefrom (u., 5 cG, follows easily from

the imbedding in the respective side in the equivalence

G,( f .6)p .- I i (6@, f : L,,$,")

(compare the preliminaries section).

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Note, however, tha t for 0 < p < 1, w, and G, are not equivalent. As a version

of the generalized Minkowski inequality is unavailable here, w, is a much better

estimation tool for 0 < p < 1. In particular, the estimation technique using Steklov-

means intermediate approximation is available for w, but not for G,.

Steklov's function f,,h(z) is given in the Preliminaries. The properties (3.1)

and (3.2) are known to hold true for 1 < p 5 cc (see [12, Section 31). For 0 < p < 1.

(3.1) and (3.2) fail to hold for G, as defined in (3.23). Kext, we show that (3.1)

and (3.2) do have their analogs for 0 < p < 1, if w, is defined by (3.22). Let us see

first that for 0 < p < 1 for any Lebesgue measurable function f ( x ) and p E N there

exists a constant ~ ( p ) such that:

holds true. Changing the varialbe Q1 to cu = $ z:=l Ox and utilizing the inequality

we obtain

I PI I J1 I A : ~ ~ ( . ) I ~ ~ I L ~ I I I ~ w , ( i , h) , -1

which proves (3.24).

Next we shall show that for 0 < p < 1 and for w, as defined in (3.22) the following inequalities hold true:

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Indeed, for u = 1 , 2 , . . . , u - 1 the derivative of the Steklov function can be written

as ([12, p. 511:

Hence

P - 1 - ,!, I L - V

" 1 1 Ih . . J h / n ~ P - * ) h / P / ( ~ + - c 0n)idOl . . , d o u - u ~ ~ p ~ ~ ~ 5 ( : ) p (5) 0

X=O 0 I-" ,=I

5 h-"? (:)' (A) " [I, l h lh l ~ L - , ~ ) h , , f ( ~ ) l d Q ~ . . . ~ ~ p - u ~ L p l l p + X=O I-" -

Since '-*h 5 h and x:Ir 0 , 5 ( g - v ) h holds, changing variables 8, to I-"

p, = B,/h and evaluating both integrals in the last expression from above yields

from which (3.25) follows with

For Y = p . the derivative fby? is given by

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and the proof of (3.25') is analogous to the one of (3.25) (but simpler).

Inequalities (3.24), (3.25) and (3.25') help us to proceed in case ii) (0 < p < 1)

along the same lines as in part i) of the proof. The same steps are repeated until one

obtains (3.7). Note, however, that in the case 0 < p < 1 the inequality laYl 5 p= 1

l l P ( a ) holds SO that in (3.7) we have the factor 1 instead of v:Ip'. The next

step in the evaluation is the following :

(3.28)

Changing the variable t to CY = a-12k-1(t - x) in (3.28), we arrive a t

Note, however, that for x E [2-k( j - a), 2-"(j + a)], it follows that

[ ( j - 2 k x - a ) / ( 2 a ) , ( j - 2 k x + a ) / ( 2 a ) ] E [-1,1].

Hence

Like in case i), we substitute f = F - Fk (Fk being the Steklov function Fp,t, for

p = 2, h = 21-ka) into (3.29) to get Dow

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Proceeding like in the proof of (3 .25) , we obtain from the last inequality

l iEk(F)lLpl lP 5 IIEk(F - Fk)ILpllP $. IIEk(Fk)lLpllP ( 3 . 3 1 )

holds. We shall evaluate IIEk(Fk)ILpIIP from above.

Analogously to the respective evaluation in case i), we have

Ysing ( 3 . 2 5 ) (with the constant c, := c ( 2 , l , p ) = (1 +2?p) ' /p from ( 3 . 2 6 ) ) we obtain

( 3 . 3 2 )

Proceeding like in ( 3 . 2 8 ) , we have:

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(3.33)

But applying (3.27) for p = 2, h = 21-ka, one can write

Hence the RHS of (3.33) can be further evaluated from above by the same method

we applied to get (3.29). Changing the variable to a = a-' . 2k-1(t - a ) we obtain

easily the following bound from above:

Note that we also used the property w,(F, 6 1 ) p 5 w,(F, 6")p , 6' 5 6" when deriving

(3.34).

Now it remains to insert the estimate from (3.34) into (3.32) to get:

Substitut,ing the last result and the result from (3.30) in (3.31), one finally gets the

estimate in part ii) of Theorem 2.1.1. Dow

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Proof o f Lemma 2.1.1.

This assertion is proved in Anastassiou and Yu [I] for the case p (x ) = $(I). In

the case p(x) + $(x) the proof goes along the same lines, using properties (2.1.2) of

$(x) only. Since we are imposing the same conditions on the function p(x) as in [I],

the only condition to be checked is that for fixed resolution level k , the coefficients

{cj):=y, with

form a non-decreasing sequence. But this is an immediate consequence of the fact

that $(x) 2 0 has a compact support and that F ( x ) is non-decreasing, since cj is

* tco aIso equal to c j = 2--

a L ~ ( 2 - ~ ( t + j)$(t)dt. Let us note that the properties

lim Ak(F)(x) = 1 and lim Ak(F)(x) = 0 are an immediate consequence of the x-m 2--02

fact that if p = 1, then the right hand side in (3.21) is bounded for any cdf F (see

the proof of corollary 2.1.2 below).

Proof o f Corollarw 2.1.2. It is immediate consequence from theorem 2.1.1 and the -t-02

fact that any cdf F E BV*. Using the properties wl(F, h)l < V F and u k ( F , h)l 5 -m 1

2wkW1(F, h)l (see (12, Section 3.11) leads to the desired estimate.

Proof o f Corollaru 2.1.3. It follows from Corollary 2.1.2, from the fact that BVl c BV, for 1 < p < cc and from (v,F)lIp 5 VIF.

Proof o f Corollarv 2 . l . l . The proof utilizes the properties wl(F, h)p < hll fILpJl and

wk(F, h)p 5 2i3k-1(Fl h )p ,p E [ I , m] of the integral p-modulus of smoothness ([12],

Section 3.11)

Proof o f Corollary 2.1.5. It follows from the property wl(F, h), < EhSl( Fj ~ , & l l for

some positive constant E (see [15]).

Proof o f Corollarw 2.1.6. I t follows from the property dl(F, h), < ~ h ~ j l ~ ~ B ; , l l for

4 < p < 1, - 1 < s < 1 for some positive constant ? (see Preliminaries).

Proof o f Corollarv 2.1.7. This is a consequence of the embeddings B,S, c B;;,, for

Proof o f Corollaru 2.1.8. This follows from the fact that F E B;, with 1 5 s < 2

iff f E B,S;';'.

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Proof of Corollaru 2.1.9, 2.1.10. 2.1.11. Omitted since similar to 2.1.6, 2.1.7.

Proof of Corollaru 2.1.1 2. Follows from the property w ~ ( F , h)p 5 hwk-l( f , h)p , k 2 2 ([12, Section 3.11).

Proof of Corollaru 2.1.13. By the equivalence of w,( f , 21-ka) to 1 < ( ( 2 l - ~ a ) ~ , f ; L,,

w;), p = 1,2, we obtain ((Ek(g)IL,II I c(lglLqlI for d l g E Lq, and llEdg)lL,ll I ~ ( 2 ~ - ~ a ) f i l l ~ j ~ [ l I for all g E w:. Recalling that (see Preliminaries) Lq = F$, W: =

k g , taking p = 2 and using the complex interpolation method (see [2],[15]), we

interpolate between the estimates and obtain J(Ek(g)(LqII 5 c(21-ka)s+~llg(~d$311,

I(Ek(g)IL,II 5 ~ ( 2 ' - ~ a ) ~ .1<(2'-~a, F; ~;?,k:Szfl) = S

Using the lifting property !(ISF~F:~!( ( ( F I ~ , " ~ + ~ I I (see Preliminaries) we obtain

and the proof follows from the respective property of the integral modulus.

Proof of (?orollaru 2.1.1.4. Follows from Corollary 2.1.13 and the imbedding theorem

about Triebel- Lizorkin spaces with different metric index ([15]).

Proof of Corollaru 2.1.15. Follows from Corollary 2.1.12 and the property wl( f , h ) l _<

h f. -03 1

Proof of Lemma 2.2.1. The condition Ak(f)(x) 2 0 almost everywhere is obvious

since y ( x ) , $(x) are non-negative. Further, on applying Fubini's theorem, we have

t m

Changing the variable x to ( = 2*z-j, applying Fubini again and using y(()d( =

1, one obtains

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ACKNOWLEDGEMENT

This work was supported by the Australian Research Council. I t was initiated during a visit of the first author a t the Department of Statistics, University of New South Wales.

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4. I. Daubechies, "Ten Lectures on Wavelets," SIAM, Philadelphia, 1992.

5. L.T. Dechevsky, and S.I. Penev, On shape- preserving wavelet estimators of cumulative distribution functions and densities, preprint No. 11, Department of Statistics, School of Mathematics, The University of New South Wales (1993).

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7. H. Johnen, and K . Scherer, On the equivalence of the Ii-functional and moduli of continuity and some applications, in Tonstructive theory of functions of several variables, Oberwolfach '76" (W. Schempp, and K. Zeller, Eds), Lect. Notes in Math. 571, pp. 119-140, Springer, Berlin Heidelberg New York, 1977.

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9. Y. Meyer, "Ondelettes e t op6rateurs I," Hermann, Paris, 1990.

10. SA4. Nikol'skii, "Approximation of functions of several variables and imbedding theorems," Springer, Berlin New York, 1975.

11. J . Peetre, "New thoughts on Besov spaces," Duke University Math. Series, Durham, 1976.

12. P.P. Petrushev, and V.A. Popov, "Rational approximation of real functions," Cambridge University Press, Cambridge, 1987.

13. B1. Sendov, and V.A. Popov, "Averaged moduli of smoothness," Bulgarian Academy of Science, Sofia, 1983.

14. A.F. Timan, "Theory of approximation of functions of a real variable", Hindus- tan Publ. Corp., Delhi, 1966.

15. H. Triebel, "Theory of function spaces." Monographs in Mathematics 78, Birk- hauser, Basel, 1983.

16. H. Triebel "Theory of function spaces 11," hlonographs in Mathematics 84, Birkhauser, Basel, 1992.

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on shape-preserving probabilistic wavelet approximators

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