On some properties of weighted Hilbert spaces Текст научной статьи по специальности «Математика»

УДК 517.5:519.2

On some Properties of Weighted Hilbert Spaces

Vladislav V. Branishti*

Institute of Computer Science and Telecommunications Reshetnev Siberian State Aerospace University Krasnoyarskiy rabochiy, 31, Krasnoyarsk, 660014

Russia

Received 14.02.2017, received in revised form 17.05.2017, accepted 10.08.2017 We describe the weighted Hilbert spaces L2,w (H) with positive weight functions w(x) which are summable on every bounded interval. We give sufficient condition for L2,wi (H) space to be extension of L2w2 (H) space. We also describe how to use given result in statistical probability density estimation.

Keywords: integrable function spaces, Hilbert spaces, weighted function spaces, second order splines, probability density function estimating. DOI: 10.17516/1997-1397-2017-10-4-410-421.

Introduction

Let L2,w (Q), where w(x) is measurable positive function, Q C R is measurable subset, be a space of real functions f : Q ^ R for which the integral

/ f2 (x)w(x)dx JQ

is finite. The measure and the integral are comprehended in Lebesgue sense. Given space is often described (e.g. [1,2]) as a case of L2(Q, £, ¡) space, where £ is a a-algebra of measurable subsets of Q, is a measure on S, which is defined by:

¡(X) = / w(x)dx. (1)

Jx

Particularly, the space L2,w(Q) is Euclidean space with scalar product

(f,g)w = / f(x)g(x)w(x)dx, J Q

which induces the norm

\\f ||2,w = VUJjW-

It is also known that if the measure (1) has countable basis then the space L2,w(Q) is separable. If w(x) = 1 then the space L2,w(Q) is denoted as L2(Q), and if Q = R then as L2,w. In [3, 7.1.3] they consider a weighted space Lp(Rn, p(x)), where p € (0;+ro), Rn = Rn is n-dimensional arithmetic space, and p(x) > 0 is Borel measurable function on Rn. The paper [4] gives conditions on weight function w(x), which makes wavelet spline system be a conditional

* branishti@sibsau.ru © Siberian Federal University. All rights reserved

or unconditional basis in Lp,w (R) space, p G [1;+œ). In the paper [5] they are studying similar problem for Haar wavelet system.

The L2,w (Q) space finds its application in the problem of statistical estimating of probability density function fç(x) of continuous random variate £. Indeed, in a case of completeness and separability of the space L2,w(Q) there is countable complete orthonormal system [pj(x)}°=0, i.e. the system which for all function f G L2,w (Q) satisfies the limiting relation

lim

(f, V3 )w V3 - f

3=0

Thus, if f G L2,w(Q) then its projective estimate fl (x) defined by

fl(x) =53 a3 V3 (x) = (f ' V3 )w V3 (x)

3=0

3=0

converges to f (x) in norm of the space L2,w(Q).

The paper [6] shows that for each continuous random variate £ and for appropriate weight function w(x) there exists the space L2,w(R) including it. In so doing, the choice of the function w(x) is important for convergence speed of projective estimate. In connection with it there is a necessity to investigate the properties of the L2,w (Q) spaces in the context of weight function choice.

0

2

1. Main results

By virtue of a-additivity of Lebesgue integral, for each measurable positive function w(x) the measure defined by (1) is also a-additive. But if the function w(x) is not summable, i.e.

/ w(x)dx = JQ

then it is possible the pathological behavior of the measure n in particular cases.

For instance, let Q = [0; 1], £ is a a-algebra of measurable subsets of Q and w(x) = —. Then

x

each segment from £ containing 0 has infinite measure:

ra a

yu,[0; a] = I w(t)dt = lim lnt\ax = a > 0.

J 0

Now we take a sequence an I 0. Then

lim ^[0; an] =

On the other hand,

We receive that

(f] [0; an] = = / w(x)dx = 0.

n=i / J{0}

M Pi [0; an] = lim m[0; an],

n

n=i

i.e. built measure p is not continuous.

Further we assume that the function w(x) is summable on each bounded interval X c R:

/ w(x)dx <

Jx

Now, it is obvious that the measure ¡ induced by the function w(x) is a-finite. Then the theorem about completeness of Lp(Q, spaces with p € [1;+ro) and a-finite measure ¡ [2, IV, 3.3] leads to completeness of the space L2,w (Q).

Besides the measure ¡ has a countable basis consisted of, for example, elements from aring generated by semiring of half-intervals on real axis with rational endpoints. This leads to separability of the space L2,w(Q).

Thus, for each positive function w(x) which is summable on every bounded interval the L2,w (Q) space is separable Hilbert space.

Present paper considers relationship between L2,w (Q) spaces with common set Q and different weight functions w(x).

Definition. The space L2wi (Q) is called an extension of the space L2w2 (Q) if the strict inclusion

L2 (Q) c L2 ,wi (Q)

holds.

Let us denote obvious proposition. Proposition 1. If the inequality wi(x) ^ w2(x) holds for all x from Q, then

L2,w2 (Q) C L2wi (Q).

Particularly, if w(x) < 1 then the space L2,w(Q) includes the space L2(Q). Proof. It follows from w\(x) ^ w2(x) that for all functions f : Q ^ R we have

f2(x)w1(x) < f2(x)w2(x),

and

/ f2(x)w\(x)dx ^ f2(x)w2(x)dx. QQ

Now, convergence of the integral at the left side follows from convergence of the integral at the right one. Thus, for all function f : Q ^ R we have that f € L2,w2 (Q) involves f € L2,wi (Q), i.e.

L2,w2 (Q) C L2,wi (Q).

□

Remark 1. The conclusion of the Proposition 1 remains true even when the inequality wi(x) ^ w2(x) holds almost everywhere on Q.

Proposition 2. With introduced assumptions on weighted functions wi(x) and w2(x) it is true that

L2,wi (Q) CI L2,w2 (Q) = L2,wmax (Q) ,

where wmax(x) = max{wi(x),w2(x)}.

Proof. It is obvious that the function wmax(x) is also summable on each bounded interval. Then the space L2,Wmax(Q) is defined and separable Hilbert. At the same time wmax(x) > wi(x) and wmax(x) > w2(x). Then from Proposition 1 we have that

L2,Wmax (Q) ^ L2,Wl (Q) and L2,Wmax (Q) ^ L2,w2 (Q),

i.e.

L2,wmax (Q) Ç L2w (Q) n L2w (Q).

To prove inverse inclusion we can take arbitrary function f (x) from the set L2,wi (Q) n L2,w2 (Q). From definition of the space L2,w (Q) we will have:

/ f2 (x)wi(x)dx < and / f2(x)w2(x)dx <

JQ JQ

Let us split the space Q by two subsets Q1 and Q2, where

Qi = [x G Q | wi(x) ^ w2(x)}, Q2 = Q \ Qi = [x G Q | wi(x) < w2(x)}.

Then

/ f2(x)wi(x)dx = / f2(x)wi(x)dx + / f2(x)wi(x)dx. JQ JQ1 ./Q2

We have got that both of the integrals

i f2(x)wi(x)dx and i f2(x)wi(x)dx •/Qi ,/02

exist and are finite.

Similarly, folowing integrals exist and are finite:

/ f2(x)w2(x)dx and / f 2(x)w2(x)dx. Q1 Q2

Now we will consider the sum of the integrals fQi f2 (x)wi(x)dx and JQ2 f 2(x)w2(x)dx:

/ f 2(x)wi(x)dx + / f2(x)w2(x)dx = Q1 Q2

= f2(x)wmax(x)dx + / f2(x)wmax(x)dx = f 2(x)wmax(x)dx < +œ. Q1 Q2 Q

We have from this that f G L2,wmax (Q). So the inclusion

L2,wi (Q) n L2,w2 (Q) Ç L2,wmax (Q),

is proved and the conclusion of the proposition as well. □

The paper [6] gives necessary condition on weight functions wi(x) and w2(x) to spaces L2,wi (Q) and L2,w2 (Q) not be equal. We express here a stronger proposition.

Proposition 3. If L2,wi (Q) = L2,w2 (Q), tften at /east one of the inequalities holds:

wi (x) wi (x)

ess ini —-— =0 or ess sup —-— = +œ. (2)

ïEQ w2 (x) œ£Q w2(x)

Proof. On the contrary we assume that all inequalities (2) do not hold. Then

wi (x) wi(x)

ess mi —-— = m> 0, ess sup —-— = M < x>;

xen w2(x) xen w2(x)

i.e. almost everywhere on Q

W1 (x)

0 <m< , { < M <

W2(x)

It follows from the given inequalities that almost everywhere on Q

wi(x) ^ Mw2(x), w2(x) ^ — wi(x).

m

Then

f2(x)wi(x)dx ^ f2(x)Mw2(x)dx = M f2(x)w2(x)dx; (3)

in Jn Jn

[ f2(x)w2(x)dx ^ [ f2(x)— wi(x)dx =— [ f2(x)wi(x)dx. (4)

Jn Jn m m Jn

We have now that (3) leads to inclusion L2,wi (Q) C L2,w2 (Q), and (4) leads to L2,w2 (Q) C

L2,wi (Q). □

It follows from the Proposition 3 that if L2,wi (Q) is an extension for L2,w2 (Q), then

• f wi(x) n

ess mi —— = 0.

xen w2(x)

Let we give sufficient condition for L2,wi (Q) to contain elements which are outside of L2,w2 (Q).

Theorem 1. Let Q C R contains right-side or left-side neighborhood of some point a € R, wi (x) and w2(x) are positive on Q functions which are summable on every bounded interval and for which at least one of one-sided limits

wi (x) wi (x)

iim —— or iim ——

x^a+0 w2 (x) x^a-0 w2 (x)

is equal 0. Then

L2,wi (Q) \ L2 ,w2 (Q) = 0.

Proposition 1 and Theorem 1 lead to convenient sufficient condition for extension of the space L2,w(Q). Let

1) Q contains right-side or left-side neighborhood of some point a € R;

2) wi(x) < w2(x) holds almost everywhere on Q;

5> wl(x) n wl(x) n

3) iim —-— =0 or iim —-— = 0.

x^a+0 w2(x) x^a-0 w2(x)

Then

L2 ,w2 (Q) c L2 ,wi (Q). - 414 -

2. Proof of the Theorem 1

We have to prove some intermediate propositions before we prove the Theorem 1.

Lemma 1. Let Q = (A;+ro), where A € and f (x) is differentiable positive non-

increasing on Q function which satisfies

lim f( x) = 0.

Then there exists non-negative on Q function g(x), for which

/ g(x)dx = and / f (x)g(x)dx <

JQ JQ

Proof. We define the function g(x) on Q in this way:

t \ f'(x) g(x) = —^r.

f(x)

Because of f (x) > 0 and f' (x) < 0 then g(x) > 0. Further,

r rf' (x)

g(x)dx = — dx = ln f(A) — lim ln f(x) =

jq j a J(x)

/ f (x)g(x)dx = — f'(x)dx = f (A) — lim f (x) = f (A) < JQ Ja

Thus, function g(x) satisfies the conclusion of the lemma. □

Lemma 2. The conclusion of the lemma 1 remains true if in the condition we change differentiability of the function f (x) by its piecewise constancy on Q.

Proof. Let the function f(x) is piecewise constant, positive and does not increase on Q. Then Q can be split by points

A = xo < xi < • • • < xn < • • •

to intervals

(xo; xi), (xi; x2),..., (xn-i; xn),... (5)

in which the function f (x) is constant:

f (x) = yn, x € (xn-i; xn), n = 1, 2,...

In this case

yi > y2 > • • • > yn > • • •

and

lim yn = 0.

We are going to prove that for the function f (x) there exists a majorizing function f0(x), i.e.

f (x) < fo(x), x € Q, (6)

which satisfies the condition of the Lemma 1.

We can build the function f0(x) in the form of 2nd order infinity spline passing through the points (xi,yi), (x2,V2), .. . :

so(x), x G (xo; x\] si(x), x G (xi; x2]

fo(x) = < :

Sn(x) x G (xn; ^^ ]

Each of the functions sn(x) is a 2nd order polynomial:

(x) = n J + n x G (x n; xn+\] -

To reach a continuity and smoothness of the function f0(x) over all set Q we submit the functions sn(x) to next conditions:

sn(xn) yn Sn(xn+l) = Vn+l sn(xn) sn-1(xn)

n = 1, 2,...

(7)

At that for so(x) we can take

so (x) = vi.

We are going to show that the system (7) defines unique 2nd order polynomial sn(x) for all xn, xn+i, yn, yn+i and s'n_ 1(xn) = y'n satisfying the conditions:

xn < xn+i, Vn > Vn+i-

Indeed, the system (7) leads to system of linear equations with variable coefficients an, bn and cn:

anxn + bn xn + Cn yn anxn+i + bn xn+i + cn yn+i 2anxn + bn yn

The determinant of basic matrix of this system is

\+1 xn+1

2x

2xn

so the system has a unique solution:

= Ai

an - / NO 5

(x2 - xi)2

= (x2 — xi)2 > 0,

A*

A3

where

Ai

Vn xn 1

Vn+l xn+1 1

vn 1 0

A

xn+1 2xn

(x* — x1)2 ''

Vn 1

Vn+1 1

vn 0

(x* — x1)2 ''

(8)

A3

^n xn yn

)

\+1 xn+1 yn+1

2xn

1

Vn

2

x

x

n

n

x

1

b

c

n

n

2

x

n

2

x

Further, in order to make the spline fo(x) satisfy the condition of majority (6) it is necessary and sufficient to satisfy

sn(x) > Vn+1, x G (xn; xn+i), n = 0,1,...

Last condition will hold if sU(xn+1) ^ 0, i.e. 2anxn+1 + bn < 0. When we substitute in this inequality the solution (8), we will have

, . 2(Vn+1 - Vn) Vn >

In the case

Vn <

xn+1 xn

2(Vn+i ~ Vn) xn+1 xri

we will build the function sn(x) by this way:

( ) I Sn^x) x G (xn;t] Sn(x) — S (2) ( ) (t;

I Sn (x) x G (t; xn +

(9)

where sn (x) and sn (x) are 2nd order polynomials

sU )(x) = «U^x + U x + cui, su )(x) = «U^x + bU x + cu2 ,

which are defined by this conditions:

(i) ( ) = sn (xn ) - yn

sn (xn d

dx

^S(n1)(x)

— Vn

(2) 1 sn (t) — 2(Vn + Vn+i) (2)2

^ (t) — l(Vn + Vn+1)

ds(n1](x) dx

0

Sn (xn+i)— Vn+1 —0

dsn2)(x)

dx

n <t< xn+1

(10)

(see Fig. 1).

The second system in (10) is similar to the system (7), therefore it defines unique function

(2)

(x). (i)

If we substitute the expression for sU1)(x) in (10), we will get (after exclusion t):

an xn + bn __ cn - yn

2an' xn + b() — v'n

4a(1)c(1) - (b«

(b"^ — 2 an\vn + Vn+1)

The last system is not linear but we can get unique solution by elementary simplifying:

' = (vU)2

n 2(vn - Vn+i)

^ U = V' -2«{1)x .

bn = Vn - 2«n xn

(1) = _i (1) 2 _ '

cn — yn + «n xn xnyn

x=x

x=t

x=t

n

yn { K \\ ; ; y = W L / 1 A / y = 2^n + yn+i)

1 y = 42)w L.

i i j

Xn t xn+l

Fig. 1. Building of the function sn (x) in the case (9)

Now we check whether found solution satisfies to inequality in (10). From the first system we find t:

Vn - Vn+l

t — -

b{1)

2a(1

— xn

V'n

At the same time because of yn — Vn+1 > 0 and y'n < ^Vn+1-Va) < o, then t > xn. Further,

xn+1 xn

_ Vn — Vn+l < Vn — Vn+l _ Xn + Xn+l <

t — Xn , < Xn 2(y , „ ) — O < Xn+1 •

V'n

2(yn+i-Vn)

xn + 1 xn

2

Thus, part sn (x) of the spline f0 (x) in the case of (9) is also built. We have that whole spline f0(x) is smooth on O, passes through the points (x1 ,v1 ), (x2,V2), • • • and satisfies (6).

We will show that the function f0(x) satisfies the condition of the Lemma 1. First, f0(x) is differentiable on O. Second, f0(x) is positive because of f0(x) > f (x) > 0. Third, according to building we have f0 (x) < 0, therefore the function f0(x) does not increase. Last, for all x E (xn; xn+1) the following holds: f0(x) ^ Vn, and so

0 < lim fo(x) < lim Vn _ 0;

X^ + tt n^tt

lim fo (x) _ 0.

X^ + tt

Then it follows from Lemma 1 that there exists non-negative function g(x), for which / g(x)dx _ and / f0(x)g(x)dx <

JQ JQ

Finiteness of the first integral and the inequality (6) lead to that the integral

/ f (x)g(x)dx J Q

is finite.

Thus, the function g(x) satisfies the conclusion of the Lemma 2. □

Lemma 3. The conclusion of the Lemma 1 is true for all positive function f (x), for which

lim f(x) — 0.

Proof. Let function f (x) satisfies to the condition of the Lemma 3. According to definition of limit of function, for all e > 0 there exists M G 0 for which for all x > M following holds:

f (x) < e.

1

Now we take a sequence en = —. Some sequence Mn corresponds to it. Let us to consider a

function

1, x G [M1; M2) 1, x G [M2; M3)

fo(x) — < .

n , x G [Mn; Mn+1)

This function satisfies the condition of the Lemma 2. Therefore, there exists non-negative function g(x), for which

/ g(x)dx = and / fo(x)g(x)dx <

JQ JQ

It is obvious that on Q the inequality f (x) < fo(x) holds. Then the integral

/ f(x)g(x)dx

is finite.

□

Lemma 4. Let Q = (a; b) and f (x) is positive on Q function for which at least one of single-sided limit

lim f(x) or lim f(x) is equal 0. Then there exists non-negative on Q function h(x), for which

/ h(x)dx = / f (x)h(x)dx <

JQ JQ

Proof. Let us to consider the case of right-sided limit. We define a variable y Then

x ^ a + 0 is equivalent to y ^

1

x — b is equivalent to y —

ba

f (x) — f[a +-V

(a+ï) '

1

and the function f a +— (from variable y) defined on Q '

y

tion of the Lemma 3. Then there exists the function g(x), for which

r

Hi' JQ' \ y,

; satisfies the condi-

I g(v)dy — and I f (a + M g(y)dy < +<x>.

Jw Jw \ y)

o

1

xa

Because of

g(y)dy = g - 7-^ dx =

Jq' JQ' \x — a) (x — aY

i f(a +1) g(y)dy =i f(x)g ( —1— ) 7—1 Zn dx < JQ' \ yj JQ' \x - a J (x — a)2

we can take for function h(x)

h(x) = g (J t—•

\x — a J (x — a)2

The case of left-sided limit is considered similar: y = ———, and

b — x

h(x) = —g{bh) 1

b — x J (b — x)2

□

Proof of Theorem 1. We are going to prove that in the L2,W1 (Q) space there is a function f which does not belong the L2,W2 (Q) space, i.e.

/ f2(x)wi(x)dx < and / f2(x)w2(x)dx =

QQ

W\ (x)

The function —— satisfies the condition of the Lemma 4. Then there exists non-negative

w2 (x)

on Q function h(x), for which

x)dx = I h(xf

W2(x)

f h(x)dx — +œ, f h(x) —1-(—^dx < +œ.

Jo. Jo r'

h(x )

We define the required function f (x) by this way: f (x) — '

I h(x)

y — 2(x)

—2 (x)

We get:

/ f2(x)—2(x)dx — / h(x)dx — +œ; oo

C C (x)

f2 (x) — 1 (x) dx — h (x)-—— dx < +œ.

JO JO —2 (x)

1

o o —2 (x)

□

Conclusion

Present paper describes the properties of weighted functional Hilbert spaces of L2,w(Q) kind in the context of building probability density function estimate for continuous random variable £. Proposition about convergence of probability density function projective estimate is true in assumption that the probability density belongs to the space L2 w (Q) with appropriate weight function w(x). However, the situations when that information is absent can appear in applications. The Theorem 1 of present paper suggests particularly the method of choice required function w(x). For instance, if according to the received values of random variate £ being investigated we have reasons to assume that for the chosen weight function w2(x) the equality

\\f\\W2 = f f2(x)w2(x)dx = +œ,

Jq,

holds, i.e. f i L2,w2(Q), then we can try to extend the space f f2(x)w2(x)dx to space

Jq

/ f 2(x)wi(x)dx by taking the function wi(x) satisfied condition:

e.g.

{

W1 (x) Иш ——- = 0,

x^a W2(x)

\x — a\a W2(x), x G (a — e; a + e)

wi(x) = ^ , a > 0, e G (0; 1],

w2(x), else

and the point a is chosen from the condition

,-a+S

t-a+o

/ f 2(x)w2(x)dx = for all S > 0.

ao

' a —6

References

[1] A.N.Kolmogorov, S.V.Fomin, Elements of the Theory of Functions and Functional Analysis, Dover Publications, Inc., 1999.

[2] L.V.Kantorovich, G.P.Akilov, Functional Analysis, Pergamon Press, 1982.

[3] H.Triebel, Theory of function spaces, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1983.

[4] J.Garcia-Cuerva, K.S.Kazarian, Spline wavelet bases of weighted Lp spaces, 1 < p < ж, Proceedings of the American Mathematical Society, 123(1995), no. 2, 433-439.

[5] K.S.Kazarian, S.S.Kazaryan, A.San Antolin, Wavelets in weighted norm spaces, arXiv: 1410.4888, 2014.

[6] V.V.Branishti, Introducing the L2,w space for building the projective estimation of probability density function, Vestnik SibGAU., 17, 1(2016), 19-26 (in Russian).

О некоторых свойствах весовых гильбертовых пространств

Владислав В. Браништи

Институт информатики и телекоммуникаций Сибирский государственный аэрокосмический университет Красноярский рабочий, 31, Красноярск, 660014

Россия

В работе рассматриваются весовые гильбертовы пространства L2>w (П) при поло^^ительных и суммируемых на любом ограниченном интервале весовых функциях w(x). Приводится достаточное условие, при котором пространство L2,wi (О) является'расширением пространства L2w2 (О). Описывается применение полученного результата при статистическом оценивании функции плотности вероятности случайной величины.

Ключевые слова: пространства интегрируемых функций, гильбертовы пространства, весовые функциональные пространства, сплайны второго порядка, оценивание функции плотности вероятности.

Î2