Intelligent Computational Methods for Financial Engineering

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SOME EXACT INEQUALITIES OF HARDY-LITTLEWOOD-POLYA TYPE FOR PERIODIC FUNCTIONS

LAITH EMIL AZAR Received 28 September 2003

We investigate the following problem: for a givenA≥0, find the infimum of the set ofB≥0 such that the inequalityx^(k)22≤Ax^{(r )2}2+Bx²2, fork,r∈N∪{0}, 0≤k < r, holds for all sufficiently smooth functions.

2000 Mathematics Subject Classification: 41A17, 42A05.

1. Introduction. LetG=RorG=T=[0,2π). ByL2(G), we will denote the spaces of all measurable functionsx:G→Rsuch that

x2= xL2(G):=

G

x(t)²dt 1/2

<∞. (1.1)

Denote byL^r2(G) (r ∈N)the space of all functionsx such that x^(r−1) are locally absolutely continuous andx^{(r )}∈L2(G), and setL^r_2,2(G)=L2(G)∩L^r2(G)(in the case G=T, we mean that spacesL2(G)andL^r2(G)consist of 2π-periodic functions). Note thatL^r2(G)⊂L2(G)ifG=T.

It is well known that the exact inequality of Hardy [3]

x^(k)2₂≤ x^{2(1−k/r )}2 x^{(r )2(k/r )}₂ , k∈N,0< k < r , (1.2) holds for every functionx∈L^r2,2(R).

For anyA >0 and anyx∈L^r2,2(R), from inequality (1.2), we get x^(k)2₂≤

k Ar

_k/(r−k) x²2

(r−k)/r Ar

k x^{(r )2}₂k/r

. (1.3)

Using Young’s inequality

ab≤a^p p +b^p

p, 1 p+ 1

p=1, 1≤p <∞, a,b >0, (1.4) withp=r /(r−k)andp=r /k, we get, for anyA >0 and anyx∈L^r2,2(R), the following inequality:

x^(k)2₂≤Ax^{(r )2}₂+r−k r

k Ar

k/(r−k)

x²2. (1.5)

This inequality is the best possible in the next sense: for a givenA >0, the infimum of constantsBsuch that the inequality

x^(k)2₂≤Ax^{(r )2}₂+Bx²2 (1.6)

holds for all functionsx∈L^r2,2(R)is equal to r−k

r k

Ar _k/(r−k)

. (1.7)

As is well known, inequality (1.2) (and consequently (1.5)) holds true for any func- tionx∈L^r2,2(T). However, the constant (1.7) is not the best possible in general (for a given constantA). Therefore, the main problem which we will study in this paper is the following.

For a givenA≥0, find the infimum of constantsBsuch that inequality (1.6) holds for all functionsx∈L^r_2,2(T).

We will denote this infimum byΨ(T;r ,k;A). We will investigate also the analogous problem in the presence of some restrictions on the spectrum of functionsx∈L^r2,2(T). Note that Babenko and Rassias [1] investigated the problem on exact inequalities for functionsx∈L^r2,2(T). They have found, for a givenA≥0, the infimum of constantsB such that the inequality

x^(k)2₂≤Ax²2+Bx^{(r )2}₂ (1.8) holds for all functionsx∈L^r_2,2(T).

For more information related to this subject, see, for example, [2,4,5,6].

2. Main results

Theorem2.1. Letk,r∈N,k < r. Then for anyA≥0and anyx∈L^r_2,2(T), x^(k)2₂≤Ax^{(r )2}₂+

v0^2k−Av0^2r

x²2=Ax^{(r )2}₂+ϕ A,v0

x²2 (2.1)

holds ifv0is such thatη(v0+1)≤A≤η(v0), where η(v)=v^2k−(v−1)^2k

v^2r−(v−1)^2r. (2.2)

GivenA, the constantϕ(A,v0)in (2.1) is the best possible; that is, Ψ(T;r ,k;A)=

v0^2k−Av0^2r

, (2.3)

wherev0is such thatη(v0+1)≤A≤η(v0). Proof. Let

ev(t):= 1

2πe^ivt, v∈Z, t∈R, cv(x) =

_2π

0 x(t)ev(t)dt

(2.4)

...

be Fourier coefficients of a functionx, and let

v∈Z

cv(x)ev(t) (2.5)

be the Fourier series of a functionx.

For anyx∈L^r_2,2(T), 0< k < r, and anyA≥0, using Parseval’s equality, we get x^(k)2₂=

v∈Zv≠0

cv(x)²v^2k

=A

v∈Zv≠0

cv(x)²v^2r+

v∈Zv≠0

cv(x)²v^2r v^2k

v^2r−A

=Ax^{(r )2}₂+

v∈Zv≠0

cv(x)²

v^2k−Av^2r

≤Ax^{(r )2}₂+max

v∈N

v^2k−Av^2r

v∈Zv≠0

cv(x)²

=Ax^{(r )2}₂+max

v∈N

v^2k−Av^2r x²2.

(2.6)

Set

ϕ(A,v):=v^2k−Av^2r; (2.7) then the last inequality can be written in the form

x^(k)2₂≤Ax^{(r )2}₂+max

v∈Nϕ(A,v)x²2. (2.8)

Our goal now is to find for a givenA≥0 the value of

maxv∈Nϕ(A,v). (2.9)

We consider the difference

δv=ϕ(A,v)−ϕ(A,v−1)

=v^2k−Av^2r−(v−1)^2k+A(v−1)^2r

=A

(v−1)^2r−v^2r

−

(v−1)^2k−v^2k

=

(v)^2r−(v−1)^2r

v^2k−(v−1)^2k v^2r−(v−1)^2r−A

.

(2.10)

Set, forv∈N,

η(v):=v^2k−(v−1)^2k

v^2r−(v−1)^2r; (2.11)

then the last equality can be written in the form δv=

(v)^2r−(v−1)^2r

η(v)−A

. (2.12)

It is not difficult to see that

sgnδv=sgn

η(v)−A

. (2.13)

We now study the functionη(v).

Note thatη(1)=1,η(v)→0 asv→ ∞(sincek < r), and, forv≥1,

η(v) > η(v+1). (2.14)

Indeed, using Cauchy’s theorem, η(v)=k

r θ^2kv

θ^2rv , v−1< θv< v. (2.15) Thus, inequality (2.14) is equivalent to the inequality

k r

θ^2kv

θ^2rv >k r

θ_v+1^2k θ_v+^2r1

(2.16) or

θv

θv+1

2r−2k

<1. (2.17)

The last inequality is true sinceθv< θv+1and 2r−2k >0.

If, for a givenA≥0, the valuev0is such thatη(v0+1)≤A≤η(v0), then forv≤v0, taking into account equality (2.13), we obtain thatδv≥0, and consequently,

ϕ(A,1)≤ϕ(A,2)≤ ··· ≤ϕ A,v0

. (2.18)

In the casev > v0, we getδv≤0 and then ϕ

A,v0

≥ϕ

A,v0+1

≥ ···. (2.19) Therefore,

maxv∈Nϕ(A,v)=max

v∈N

v^2k−Av^2r

=ϕ A,v⁰

(2.20) ifη(v0+1)≤A≤η(v0). Thus inequality (2.1) is proved.

We now show the evidence of equality (2.3). Letx(t)=cosv0t. Then the inequality becomes an equality since

x^(k)2₂=πv0^2k, x²2=π, x^{(r )2}₂=πv0^2r. (2.21)

The functionΨ(T;r ,k;A)defined by (2.3) is continuous, linear on any interval [η(v+ 1),η(v)], and for anyv≥1,

Ψ

T;r ,k;η(v+1)

=v^2k(v+1)^2r−v^2r(v+1)^2k

(v+1)^2r−v^2r . (2.22)

...

Claim that

v0^2k−Av0^2r<r−k r

k Ar

k/(r−k)

. (2.23)

To do this, we will consider the function f (A)=r−k

r k

Ar k/(r−k)

−v^2k+Av^2r. (2.24)

Differentiating the functionf, we get f(A)=v^2r−

k r

r /(r−k)1 A

r /(r−k)

(2.25) and the conditionf(A)=0 implies

A0=k

rv^2k−2r. (2.26)

Now we havef (A0)=0 and our statement is proved.

LetΠ2n+1be the set of trigonometric polynomials of order less than or equal ton. Then in view of the Bernstein-type inequality, we have, for anyτ∈Π2n+1and anyk∈N,

τ^(k)2₂≤n^2kτ²2. (2.27)

Therefore, forx=τ, inequality (1.6) holds withA=0 andB=n^2k. Let nowA >0. By repeating (with obvious modifications) the proof ofTheorem 2.1, we obtain that for any k,r∈N,k < r, and anyτ∈Π2n+1, the following holds:

τ^(k)2₂≤Aτ^{(r )2}₂+Bτ²2=Aτ^{(r )2}₂+max

v∈Nv≤n

ϕ(A,v)τ²2. (2.28)

We now compute the value

maxv∈N v≤n

ϕ(A,v). (2.29)

Letη(v0+1)≤A≤η(v0), wherev0≤n. Then maxv∈N

v≤n

ϕ(A,v)=ϕ A,v0

=max

v∈Nϕ(A,v). (2.30)

Ifη(v0+1)≤A≤η(v0), wherev0≥n+1, we get, taking into account the relations ϕ(A,1)≤ϕ(A,2)≤ ··· ≤ϕ(A,n)≤ ··· ≤ϕ

A,v0

, (2.31)

that

maxv∈N v≤n

ϕ(A,v)=ϕ(A,n)=n^2k−An^2r (2.32)

ifA≤η(n). Therefore, we have proved the following theorem.

Theorem2.2. For anyk,n,r∈N,k < r, anyτ∈Π2n+1, and anyA≥0,

τ^(k)2₂≤Aτ^{(r )2}₂+Bτ²2, (2.33) where

B=ϕ A,v0

(2.34) ifη(v0+1)≤A≤η(v0),v0≤n, and

B=ϕ(A,n) (2.35)

ifA≤η(n). Inequality (2.33) is the best possible for anyA≥0.

Consider the set of functionsx∈L^r_2,2(T)such thatcv(x)=0 for|v| ≤n−1 (we will denote this set of functions byL^r2,2(T;n)). The following inequality is well known for functionsx∈L^r2,2(T;n):

x²₂≤ 1

n^2rx^{(r )2}₂. (2.36)

Thus, for anyk < r,

x^(k)2₂≤ 1

n^2r−2kx^{(r )2}₂. (2.37)

Then inequality (1.6) for functionsx∈L^r2,2(T;n)holds withB=0 andA≥1/n^2r−2k. By repeating (with obvious modifications) the proof ofTheorem 2.1, we obtain that for anyk,r∈N,k < r, anyx∈L^r_2,2(T;n), and any 0≤A≤1/n^2r−2k,

x^(k)2₂≤Ax^{(r )2}₂+max

v∈Nv≥n

ϕ(A,v)x²2. (2.38)

We need to find the value of

maxv∈N v≥n

ϕ(A,v). (2.39)

Note that

η(n)=n^2k−(n−1)^2k n^2r−(n−1)^2r ≤ n^2k

n^2r. (2.40)

To show this, assume that

η(n) >n^2k

n^2r, (2.41)

then we get

n n−1

2r

<

n n−1

2k

(2.42) which is impossible sincen/(n−1) >1 andk < r.

...

First letη(v0+1)≤A≤η(v0)wherev0≤n. Then

ϕ(A,n)≥ϕ(A,n+1)≥ ··· (2.43)

and therefore

maxv∈N v≥n

ϕ(A,v)=ϕ(A,n) (2.44)

ifη(v0+1)≤A≤n^2k−2r.

Let nowη(v0+1)≤A≤η(v0)wherev0≥n+1. In this case, we get maxv∈N

v≥n

ϕ(A,v)=max

v∈Nϕ(A,v)=ϕ A,v0

. (2.45)

Thus we have proved the following theorem.

Theorem2.3. For anyk,n,r∈N,k < r, anyx∈L^r_2,2(T;n), and any0≤A≤n^2k−2r, inequality (1.6) holds where B=ϕ(A,n) ifη(v0+1)≤A≤n^2k−2r,v0≤n, and B= ϕ

A,v0

ifη(v0+1)≤A≤η(v0),v0≥n+1. Inequality (1.6) is the best possible for any 0≤A≤n^2k−2r.

References

[1] V. F. Babenko and T. M. Rassias,On exact inequalities of Hardy-Littlewood-Polya type, J. Math.

Anal. Appl.245(2000), no. 2, 570–593.

[2] B. D. Bojanov and A. K. Varma,On a polynomial inequality of Kolmogoroff’s type, Proc. Amer.

Math. Soc.124(1996), no. 2, 491–496.

[3] G. H. Hardy, J. E. Littlewood, and G. Polya,Inequalities, Cambridge University Press, Cam- bridge, 1934.

[4] N. P. Korne˘ıchuk,Exact Constants in Approximation Theory, Encyclopedia of Mathematics and Its Applications, vol. 38, Cambridge University Press, Cambridge, 1991.

[5] S. Z. Rafal’son,An inequality between norms of a function and its derivative in integral metrics, Mat. Zametki33(1983), no. 1, 77–82.

[6] A. K. Varma,A new characterization of Hermite polynomials, Acta Math. Hungar.49(1987), no. 1-2, 169–172.

Laith Emil Azar: Department of Mathematics, Al Al-Bayt University, Mafraq 25113, Jordan E-mail address:[email protected]

Journal of Applied Mathematics and Decision Sciences

Special Issue on

Intelligent Computational Methods for Financial Engineering

Call for Papers

As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.

However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used e

ectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)

This special issue will include (but not be limited to) the following topics:

• Computational methods

: artificial intelligence, neu- ral networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learn- ing, multiagent learning

• Application fields

: asset valuation and prediction, as- set allocation and portfolio selection, bankruptcy pre- diction, fraud detection, credit risk management

• Implementation aspects

: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, imple- mentation

Authors should follow the Journal of Applied Mathemat- ics and Decision Sciences manuscript format described at the journal site

Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Track- ing System at

lowing timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Lean Yu,

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;

Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong;

[email protected]

Shouyang Wang,

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; [email protected]

K. K. Lai,

Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; [email protected]

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