PYTHAGOREAN IDENTITY FOR POLYHARMONIC POLYNOMIALS

Polyharmonic polynomials in n variables are shown to satisfy a Pythagorean identity on the unit hypersphere. Application is made to establish the convergence of series of polyharmonic polynomials.


Introduction. Let L k
n denote the vector space of real homogeneous polynomial solutions of degree k of Laplace's equation where Such polynomials are called spherical harmonics.As shown in [9, pages 140-141], Suppose that {y k j (x)} d k n j=1 is an orthonormal basis for L k n , where orthonormality is with respect to the inner product f ,g = 1 f (x)g(x)dx (1.4) on the unit sphere 1 : It is well known (cf.[9, page 144]) that for all s ∈ 1 , where ω n is the surface area of the unit sphere 1 in R n .We call (1.5) the Pythagorean identity for spherical harmonics, since it generalizes the Pythagorean theorem Solutions of partial differential equation where ∆ is the Laplacian (1.2) and m is a positive integer, are called polyharmonic functions.In the case m = 2, such functions are called biharmonic and are used to model the bending of thin plates (for a brief history of this application, see [7, pages 416 and 432-443]).
We show here that homogeneous polyharmonic polynomials satisfy a Pythagorean identity on 1 and use this identity to establish the convergence of polyharmonic polynomial series.

Pythagorean identity. Let J k
n denote the vector space of real homogeneous polynomial solutions of the partial differential equation (1.7).Since ∆ m is a homogeneous differential operator of order 2m, using a standard argument (cf.[5, Theorem 1]) we find that In the vector space J k n , we introduce the Calderón inner product [1] (p, q) = p ∂ ∂x q(x), ( where j=1 is an orthonormal basis for the vector space J k n of homogeneous polyharmonic polynomials of degree k, where orthonormality is with respect to the inner product (2.2).Then for all s = (s 1 ,s 2 ,...,s n ) ∈ 1 , the unit where γ k n is a constant depending only on n and k.
Proof.A modification in the argument used for spherical harmonics suffices: fix a point y ∈ R n and consider the linear functional L : J k n → R defined by for all p ∈ J k n (i.e., all finite-dimensional inner product spaces are self-dual).Further, since {Q But, by the defining property of Z y , (2.9) Since the choice of y ∈ R n was arbitrary, Z y (x) is a function of the two variables x, y ∈ R n ; thus, we write (2.10) The Calderón inner product (2.2) is invariant with respect to rotations; that is, if where q(x) = p(O −1 x).Since rotations are invariant transformations for the Laplacian, it follows that q(x) ∈ J k n .Thus, by the defining property of Z(x, y), q(x),Z(x,Oy) = q(Oy). (2.12) But q(Oy) = p(O −1 Oy) = p(y).Thus, we have shown that p(x), Z(Ox, Oy) = p(y). (2.13) From the uniqueness of the representation of linear functionals, it follows that for all x, y ∈ R n .In particular, for every rotation O. Since every point on the unit sphere 1 is the image under rotation for some fixed point on 1 , the equality (2.15) implies that a constant, for all s ∈ 1 .

Polyharmonic polynomial series.
Pythagorean identities have been used to establish the convergence of series of spherical harmonics [4], as well as series of orthonormal homogeneous polynomials in several real variables in general [3].We obtain here convergence for series of polyharmonic polynomials.
and γ k n is the Pythagorean constant appearing in (2.4).

Proof. Since each of the polynomials
by the Cauchy-Schwarz inequality

.4)
Appealing now to the Pythagorean identity (2.4), we find that from which the desired result is immediate.
Let H k n denote the vector space of homogeneous polynomials of degree k in R n .Since every orthonormal basis of J k n be extended to an orthonormal basis of H k n , it follows from [2, Theorem 3] that Thus, and appealing to the result of Theorem 3.1 we find that the polyharmonic polynomial series (3.1) converges absolutely and uniformly on compact subsets of the open ball |x| < ρ.We predict that the evaluation of the Pythagorean constant γ k n will show that such convergence actually obtains within a somewhat larger ball.
In [11], it was shown that, in the space of homogeneous harmonic polynomials L k n , the Calderón inner product (2.2) is a constant multiple of the inner product (1.4).That is, for all p, q ∈ L k n , where c k n is a constant depending only on n and k.Thus, the Pythagorean identity for spherical harmonics (1.5) is a special case (m = 1) of the result of Theorem 2.1.
The Pythagorean identity for spherical harmonics is also a special case of the addition formula for spherical harmonics [9, page 149] and [8, page 268].This leads us to conjecture that the homogeneous polyharmonic polynomials satisfy a similar addition formula, from which Theorem 2.1 might follow as an immediate consequence.Such a development could include a significant generalization of the ultraspherical polynomials [6,10].

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need 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 distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (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, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Theorem 3 . 1 . 1 )
Suppose that {Q j k (x)} b k n j=1 are sets of orthonormal polyharmonic polynomials in R n of degree k, k = 0, 1, 2,.... Then the series converges absolutely and uniformly on compact subsets of the open ball |x| =

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation