STRICT POSITIVE DEFINITENESS ON SPHERES VIA DISK POLYNOMIALS

We characterize complex strictly positive definite functions on 
spheres in two cases, the unit sphere of ℂ q , q ≥ 3 , and the unit sphere of the complex l 2 . The results depend upon 
the Fourier-like expansion of the functions in terms of disk 
polynomials and, among other things, they enlarge the classes of 
strictly positive definite functions on real spheres studied in 
many recent papers.


1.
Introduction.This paper is concerned with positive definite kernels that are useful to perform interpolation on spheres.Its main purpose is to study strictly positive definite kernels on the unit sphere Ω 2q of C q , q ≥ 2, and the unit sphere Ω ∞ of 2 .Specifically, we will provide an elementary description of strict positive definiteness of kernels which are inner-product dependent, that is, of the form (z, w) ∈ Ω 2q × Ω 2q → f z, w , (1.1) for some continuous complex function f (the shape function) defined at least on the disk B 2 := {z ∈ C : |z| ≤ 1}.Hereafter, •, • will denote the usual inner product in C q or 2 .
A shape function whose associated kernel is positive definite on Ω 2q will be termed a positive definite function on Ω 2q .Thus, a positive definite function f on Ω 2q is a continuous function such that for N ≥ 1, {ξ 1 ,...,ξ N } ⊂ Ω 2q , and {c 1 ,...,c N } ⊂ C. The reader can refer to [1] for the basic facts about positive definite and related kernels.If N has been fixed, we say that the positive definite function f is strictly positive definite of order N on Ω 2q if the inequality above is strict when the points ξ µ are pairwise distinct and N µ=1 |c µ | > 0. Finally, f is strictly positive definite on Ω 2q if it is strictly positive definite of order N on Ω 2q , N = 1, 2,....If we are given distinct points ξ 1 ,...,ξ N on Ω 2q , complex numbers λ 1 ,...,λ N , and a strictly positive definite function f of order at least N on Ω 2q , then there is a unique function of the form ξ ∈ Ω 2q → N ν=1 c ν f ξ, ξ ν (1.3) that interpolates the data {(ξ µ ,λ µ ) : µ = 1,...,N}.This interpolation procedure is a complex version of what is often called spherical radial basis function interpolation (see [4,6,8,12]).In the latter, the ξ µ belong to a sphere in some R m and the function f used in the generation of solutions is usually real valued.The method is well established and has many applications in the handling of problems of global nature that emerge in several areas such as meteorology, oceanography, satellite-related sciences, and so forth, where the unit sphere in R 3 is the standard model.
A strictly positive definite function on Ω 2q is also strictly positive definite on every real sphere embedded in it, in the sense considered in the references quoted above.The results in this paper will then provide a concise way of constructing complex strictly positive definite functions on the real spheres.As a matter of fact, since the definitions of positive definiteness used here encompass those used in the real setting, our results will amplify the classes of positive definite functions that can be used to perform interpolation on the real spheres.
The outline of the paper is as follows.In Section 2, we collect basic results about positive definite functions on spheres and disk polynomials.This is crucial in the paper, because the analysis of strict positive definiteness is based on the expansion of positive definite functions on Ω 2q in terms of certain disk polynomials.Section 3 contains two technical results, one of them being an alternative to the concept of strict positive definiteness on Ω 2q .Section 4 contains the main part of the paper, including characterizations of strict positive definiteness on Ω 2q , q ≥ 3.

Prerequisites.
The strict positive definiteness of a positive definite function on Ω 2q , q < ∞, can be detected by looking at the series expansion of the function in terms of certain disk polynomials.This section contains the basic facts about them along with some basic properties needed in the sequel.
Positive definite functions on Ω 2q are representable in the form in which a m,n (f ) ≥ 0 for all m and n and (m,n stands for the disk polynomial of degree m +n associated to the integer q −2 when q is finite while R ∞ m,n (z) = z m z n (see [7,11]).The disk polynomial of degree m + n in x and y associated with a nonnegative real number α is the polynomial R α m,n given by where R (α,|m−n|) m∧n is the usual Jacobi polynomial of degree m ∧ n := min{m, n} associated with the numbers α and |m − n|.The normalization for the Jacobi polynomials adopted here is R (α,|m−n|) m∧n (1) = 1.The expansion (2.1) is then possible due to the following two facts (see [2,9]): R q−2 m,n is a zonal surface harmonic of degree m + n in 2q real variables in the classical sense and the set {R α m,n : 0 ≤ m, n < ∞} is a complete orthogonal set in L 2 (B 2 ,dw α ), where dw α is the measure on B 2 given by 3) The following properties of disk polynomials are straightforward [9].
Lemma 2.1.Let m and n be nonnegative integers and α a nonnegative real.The following properties hold: m,n is an even function if m − n is even, and an odd function otherwise.
Next, we list the less elementary properties of disk polynomials to be used in this work.The first one is the so-called addition formula (see [9,14]), which is quoted here in a version adapted to our purposes.Lemma 2.2.If m and n are nonnegative integers, q is an integer at least 3, and the constants b k,l m,n,q−2 are all positive.
Before stating the next lemma, we need some additional notation.We need to deal with elements from the space Π q composed of complex polynomials in the independent complex variables z ∈ C q and z.A typical element of Π q is of the form where standard multi-index notation is in force.The subset of Π q , composed of polynomials with the property that becomes a subspace.The elements of this subspace which are in the kernel of the complex Laplacian also form a subspace of Π q .The set of restrictions of elements of this subspace to Ω 2q will be denoted by Ᏼ q m,n .It is also a vector space over C and its dimension will be written as N(q; m, n).
Lemma 2.3.Let q be an integer at least 2 and {Y where ω 2q denotes the surface area of Ω 2q .

Basic technical results.
In this section, we present a crucial formulation of strict positive definiteness (SPD) on Ω 2q to be used in the proofs of the main results of the paper.Before that, we formulate two elementary technical lemmas.Lemma 3.1.Let q be an integer at least 2 and ζ an element of Ω 2q .Then, every element ξ of Ω 2q is uniquely representable in the form Before stating our first result, we introduce a new terminology.It comes from the fact that the strict positive definiteness of a positive definite function f on Ω 2q does not depend on the actual values of the coefficients a m,n (f ) (see [10]).We say that a subset K of Z 2 + induces SPD of order N on Ω 2q if every positive definite function f for which is strictly positive definite of order N on Ω 2q .Theorem 3.3 below is the major result in this section and it yields different formulations for the concept of SPD on Ω 2q .The results in Section 4 will only require the fact that condition (i) implies condition (iv).The other implications in Theorem 3.3 are of independent interest at this moment.

Main results.
In this section, we use Theorem 3.3 to obtain an elementary characterization of SPD of all orders on Ω 2q , q ≥ 3. We begin with a necessary condition for SPD proved in [10].Lemma 4.1.Let q be an element of {2, 3,...}∪{∞}, K a subset of Z 2 + , and N a positive integer.If K induces SPD of order N on Ω 2q then {m − n : (m, n) ∈ K} intersects the sets NZ + j, j = 0, 1,...,N − 1.
Theorem 4.2.Let q be an integer at least 3 and K a subset of Z 2 + .Then K induces SPD of all orders on Ω 2q if and only if the set {m − n : (m, n) ∈ K} contains infinitely many even and infinitely many odd integers.
we may use Theorem 3.3 to conclude that We now split the proof into two cases.If T j (ξ µ ), T j (ξ j ) ≠ −1, µ ≠ j, we use our hypothesis to select a sequence and the fact that Using this information with η = ξ j , we obtain If T j (ξ j ), T j (ξ s ) = −1 for some s then θ s = π/2 and ξ s = (0, −1, 0,...,0).Repeating the procedure used in the previous case and recalling that R It is now clear that c j = 0.
We conclude the paper by proving a version of Theorem 4.2 for the case q = ∞.A result of this sort was the main goal intended, but not reached, in [15].The following lemma is the only additional result needed.Lemma 4.3.If q is a positive integer at least 2 and m and n are nonnegative integers, then there are positive constants c j q,m,n , j = 0,...,m ∧ n, such that Proof.One half follows from Lemma 4.1.For the other, let K be a subset of Z 2 + such that {m − n : (m, n) ∈ K} contains infinitely many even and infinitely many odd integers.We will show that K induces SPD of order N ≥ 3 on Ω ∞ .Let ξ 1 ,...,ξ N be distinct points on Ω ∞ and f a positive definite function on Without loss of generality, we can assume that {ξ 1 ,...,ξ N } ⊂ Ω 2N .By Lemma 4.3, where L := (m,n)∈K {(m − j, n − j) : j = 0,...,m ∧ n} and g is a positive definite function on L contains infinitely many even and infinitely many odd integers.Thus, Theorem 4.2 implies that L induces SPD of all orders on Ω 2N .This guarantees that (g( ξ µ ,ξ ν )) is positive definite and, consequently, so is (f ( ξ µ ,ξ ν )).Therefore, K induces SPD of order N on Ω ∞ , completing the proof.

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 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 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 ) for (m, n) ∈ K, ψ ∈ [0, 2π), and η ∈ Ω 2q−2 .Next, we use Lemma 3.2 to see that, for each (m, n) ∈ K, the functions ψ → e −i(m−n−k+l)ψ , k= 0,...,m, l = 0,...,n, k+ l = p (3.13) form a linearly independent set over [0, 2π).It follows that b

. 11 )Theorem 4 . 4 .
Proof.A proof, including exact values of the constants c j q,m,n , can be found in [2, page 28].A subset K of Z 2 + induces SPD of all orders on Ω ∞ if and only if the set {m − n : (m, n) ∈ K} contains infinitely many even and infinitely many odd integers.
.1)in which ξ is an element of Ω 2q in the hyperplane of C q orthogonal to ζ.The coefficient te iϕ is precisely ξ, ζ .

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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