Examples of Orthogonal Polynomials in Several Variables

Dunkl, Charles F.; Xu, Yuan

doi:10.1017/cbo9780511565717.003

Cited by 97 publications

(173 citation statements)

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“…We collect a few preliminary results from the refrences [13] or [18]. Let s = (s n ) be a Stieltjes moment sequence and µ ∈ S representing measure of s. We assume in this section that the support of µ has N(µ) ≥ q elements.…”

Section: Preliminary Resultsmentioning

confidence: 99%

The Polyanalytic Reproducing Kernels

Hachadi¹,

Youssfi

2019

Complex Anal. Oper. Theory

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Let ν be a rotation invariant Borel probability measure on the complex plane having moments of all orders. Given a positive integer q, it is proved that the space of ν-square integrable q-analytic functions is the closure of q-analytic polynomials, and in particular it is a Hilbert space. We establish a general formula for the corresponding polyanalytic reproducing kernel. New examples are given and all known examples, including those of the analytic case are covered. In particular, weighted Bergman and Fock type spaces of polyanalytic functions are introduced. Our results have a higher dimensional generalization for measure on C p which are in rotation invariant with respect to each coordinate.

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Section: Preliminary Resultsmentioning

confidence: 99%

The Polyanalytic Reproducing Kernels

Hachadi¹,

Youssfi

2019

Complex Anal. Oper. Theory

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“…The way around this problem is to find a pair of bases with the property that, for each basis, each axis in it is orthogonal to all but one of the axes in the other basis. A pair of bases that have this property are said to be "biorthogonal" to one another (Dunkl and Xu, 2001), and it is relatively easy to construct a pair of biorthogonal bases for our phenotype space (see Supplementary Material). Figure 18 shows a simple example of a pair of bases that are biorthogonal to one another.…”

Section: Joint Evolution Of Multiple Traitsmentioning

confidence: 99%

“…Multivariate orthogonal polynomials pose a substantial challenge that does not appear in the univariate case. As shown above, finding a set of orthogonal polynomials for a multivariate distribution is relatively easy, the problem is that there are many such sets, and which one we get depends on how we choose to order our initial variables (Dunkl and Xu, 2001). Figure 17 illustrates this: The two sets of colored lines on the left represent two different bases that are orthogonal with respect to the two dimensional distribution of points shown.…”

Section: Biorthogonal Basesmentioning

confidence: 99%

Evolution with complex selection and transmission

Rice¹

2019

Preprint

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Inheritance is the key factor making biological evolution possible. Despite this central role, transmission is often bundled into the simplifying assumptions of evolutionary models, making it difficult to see how changes in the patterns of transmission influence evolutionary dynamics. We present a mathematical formalism for studying phenotypic evolution, under any selection regime and with any transmission rules, that clearly delineates the roles played by transmission, selection, and interactions between the two.To illustrate the approach, we derive models in which heritability and and fitness are influenced by the same environmental factors -producing a covariation between selection and transmission. By itself, variation in heritability does not influence directional evolution. However, we show that any covariation between heritability and selection can have a substantial effect on trait evolution. Moderate differences in heritability between environments can lead to organisms adapting much more to environments with higher heritability, and can pull a population off of an "adaptive peak". When habitat preference is allowed to evolve as well, variation in heritability between environments can lead to organisms exclusively using the environment in which heritability is highest. This effect is most pronounced when initial habitat selection is weak.

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“…A general approach to construct orthogonal polynomials of several variables with an arbitrary probability distribution for the variables has been developed. 15 However, the direct application of multivariate orthogonal polynomials to construct HDMR component functions has two drawbacks: (1) to construct the basis functions used for a high order HDMR component function, the degree of the polynomial basis functions used for its nested lower order HDMR component functions must be known in advance. Improper setting of the degree may cause a large error for the HDMR model; (2) for large n the number of polynomial basis functions will in turn also be large.…”

Section: Principles Of Hdmrmentioning

confidence: 99%

Experimental Design of Formulations Utilizing High Dimensional Model Representation

Bastian

Welsh

et al. 2015

J. Phys. Chem. A

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Many applications involve formulations or mixtures where large numbers of components are possible to choose from, but a final composition with only a few components is sought. Finding suitable binary or ternary mixtures from all the permissible components often relies on simplex-lattice sampling in traditional design of experiments (DoE), which requires performing a large number of experiments even for just tens of permissible components. The effect rises very rapidly with increasing numbers of components and can readily become impractical. This paper proposes constructing a single model for a mixture containing all permissible components from just a modest number of experiments. Yet the model is capable of satisfactorily predicting the performance for full as well as all possible binary and ternary component mixtures. To achieve this goal, we utilize biased random sampling combined with high dimensional model representation (HDMR) to replace DoE simplex-lattice design. Compared with DoE, the required number of experiments is significantly reduced, especially when the number of permissible components is large. This study is illustrated with a solubility model for solvent mixture screening.

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Examples of Orthogonal Polynomials in Several Variables

Cited by 97 publications

References 0 publications

The Polyanalytic Reproducing Kernels

The Polyanalytic Reproducing Kernels

Evolution with complex selection and transmission

Experimental Design of Formulations Utilizing High Dimensional Model Representation

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