Diagonal Distributions Via Orthogonal Expansions and Tests of Independence

Author(s):  
Carles M. Cuadras
Author(s):  
Sheehan Olver ◽  
Yuan Xu

Abstract Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.


2019 ◽  
Vol 3 (Supplement_1) ◽  
Author(s):  
Abigail Gilman ◽  
Michael Bruneau ◽  
Tanja Kral ◽  
Brandy-Joe Milliron ◽  
Patricia Shewokis ◽  
...  

Abstract Objectives We examined the effect of a three year, multi-component, school-based intervention on health behaviors of elementary school students. Methods The multi-component, obesity intervention was provided by community partners to 13 schools over three years. Schools were assigned into three varying Levels of Intervention: “Core” Schools (n = 4) received weekly interventions, “Level 1” Schools (n = 5) received monthly interventions, and Control Schools (n = 5) received no intervention. Participants completed self-reported questionnaires assessing several health behaviors, including fruit and vegetable consumption, and physical activity levels. Questionnaires were administered in the Fall and Spring of each year of the intervention. For this study, responses were analyzed from Year 3 and were compared to national recommendations. Descriptive statistics and chi-square tests of independence examined Level of Intervention and questionnaire responses. Results No significant chi-square tests of independence were identified for the Level of Intervention and quantity of fruit or vegetable consumption. A significant (P = 0.003) chi-square test of independence assessing physical activity was identified in the Fall of Year 3. Descriptive analyses indicated that higher proportions of students in the Core and Level 1 Schools met national fruit, vegetable, and physical activity recommendations compared to students in Control Schools. Conclusions The higher frequency of intervention provided to students in the Core Schools did not influence the dietary behaviors of participants. However, receiving any intervention appeared to affect health behaviors of children receiving the intervention compared to the Control Schools. Funding Sources The Independence Blue Cross Foundation.


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