Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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REAL WEIGHT FUNCTIONS FOR THE CIRCLE POLYNOMIALS BY THE REGULARIZATION

  • Lee, J.K. (Department of Mathematics, SunMoon University) ;
  • Lee, C.H. (Department of Mathematics, Hoseo University) ;
  • Han, D.H. (Department of Mathematics, SunMoon University)
  • Published : 2010.01.30
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Abstract

We consider the differential equation $$(x^2\;-\;1)u_{xx}\;+\;2xyu_{xy}\;+\;(y^2\;-\;1)u_{yy}\;+\;gxu_x\;+\;gyu_y\;=\;\lambda_nu,\;(*)$$ where $\lambda_n\;=\;n(n\;+\;9\;-\;1)$. We show that the differential equation (*) has a polynomial set as solutions if $g\;{\neq}\;-1$, -3, -5, $\cdots$. Also, we construct an orthogonalizing distributional weight for g < 1 and $g\;{\neq}\;1$, 0, -1, $\cdots$ by regularizing a one-dimensional integral with a singularity on the endpoint of the interval.

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References

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