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On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets

Year 2018, Volume: 1 Issue: 2, 156 - 162, 24.12.2018
https://doi.org/10.33434/cams.439977

Abstract

The regular, real-valued solutions of the second-order elliptic partial differential equation\beq \frac{\prt^2F}{\prt x^2} + \frac{\prt^2F}{\prt y^2} + \frac{2\alpha+1}{x} \frac{\prt F}{\prt y} + \frac{2\beta+1}{y} \frac{\prt F}{\prt x} =0, \alpha,\beta>\frac{-1}{2}\eeq are known as generalized bi-axially symmetric potentials (GBSP's). McCoy \cite{17} has showed that the rate at which approximation error $E^{\frac{p}{2n}}_{2n}(F;D),(p\ge 2,D$ is parabolic-convex set) tends to zero depends on the order of $GBSP$ F and obtained a formula for finite order. If $GBSP$ F is an entire function of infinite order then above formula fails to give satisfactory information about the rate of decrease of $E^{\frac{p}{2n}}_{2n}(F;D)$. The purpose of the present work is to refine above result by using the concept of index-q. Also, the formula corresponding to $q$-order does not always hold for lower $q$-order. Therefore we have proved a result for lower $q$-order also, which have not been studied so far.

References

  • [1] P. A. McCoy, Approximation of generalized biaxially symmetric potentials on certain domains, J. Math. Anal. Appl., 82 (1981), 463-469.
  • [2] R. P. Gilbert, Integral operator methods in biaxially symmetric potential theory, Contrib. Diff. Eqns., 2 (1963), 441-456.
  • [3] G. S. Srivastava, On the growth and polynomial approximation of generalized biaxisymmetric potentials, Soochow J. Math., 23(4) (1997), 345-358.
  • [4] P. A. McCoy, Best Lp-approximation of generalized biaxisymmetric potentials, Proc. Amer. Math. Soc., 79 (1980), 435-440.
  • [5] D. Kumar, A. Basu, Growth and approximation of generalized biaxially symmetric potentials, J. Math. Res. Appl., 35(6) (2015), 613-624.
  • [6] D. Kumar, P. Bishnoi, On the refined measures of growth of generalized biaxially symmetric potentials having index-q, Fasciculi Matematici, 48 (2012), 61-72.
  • [7] M. Harfaoui, Generalized order and best approximation of entire function in Lp-norm, Intern. J. Math. Math. Sci., 2010 (2010), 1-15.
  • [8] D. Kumar, Growth and approximation of solutions to a class of certain linear partial differential equations in RN, Math. Slovaca, 64(1) (2014), 139-154.
  • [9] M. E. Kadiri, M. Harfaoui, Best polynomial approximation in Lp-norm and (p;q)-growth of entire functions, Abstr. Appl. Anal., 2013 (2013), 1-9.
  • [10] H. S. Kasana, D. Kumar, Approximation of generalized bi-axially symmetric potentials with fast rates of growth, Acta Math. Sinica (Wuhan-China), 15(4) (1995), 458-467.
  • [11] H. S. Kasana, D. Kumar, The Lp-approximation of generalized biaxisymmetric potentials, Int. J. Diff. Eqns. Appl., 9(2) (2004), 127-142.
  • [12] H. S. Kasana, D. Kumar, Lp-approximation of generalized bi-axially symmetric potentials over Caratheodory domains, Math. Slovaca, 55(5) (2005), 563-572.
  • [13] G. P. Kapoor, A. Nautiyal, Growth and approximation of generalized bi-axially symmetric potentials, Indian J. Pure Appl. Math., 19 (1980), 464-476.
  • [14] P. A. McCoy, Polynomial approximation of generalized biaxisymmetric potentials, J. Approx. Theory, 25(2) (1979), 153-168.
  • [15] D. Sato, On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc., 69 (1963), 411-414.
  • [16] J. M. Whittaker, The lower order of integral functions, J. London Math. Soc., 8 (1973), 20-27.
  • [17] A. Gray, S. M. Shah, Holomorphic functions with gap power series, Math. Z., 86 (1965), 375-394.
  • [18] O. P. Juneja, G.P. Kapoor, On the lower order of entire functions, J. London Math. Soc., 5(2) (1972), 310-312.
  • [19] S. N. Bernstein, Lecons sur les proprietes extremales et la meilleure approximation des fonctions analytiques d’une variable reille, Gauthier-Villars, Paris, 1926.
  • [20] A. Giroux, Approximation of entire functions over bounded domains, J. Approx. Theory, 28 (1980), 45-53.
  • [21] O. P. Juneja, On the coefficients of an entire series, J. Anal. Math., 24 (1971), 395-401.
Year 2018, Volume: 1 Issue: 2, 156 - 162, 24.12.2018
https://doi.org/10.33434/cams.439977

Abstract

References

  • [1] P. A. McCoy, Approximation of generalized biaxially symmetric potentials on certain domains, J. Math. Anal. Appl., 82 (1981), 463-469.
  • [2] R. P. Gilbert, Integral operator methods in biaxially symmetric potential theory, Contrib. Diff. Eqns., 2 (1963), 441-456.
  • [3] G. S. Srivastava, On the growth and polynomial approximation of generalized biaxisymmetric potentials, Soochow J. Math., 23(4) (1997), 345-358.
  • [4] P. A. McCoy, Best Lp-approximation of generalized biaxisymmetric potentials, Proc. Amer. Math. Soc., 79 (1980), 435-440.
  • [5] D. Kumar, A. Basu, Growth and approximation of generalized biaxially symmetric potentials, J. Math. Res. Appl., 35(6) (2015), 613-624.
  • [6] D. Kumar, P. Bishnoi, On the refined measures of growth of generalized biaxially symmetric potentials having index-q, Fasciculi Matematici, 48 (2012), 61-72.
  • [7] M. Harfaoui, Generalized order and best approximation of entire function in Lp-norm, Intern. J. Math. Math. Sci., 2010 (2010), 1-15.
  • [8] D. Kumar, Growth and approximation of solutions to a class of certain linear partial differential equations in RN, Math. Slovaca, 64(1) (2014), 139-154.
  • [9] M. E. Kadiri, M. Harfaoui, Best polynomial approximation in Lp-norm and (p;q)-growth of entire functions, Abstr. Appl. Anal., 2013 (2013), 1-9.
  • [10] H. S. Kasana, D. Kumar, Approximation of generalized bi-axially symmetric potentials with fast rates of growth, Acta Math. Sinica (Wuhan-China), 15(4) (1995), 458-467.
  • [11] H. S. Kasana, D. Kumar, The Lp-approximation of generalized biaxisymmetric potentials, Int. J. Diff. Eqns. Appl., 9(2) (2004), 127-142.
  • [12] H. S. Kasana, D. Kumar, Lp-approximation of generalized bi-axially symmetric potentials over Caratheodory domains, Math. Slovaca, 55(5) (2005), 563-572.
  • [13] G. P. Kapoor, A. Nautiyal, Growth and approximation of generalized bi-axially symmetric potentials, Indian J. Pure Appl. Math., 19 (1980), 464-476.
  • [14] P. A. McCoy, Polynomial approximation of generalized biaxisymmetric potentials, J. Approx. Theory, 25(2) (1979), 153-168.
  • [15] D. Sato, On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc., 69 (1963), 411-414.
  • [16] J. M. Whittaker, The lower order of integral functions, J. London Math. Soc., 8 (1973), 20-27.
  • [17] A. Gray, S. M. Shah, Holomorphic functions with gap power series, Math. Z., 86 (1965), 375-394.
  • [18] O. P. Juneja, G.P. Kapoor, On the lower order of entire functions, J. London Math. Soc., 5(2) (1972), 310-312.
  • [19] S. N. Bernstein, Lecons sur les proprietes extremales et la meilleure approximation des fonctions analytiques d’une variable reille, Gauthier-Villars, Paris, 1926.
  • [20] A. Giroux, Approximation of entire functions over bounded domains, J. Approx. Theory, 28 (1980), 45-53.
  • [21] O. P. Juneja, On the coefficients of an entire series, J. Anal. Math., 24 (1971), 395-401.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Devendra Kumar

Publication Date December 24, 2018
Submission Date July 3, 2018
Acceptance Date October 18, 2018
Published in Issue Year 2018 Volume: 1 Issue: 2

Cite

APA Kumar, D. (2018). On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets. Communications in Advanced Mathematical Sciences, 1(2), 156-162. https://doi.org/10.33434/cams.439977
AMA Kumar D. On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets. Communications in Advanced Mathematical Sciences. December 2018;1(2):156-162. doi:10.33434/cams.439977
Chicago Kumar, Devendra. “On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets”. Communications in Advanced Mathematical Sciences 1, no. 2 (December 2018): 156-62. https://doi.org/10.33434/cams.439977.
EndNote Kumar D (December 1, 2018) On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets. Communications in Advanced Mathematical Sciences 1 2 156–162.
IEEE D. Kumar, “On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets”, Communications in Advanced Mathematical Sciences, vol. 1, no. 2, pp. 156–162, 2018, doi: 10.33434/cams.439977.
ISNAD Kumar, Devendra. “On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets”. Communications in Advanced Mathematical Sciences 1/2 (December 2018), 156-162. https://doi.org/10.33434/cams.439977.
JAMA Kumar D. On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets. Communications in Advanced Mathematical Sciences. 2018;1:156–162.
MLA Kumar, Devendra. “On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets”. Communications in Advanced Mathematical Sciences, vol. 1, no. 2, 2018, pp. 156-62, doi:10.33434/cams.439977.
Vancouver Kumar D. On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets. Communications in Advanced Mathematical Sciences. 2018;1(2):156-62.

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