Measures of Growth and Approximation of Entire Harmonic Functions in n-Dimensional Space in Some Banach Spaces

The relationship between the classical order and type of an entire harmonic function in space ℝⁿ, n ≥ 3, and the rate of its best harmonic polynomial approximation for some Banach spaces of functions harmonic in the ball of radius R has been studied.

Full text (pdf)

1. H. Beitmen, A. Erdeiy, Vysshye Transtsendentnye Funktsyy, 2nd edition, Nauka, Moscow, 1974, p. 296.
2. P.L. Duren, B.W. Romberg, A.L. Shields, Linear functionals in H_p spaces with 0 < p < 1, J. Reine Angew. Math. 238 (1969) 4-60.
3. A.J. Fryant, Growth of entire harmonic functions in ℝ³, J. Math. Anal. Appl. 66 (1978) 599-605.
4. T.B. Fugard, Growth of entire harmonic functions in ℝⁿ, n ≥ 2, J. Math. Anal. Appl. 74 (1) (1980) 289-291.
5. M.I. Gvaradze, On one class of spaces of analytic functions, Mat. Zametki 21 (2) (1977) 141-150.
6. G.H. Hardy, J.E. Littlewood, Some properties of fractional integrals, II, Math. Z. 34 (3) (1931) 403-439.
7. M. Harfaoui, Generalized order and best approximation of entire function in L_p-norm, Intern. J. Math. and Math. Sci. 2010 (2010) 1-15.
8. I.I. Ibragimov, N.I. Shikhaliev, On the best polynomial approximation in one space of analytic functions, Dokl. Akad. Nauk SSSR 227 (2) (1976) 280-283.
9. I.I. Ibragimov, N.I. Shikhaliev, On the best approximation in the mean for analytic functions in the space A_p( |z| < 1), Spets. Vopr. Teor. Funkts. 1 (1977) 84-96.
10. G.P. Kapoor, A. Nautiyal, Approximation of entire harmonic functions in ℝ³, Indian J. Pure and Appl. Math. 13 (9) (1982) 1024-1030.
11. G.P. Kapoor, A. Nautiyal, On the growth of harmonic functions in ℝ³, Demonstr. Math. 16 (4) (1983) 811-819.
12. D. Kumar, The growth of harmonic functions in Hyperspheres, Demonstr. Math. 32 (4) (1999) 717-724.
13. D. Kumar, Growth and approximation of entire harmonic functions in ℝⁿ, n > 3, Georgion Math. J. 15 (1) (2008) 1-12.
14. D. Kumar, Growth and approximation of solutions to a class of certain linear partial differential equations in ℝ^ℕ, Mathematica Slovaca, 64 (1) (2014) 139-154.
15. D. Kumar, H.S. Kasana, On maximum term, maximum modulus and approximation error of an entire harmonic function in ℝ³, Rev. Mat. Univ. Parma 6 (1) (1998) 215-223.
16. D. Kumar, H.S. Kasana, Approximation of entire harmonic functions in ℝ³ in L^β-norm, Fasciculi Math. 34 (2004) 55-64.
17. D. Kumar, G.S. Srivastava, H.S. Kasana, Approximation of entire harmonic functions in ℝ³ having index-pair (p, q), Anal. Numer. Theor. Approx. 20 (1-2) (1991) 47-57.
18. A.R. Reddy, A contribution to best approximation in the L²-norm, J. Approx. Theory 11 (11) (1974) 110-117.
19. G.S. Srivastava, Generalized growth of entire harmonic functions, Fasciculi Math. 40 (2008) 79-89.
20. G.S. Srivastava, S. Kumar, Uniform approximation of entire functions on compact sets and their generalized growth, New Zealand J. Math. 39 (2009) 33-43.
21. M. Shaker Abdu-Hussein, G.S. Srivastava, On the generalized type and approximation of entire harmonic functions in ℝ³ having index-pair (p, q), Istanbul Univ. Fem. Fak. Mat. Der. 60 (2001) 1-17.
22. Y. Stein, H. Veis, , Vvedenye v Harmonycheskyi Analyz na Evklydovykh Prostranstvakh, Moskow, Myr. 1974, p. 336.
23. A. Tyman, V.N. Trofymov, Vvedenye v Teoryiu Harmony Ches Kykh Funktsii, Nauka, Moscow, 1968, p. 208.
24. S.B. Vakarchuk, On the best polynomial approximation of analytic functions in the space B_p,q,λ, Dokl. Akad. Nauk Ukr. SSR, Ser. Fiz-Mat. Tekh. Nauk. 8 (1989) 6-9.
25. S.B. Vakarchuk, S.I. Zhir, Best polynomial approximations of entire transcendental functions of the generalized order of growth in Banach spaces ε’_p(G) and ε_p(G), p ≥ 1, Ukr. Mat. Vestn. 8 (2) (2011) 255-291.
26. S.B. Vakarchuk, S.I. Zhir, On the best polynomial approximation of entire transcendental functions of many complex variables in some Banach spaces, Ukr. Math. J. 66 (12) (2015) 1793-1811.
27. O.V. Veselovskaia, Oroste tselykh harmonycheskykh v funktsyi, Yzv. Vuzov. Matem. 10 (1983) 13-17.