Abstract
In this paper, we define a parametric scale of non-uniformly Laguerre-weighted Sobolev spaces and a scale of spaces isometric to it. The second scale includes the ordinary (non-weighted) Sobolev space. Approximations by generalized Laguerre polynomials and functions in these spaces are studied. Some weighted \(L^{2}\)-analogues of Bernstein’s inequality for these polynomials and functions are obtained.
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Notes
If the domain of definition of functions coincides with \({{R}_{+}}\), then further we will not indicate it in the notation of the space of functions. So \(L^{2}_{\omega_{\alpha}}\), \({H}^{s}_{m}\) is short for \(L^{2}_{\omega_{\alpha}}({{R}_{+}})\), \({H}^{s}_{m}({{R}_{+}})\).
REFERENCES
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM-CBMS, Philadelphia, 1977).
C. Bernardi and Y. Maday, ‘‘Spectral method,’’ in Handbook of Numerical Analysis, Ed. by P. G. Ciarlet and L. L. Lions (North-Holland, Amsterdam, 1997), Vol. 5, Part 2.
C. Canuto, M. Y. Hussaini, A. Quarteronia, and T. A. Zang, Spectral Methods in Fluid Dynamics (Springer, New York, 1987).
C. Schwab, p- and hp- Finite Element Methods: Theory and Applications to Solid and Fluid Mechanics (Oxford Univ. Press, Oxford, 1999).
D. Funaro Polynomial Approximations of Differential Equations (Springer, Berlin, 1992).
J. Shen and T. Tang, Spectral and High-Order Methods with Applications (Science Press of China, Beijing, 2006).
M. R. Dorr, ‘‘The approximation theory for the \(p\)-version of the finite element method,’’ SIAM J. Numer. Anal. 21, 1180–1207 (1984).
S. Nicaise, ‘‘Jacobi polynomials, weighted Sobolev spaces and approximation results of some singularities,’’ Math. Nachr. 213, 117–140 (2000).
B.-Y. Guo and L.-L. Wang, ‘‘Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces,’’ J. Approx. Theory 128, 1–41 (2004).
J. Shen and L.-L. Wang, ‘‘Some recent advances on spectral methods for unbounded domains,’’ Commun. Comput. Phys. 5, 195–241 (2009).
R. Z. Dautov, ‘‘A sharp error estimate of the best approximation by algebraic polynomials in the weighted space \(L_{2}(-1,1)\),’’ Russ. Math. 57 (5), 51–53 (2013).
R. Z. Dautov and M. R. Timerbaev, ‘‘Sharp estimates for the polynomial approximation in weighted Sobolev spaces,’’ Differ. Equat. 51, 886–894 (2015).
C. Canuto and A. Quarteroni, ‘‘Approximation results for orthogonal polynomials in Sobolev spaces,’’ Math. Comput. 38, 67–86 (1982).
Y. Maday, ‘‘Analysis of spectral projectors in one-dimensional domains,’’ Math. Comput. 55 (192), 537–562 (1990).
C. Bernardi and Y. Maday, ‘‘Polynomial interpolation results in Sobolev spaces,’’ J. Comput. Appl. Mat. 43, 53–80 (1992).
A. Guessab and G. V. Milovanović, ‘‘Weighted L2-analogues of Bernstein’s inequality and classical orthogonal polynomials,’’ J. Math. Anal. Appl. 182, 244–249 (1994).
G. Szego, Orthogonal Polynomials, 4th ed., Vol. 23 of Amer. Math. Soc. Collog. Publ. (Amer. Math. Soc., Providence, RI, 1975).
R. Courant and D. Hilbert, Methods of Mathematical Physics (OGIZ, Moscow, 1933), Vol. 1 [in Russian].
L. D. Kudryavtsev, ‘‘Equivalent Norms in weighted spaces,’’ in Studies in the Theory of Differential Functions of Several Variables and its Applications, Tr. Mat. Inst. Steklov 170, 161–190 (1984).
ACKNOWLEDGMENTS
The author is grateful to E.M. Karchevskii for critical remarks and useful discussion.
Funding
This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program (PRIORITY-2030).
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(Submitted by A. V. Lapin)
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Dautov, R.Z. On Approximation of Functions on the Half-Line by Laguerre Polynomials and Functions. Lobachevskii J Math 43, 935–947 (2022). https://doi.org/10.1134/S1995080222070071
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DOI: https://doi.org/10.1134/S1995080222070071