On the issue of expansion of a given function in a series of Jacobi polynomials
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Abstract
The purpose of this note is to simplify the method proposed by Academician N. M. Krylov for expanding a given function in a series in Jacobi polynomials. By constructing a priori an expansion with uniform and absolute convergence, we show, without relying on the Riesz-Fischer theorem, that it can be identified with expansions in Jacobi polynomials.
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References
- V. A. Steklov. Journal für reine und angewandte Mathematik, vol. 125 (1903), p. 214.
- Zapiski Petersburgskoi Academii Nauk, vol. XV, No. 7 (1904), p. 20. (in Russian)
- N. M. Krylov. Comptes rendus des séances de l'Académie des Sciences, Paris, vol. 150 (1910), p. 316.
- Izvestia Kievskogo Universiteta (1910), No. 10. (in Russian)
- N. M. Krylov. On Expansions in Series in Fundamental Functions... and on Expansions in Lubbie Polynomials. Dissertation, Kiev (1911), pp. 94–103. (in Russian)
- Encyclopédie des Sciences mathématiques (French edition), vol. 2, no. 5, fasc. 2 (1914), p. 234.
- Comptes rendus, vol. 144 (1907), pp. 615 and 1022.
- R. Appell. Archiv der Mathematik und Physik, vol. 66 (1881), p. 238.
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