We consider a third-order differential operator with matrix coefficients. The absolute and uniform convergence of the orthogonal expansion of a vector function from the class \( {W}_{p,m}^1(G),p>1 \) , in the vector eigenfunctions of this operator is studied and the rate of uniform convergence of this expansion on \( \overline{G}=\left[0,1\right] \) is estimated.
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V. A. Il’in, “On the unconditional basis property of systems of eigenfunctions and associated functions of a differential operator of the second order on a closed interval,” Dokl. Akad. Nauk SSSR, 273, No. 5, 1048–1053 (1983).
N. L. Lazetic, “On uniform convergence on closed intervals of spectral expansions and their derivatives for functions from \( {W}_p^{(1)} \) ,” Mat. Vestnik, 56, 91–104 (2004).
V. M. Kurbanov and R. A. Safarov, “Uniform convergence of expansion responding to the Schr¨odinger operator,” Proc. Inst. Math. Mech. Nat. Acad. Sci. Azerbaijan, 20, 63–70 (2004).
V. M. Kurbanov and A. T. Garayeva, “Absolute and uniform convergence of expansion in root-vector functions of the Schrödinger operator with matrix potential,” Dokl. Math., 87, No. 3, 304–306 (2013).
V. M. Kurbanov and E. B. Akhundova, “Absolute convergence of spectral expansion in eigenfunctions of third order ordinary differential operator,” Proc. Inst. Math. Mech. Nat. Acad. Sci. Azerbaijan, Special Issue, 40, 264–274 (2014).
V. M. Kurbanov and Y. I. Huseynova, “On convergence of spectral expansion of absolutely continuous vector-function in eigenvector-functions of fourth order differential operator,” Trans. Acad. Sci. Azerb., Ser. Phys.-Tech. Math. Sci., 34, No. 1, 83–90 (2014).
M. A. Naimark, Linear Differential Operators [in Russian], Nauka, Moscow (1969).
N. B. Kerimov, “Some properties of eigenfunctions and associated functions of ordinary differential operators,” Dokl. Akad. Nauk SSSR, 291, No. 5, 1054–1055 (1986).
V. M. Kurbanov, “On the Hausdorff–Young inequality for systems of root vector functions of a differential operator of order n,” Differents. Uravn., 33, No. 3, 356–367 (1997).
V. M. Kurbanov, “On the analog of the Riesz theorem and the basis property of a system of root functions of a differential operator in L p . I, II,” Differents. Uravn., 49, No. 1, 7–19; No. 4, 437–449 (2013).
A. Zygmund, Trigonometric Series [Russian translation], Vol. 2, Mir, Moscow (1965).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 6, pp. 719–733, June, 2017.
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Abbasova, Y.G., Kurbanov, V.M. Convergence of the Spectral Decomposition of a Function from the Class \( {W}_{p,m}^1(G),p>1 \) , in the Vector Eigenfunctions of a Differential Operator of the Third Order. Ukr Math J 69, 839–856 (2017). https://doi.org/10.1007/s11253-017-1400-0
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DOI: https://doi.org/10.1007/s11253-017-1400-0