On the pointwise convergence of the eigenfunction expansion associated with some iterated boundary value problems

Authors:
Jyoti Das and Prabir Kumar Sen Gupta

Journal:
Proc. Amer. Math. Soc. **98** (1986), 593-600

MSC:
Primary 34B25; Secondary 47E05

MathSciNet review:
861757

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Abstract: Given a boundary value problem consisting of a second-order differential equation and some boundary conditions, one can derive higher-order boundary value problems, called iterated boundary value problems, provided the coefficients in the second-order differential equation are sufficiently smooth. The problem of convergence of the eigenfunction expansions associated with boundary value problems of even order has been the central attraction for mathematicians since the beginning of this century. The idea of this paper is to single out some higher-order boundary value problems, for which the question of convergence of the said expansion is completely answered by the similar problem associated with the second-order boundary value problem responsible for the generation of the iterated boundary value problem.

**[1]**Chaudhuri Jyoti,*Some problems in the theory of eigenfunction expansions*, Thesis, Oxford Univ., 1964.**[2]**J. Chaudhuri,*On the convergence of the eigenfunction expansion associated with a fourth-order differential equation*, Quart. J. Math. Oxford Ser. (2)**15**(1964), 258–274. MR**0165170****[3]**Jyoti Chaudhuri and W. N. Everitt,*On an eigenfunction expansion for a fourth-order singular differential equation*, Quart. J. Math. Oxford Ser. (2)**20**(1969), 195–213. MR**0247167****[4]**Jyoti Chaudhuri and W. N. Everitt,*On the square of a formally self-adjoint differential expression*, J. London Math. Soc. (2)**1**(1969), 661–673. MR**0248562****[5]**W. N. Everitt,*Fourth order singular differential equations*, Math. Ann.**149**(1962/1963), 320–340. MR**0152693****[6]**W. N. Everitt and M. Giertz,*On some properties of the powers of a formally self-adjoint differential expression*, Proc. London Math. Soc. (3)**24**(1972), 149–170. MR**0289841****[7]**Kunihiko Kodaira,*On ordinary differential equations of any even order and the corresponding eigenfunction expansions*, Amer. J. Math.**72**(1950), 502–544. MR**0036404****[8]**J. B. Mcleod,*Three problems in eigenfunction expansions*, Thesis, Oxford Univ., 1958.**[9]**J. B. McLeod,*Convergence of eigenfunction expansions under Fourier conditions. I*, Proc. London Math. Soc. (3)**12**(1962), 144–166. MR**0138824****[10]**D. B. Sears,*An expansion in eigenfunctions*, Proc. London Math. Soc. (2)**53**(1951), 396–421. MR**0043312****[11]**E. C. Titchmarsh,*On the convergence of eigenfunction expansions*, Quart. J. Math. Oxford Ser. (2)**3**(1952), 139–144. MR**0048672****[12]**E. C. Titchmarsh,*On the convergence of eigenfunction expansions. II*, Quart. J. Math. Oxford Ser. (2)**8**(1957), 236–240. MR**0097577****[13]**E. C. Titchmarsh,*Eigenfunction expansions associated with second-order differential equations. Part I*, Second Edition, Clarendon Press, Oxford, 1962. MR**0176151****[14]**Hermann Weyl,*Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen*, Math. Ann.**68**(1910), no. 2, 220–269 (German). MR**1511560**, 10.1007/BF01474161

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0861757-1

Keywords:
Bilinear concomitant,
boundary condition function,
boundary value problem,
convergence under Fourier conditions,
Hilbert space,
Lebesgue square-integrable solution,
limit-point case at infinity,
limit- case at infinity,
resolvent operator,
simple closed contour

Article copyright:
© Copyright 1986
American Mathematical Society