Given n functions P1,...,Pn on a region A⊂Rn, we seek n other functions p1,...,pn on A such that \( \int\limits_A {({p^{{1^2}}} + ... + {p^{{n^2}}})} \) is minimal, subject to the condition that
$${({p^j} + {p^j})_i} = ({p^i} + {p^i})j,1 \le i \le j \le n,$$
where subscripts indicate partial derivatives. The problem is solved for the case n = 2 and A, an interval. The key to the solution is the Fourier expansion of the discrepancy \(P_1^2 - P_2^1\) in terms of the eigenfunctions of the vibrating string.


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  1. [1]
    Riesz, F. and Sz.-Nagy, B. Leçons d’analyse fonctionnelle, Budapest, 1952 = Functional analysis, translated by L.F. Boron, New York, 1955.Google Scholar

Copyright information

© Springer Basel AG 1980

Authors and Affiliations

  • Arthur Sard
    • 1
    • 2
  1. 1.Queens CollegeCity University of New YorkFlushingUSA
  2. 2.Universität SiegenGermany

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