On integrals constant on congruent domains

  • V. V. Proizvolov


We prove that if a real function of two variables is defined, continuous, and bounded on the whole plane, then it is constant under the condition that its integral on each square of unit area is constant. We point out variants of this theorem. We present an example of a function that is not constant but whose integral on each circle of unit radius is constant. Such a function is sin Βx, where Β is any root of the Bessel function J1.


Unit Area Bessel Function Real Function Unit Radius 
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Copyright information

© Plenum Publishing Corporation 1977

Authors and Affiliations

  • V. V. Proizvolov
    • 1
  1. 1.Central Scientific-Research and Design-Experimental Institute of the Organization of Mechanization and Engineering Assistance of the State System Construction Project of the USSRUSSR

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