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In this paper, we prove that infinite-dimensional vector spaces of $\alpha$-dense curves are generated by means of the functional equations $f(x)+f(2x)+\cdots+f(nx)=0$, with $n\geq 2$, which are related to the partial sums of the Riemann zeta function. These curves $\alpha$-densify a large class of compact sets of the plane for arbitrary small $\alpha$, extending the known result that this holds for the cases $n=2,3$. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the $n^{th}$ power of the density approaches the Jordan content of the compact set which the curve densifies.
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참고문헌 (8)
참고문헌 신청
G. Mora / 1997 / Characterization and generation of -dense curves / Comput. Math. Appl 33 (9) : 83 ~ 91
G. Mora / 2000 / Functional equations generating space-densifying curves / Comput. Math. Appl 39 (9-10) : 45 ~ 55
G. Mora / 2008 / A new approach to the reduction of multiple integrals to simple ones using Chebyshev’s kernels / Kybernetes 37 (1) : 104 ~ 119
G. Mora / 2008 / A note on the functional equation F(z) + F(2z) + · · · + F(nz) = 0 / J. Math. Anal. Appl 340 (1) : 466 ~ 475
G. Mora / 2009 / On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function / J. Math. Anal. Appl 350 (1) : 409 ~ 415
G. Mora / 2000 / Functional equations generating space-densifying curves / Comput. Math. Appl 39 (9-10) : 45 ~ 55
G. Mora / 2008 / A new approach to the reduction of multiple integrals to simple ones using Chebyshev’s kernels / Kybernetes 37 (1) : 104 ~ 119
G. Mora / 2008 / A note on the functional equation F(z) + F(2z) + · · · + F(nz) = 0 / J. Math. Anal. Appl 340 (1) : 466 ~ 475
G. Mora / 2009 / On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function / J. Math. Anal. Appl 350 (1) : 409 ~ 415
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