A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

Let $u(t,y)$ be a polyharmonic function of order $N$ defined on the strip $(a,b)\times {\mathbb{R}}^d$ satisfying the growth condition $$\underset{t\in K}{\sup }\mid u(t,y)\mid \le o\left({\mid y\mid }^{(1-d)\mathrm{/}2}{e}^{\frac{\pi }{c}\mid y\mid }\right)$$ for $\left\vert y\right\vert \rightarrow\infty$andanycompactsubinterval$K$of$\left( a,b\right) $,andsupposethat$u\left( t,y\right) $vanisheson$2N-1$equidistanthyperplanesoftheform$\left{ t_{j}\right} \times$\mathbb{R}$^{d}$for$t_{j}=t_{0}+jc\in\left( a,b\right) $and$j=-\left( N-1\right) ,…,N-1.$Thenitisshownthat$u\left( t,y\right) $isoddat$t_{0},$i\mathrm{.}e\mathrm{.}that$u\left( t_{0}+t,y\right) =-u\left( t_{0}-t,y\right) $for$y\in$\mathbb{R}$^{d}$\mathrm{.}Thesecondmainresultstatesthat$u$ is identically zero provided that $u$ satisfies the growth condition and vanishes on $2N$ equidistant hyperplanes with distance $c\mathrm{.}$