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On existence of interference between the components of biharmonic functions. (English. Russian original) Zbl 0984.31003
Mosc. Univ. Math. Bull. 55, No. 1, 29-32 (2000); translation from Vestn. Mosk. Univ., Ser. I 2000, No. 1, 45-48 (2000).
Let $$D^m:=\{x\in \mathbb R^m\mid \|x\|<1\}$$ be the unit sphere in $$\mathbb R^m$$, $$H_n = H_n(D^m)$$ be the space of polyharmonic functions $$f$$ of $$n$$th order in $$D^m$$ $$(\Delta^nf = 0)$$, $$M^p(D^m)$$ be the class of bounded functions of $$p$$th order. The main result of the paper is as follows. Let $$m\geq 3$$, $$p\in \mathbb N$$. Then there exists a biharmonic function $$f\in H_2(D^m)\cap M^p(D^m)$$, $$f= P+f_0+f_1$$, $$f_k= (1-|x|^2)^k\Phi_k$$, $$\Phi_k$$ polyanalytic, $$P$$ a polynomial, such that $$f_0,f_1\notin M^p(D^m)$$.
##### MSC:
 31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions 32A50 Harmonic analysis of several complex variables
##### Keywords:
biharmonic function; interference