Quadratic functional equations associated with Borel functions and module actions
Bull. Korean Math. Soc. 2009 Vol. 46, No. 3, 499-510
https://doi.org/10.4134/BKMS.2009.46.3.499
Printed May 1, 2009
Won-Gil Park and Jae-Hyeong Bae
National Institute for Mathematical Sciences and Kyung Hee University
Abstract : For a Borel function $\psi:\mathbb R\times\mathbb R\to\mathbb R$ satisfying the functional equation $\psi(s+t,u+v)+\psi(s-t,u-v)=2\psi(s,u)+2\psi(t,v)$, we show that it satisfies the functional equation $$\psi(s,t)=s(s-t)\psi(1,0)+st\psi(1,1)+t(t-s)\psi(0,1).$$ Using this, we prove the stability of the functional equation $$f(ax+ay,bz+bw)+f(ax-ay,bz-bw)=2abf(x,z)+2abf(y,w)$$ in Banach modules over a unital $C^\star$-algebra.
Keywords : stability, functional equation, quadratic form, Borel function
MSC numbers : Primary 39B22, 39B82
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