Quadratic functional equations associated with Borel functions and module actions
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초록
For a Borel function ψ : R × R → R satisfying the functional equation ψ(s + t, u + v) + ψ(s − t, u − v) = 2ψ(s, u) + 2ψ(t, v), we show that it satisfies the functional equation ψ(s, t) = s(s − t)ψ(1, 0) + stψ(1, 1) + t(t − s)ψ(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^(*)-algebra.
Abstract
For a Borel function ψ : R × R → R satisfying the functional equation ψ(s + t, u + v) + ψ(s − t, u − v) = 2ψ(s, u) + 2ψ(t, v), we show that it satisfies the functional equation ψ(s, t) = s(s − t)ψ(1, 0) + stψ(1, 1) + t(t − s)ψ(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^(*)-algebra.
참고문헌 (7)
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