Let X be a linear normed space, λ≥0, n∈ℕ. Let ????λ(n) be a set defined by
????λ(n) = {g: Xn→ℂ | |g(x→)| ≤ Mg · eλΣk=1n‖xk‖, x→∈Xn},
where Mg is a constant depending on g. Moreover for all g∈????λ(n) we define
‖g‖ := supx→∈Xn {e-λΣk=1n‖xk‖ · |g(x→)|}.
In the paper norms of the d'Alembert and Lobaczevski difference operators in the ????λn spaces are calculated (their Pexider type generalizations are also considered). Moreover it is proved that if f: X→ℂ is a function such that A(f)∈????λ(2), where A is the d'Alembert difference operator, then f∈????λ or A(f) = 0.
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Vol. 20 (2006)
Published: 2006-09-29
10.1515/amsil
English