The d'Alembert and Lobaczevski difference operators in 𝓕_λ spaces
Abstract
Let X be a linear normed space, λ≥0, n∈ℕ. Let 𝓕λ(n) be a set defined by
𝓕λ(n) = {g: Xn→ℂ | |g(x→)| ≤ Mg · eλΣk=1n‖xk‖, x→∈Xn},
where Mg is a constant depending on g. Moreover for all g∈𝓕λ(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 𝓕λn 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)∈𝓕λ(2), where A is the d'Alembert difference operator, then f∈𝓕λ or A(f) = 0.
References
2. Czerwik S., Functional Equations and Inequalities in Several Variables, World Scientific, New Jersey, London, Singapore, Hong Kong 2002.
3. Czerwik S., Dłutek K., The Cauchy and Pexider difference operators in X_? spaces (to appear).
Instytut Matematyki, Politechnika Śląska Poland
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