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We define weak and ultraweak functional calculus and construct using the Fourier transform technique an ultraweak functional calculus for an unbounded k-tuple of commuting generalized scalar operators T j acting on a Banach space X. This functional calculus comprises the functions from the subset F α = {f: ℝ k → ℂ |f is a measure such that ∫ ℝ k (1+|t| α )d|f| 0. It also contains some unbounded functions. We also give examples and related results.

A functional calculus for unbounded generalized scalar operators on Banach spaces