Displaying similar documents to “Borel’s theorem for C ∞ -functions on a non-archimedean valued field”

On the power-series expansion of a rational function

D. V. Lee (1992)

Acta Arithmetica

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Introduction. The problem of determining the formula for P S ( n ) , the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, h s ₁ , . . . , h s k , of the equation hs₁ s₁ + ... + hsk sk = n, was solved in the nineteenth century (see Sylvester [4] and Glaisher [3] for detailed accounts). The solution is the coefficient of x ⁿ i n [(1-xs₁)... (1-xsk)]-1, expressions for which they derived. Wright [5] indicated a simpler method by which to find part...

Some results about Beurling algebras with applications to operator theory

Thomas Vils Pedersen (1995)

Studia Mathematica

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We prove that certain maximal ideals in Beurling algebras on the unit disc have approximate identities, and show the existence of functions with certain properties in these maximal ideals. We then use these results to prove that if T is a bounded operator on a Banach space X satisfying ∥ T n ∥ = O ( n β ) as n → ∞ for some β ≥ 0, then ∑ n = 1 ∞ ∥ ( 1 - T ) n x ∥ / ∥ ( 1 - T ) n - 1 x ∥ diverges for every x ∈ X such that ( 1 - T ) [ β ] + 1 x ≠ 0 .