On the regularization of sequences and associated weight functions

Abstract

We revisit and generalize the geometric procedure of regularizing a sequence of real numbers with respect to a so-called regularizing function. This approach was studied by S. Mandelbrojt and becomes useful and necessary when working with corresponding classes of ultradifferentiable functions defined via weight sequences and analogous weighted spaces. In this note we also study non-standard situations for the construction yielding the (log-)convex minorant of a sequence and allow a "blow-up" for the regularizing function.

30 pages; this version is accepted for publ. in the Bull. Belg. Math. Soc. - Simon Stevin. According to the comments made by the anonymous referee we have rewritten the introduction and improved the presentation of the main results; some stylistic minor changes have been applied

References in corpus (11)

• Composition in ultradifferentiable classes
• Sectorial extensions, via Laplace transforms, in ultraholomorphic classes defined by weight functions
• On subadditivity-like conditions for associated weight functions
• Optimal flat functions in Carleman-Roumieu ultraholomorphic classes in sectors
• On inclusion relations between weighted spaces of entire functions
• On the inclusion relations of global ultradifferentiable classes defined by weight matrices
• On the construction of large Algebras not contained in the image of the Borel map
• On strong growth conditions for weighted spaces of entire functions
• Solid hulls and cores of classes of weighted entire functions defined in terms of associated weight functions
• Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting
• On the equivalence between moderate growth-type conditions in the weight matrix setting

Cited by in corpus (5)

• On strong growth conditions for weighted spaces of entire functions
• On inclusion relations of weighted $L^p$-type spaces defined in terms of weight function matrices
• On the class of almost subadditive weight functions
• On the conjugate weight function and ultradifferentiable classes of entire functions
• Construction of the log-convex minorant of a sequence $\{M_{\alpha }{\}}_{\alpha \in {\mathbb{N}}_0^d}$