The affine synthesis operator Sc = P j>0 P k∈Z d c j,k ψ j,k is shown to map the coefficient space ℓ p (Z+ × Z d) surjectively onto L p (R d), for p ∈ (0, 1]. Here ψ j,k (x) = | det aj| 1/p ψ(ajx − k) for dilation matrices aj that expand , and the synthesizer ψ ∈ L p (R d) need satisfy only mild restrictions, for example ψ ∈ L 1 (R d) with nonzero integral or else with periodization that is rea...

In a first course in functional analysis, a great deal of time is spent with Banach spaces, especially the interaction between such spaces and their dual spaces. Banach spaces are a special type of topological vector space, and there are important topological vector spaces which do not lie in the Banach category, such as the Schwartz spaces. The most fundamental theorem about Banach spaces is t...

background and aims:in neurodegenerative disorders,oxidative stress mediated by reactive oxygen species is strongly associated with increased neuronal damages which can lead to apoptosis. pro-apoptotic and anti-apoptotic gene expressions are changed during the cell differentiation that affect cell viability and differentiation. therefore, this study was conducted to determine the effects of hyd...