mollifierekhez

Mollifierekhez is the dative plural form of the Hungarian noun mollifikátor, i.e., "to the mollifiers" or "for the mollifiers." As a standalone term it is primarily a grammatical form; its precise meaning in any given sentence depends on the surrounding context and the subject mollifikátorok (the mollifiers). In mathematical literature, mollifiers (mollifikátorok) refer to smooth, compactly supported functions used to regularize other functions via convolution. A mollifier is typically nonnegative, integrates to one, and has support in a small neighborhood of the origin. Convolution with a mollifier produces smooth approximations to nonsmooth functions, and, under appropriate conditions, f * η_ε converges to f as ε → 0.

Construction: A standard mollifier is built from a function η ∈ C_c^∞(R^n), η ≥ 0, ∫ η = 1. For ε > 0, η_ε(x) = ε^{-n} η(x/ε). Then η_ε ∈ C_c^∞ and ∫ η_ε = 1. For f ∈ L^p, 1 ≤ p < ∞, the convolution f * η_ε is in C^∞ and tends to f in L^p as ε → 0.

Applications include regularization of functions and distributions, proof techniques in partial differential equations, and numerical smoothing. See also mollifier, mollification, and convolution.

Note: In Hungarian mathematical prose, the form mollifierezhez would appear when indicating direction towards the mollifiers or when describing operations applied to or with respect to the mollifiers; it is primarily a grammatical variant rather than a distinct mathematical concept.