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Opsomming
As die kuns van gladstryking van data geformaliseer word, is dit nuttig om die konsep van ’n skeier (separator) in te voer as ’n operator S wat ’n gegewe ry x skei in dit wat benodig word Sx, of die sein, en dit wat verwyder moet word, of die geraas. Vir lineêre gladstrykers sal projeksies dikwels genoegsaam wees. In hierdie geval is Sx outomaties ’n beste benadering in die 2-norm tot x, vanuit die beeldruimte van S. Die Lebesgue-ongelykheid verseker naasbeste benadering in ander norme.
As die gladstrykers algemeen, of nielineêr, is, kan mens hierdie projeksies veralgemeen tot separators. In die LULU-teorie van gladstrykers is alle komposisies S van die basiese LULU-operatore sodanig dat Sx steeds naasbeste benaderings, uit die beeldruimte van S, tot x gee.
In hierdie artikel word relevante gevestigde resultate opgesom wat presies dit bewys: alle operatore soos S het Sx tussen twee benaderings ULx en LUx tot x, wat dieselfde beeldversamelings het. Beste benaderings vanuit hierdie beeldversameling bestaan in al die gewone p-norme vir rye, en Ulx, LUx en Sx lewer benaderingsfoute begrens deur dieselfde Lebesgue-agtige faktor en die beste benaderingsfout.
Die nuwe resultate wat in hierdie artikel afgelei word, is eerstens dat ULx enLUx altyd benaderingsfoute binne ’n faktor 2 van mekaar het, en tweedens dat daar ’n afleibare benadering in dieselfde beeldversameling lê, wat tussen ULx en LUx is, en nie swakker as albei benader nie. Belangrike implikasies vir keuses in Diskrete-Puls-Transforms word afgelei.
Abstract
When formalising the art of smoothing data it is convenient to introduce the concept of a separator, as an operator that separates a given sequence x of data into that which is required (Sx, or the signal) and that to be eliminated (x – Sx, or the noise). For linear smoothers projections often suffice. In this case we automatically have that Sx is a best approximation in the 2 norm to x, from the vector space that is the range of S. The Lebesgue Inequality ensures near best approximation in other norms.
When smoothers are general, or non-linear, one can generalise this projection into a separator. In the LULU theory of smoothers all compositions of the basic LULU operators are such that Sx is still a near best approximation to x from the range of S.
In this article relevant established results are summed up that prove precisely this; all operators like S have Sx between two approximations ULx and LUx to x, which have the same image sets. Best approximations from these image sets exist in all the p norms for sequences, and LUx, LUx and Sx yield approximation errors bounded by the same Lebesgue-type factor and the best-approximation error.
The new results that are derived in this article are, firstly, that ULx and LUx always have approximation errors inside a factor of two from each other and, secondly, that there is a derivable approximation, which is between these two, which does not approximate more badly. Important implications for choices in Discrete Pulse Transforms are derived.
Klik hier om die volledige artikel in PDF-formaat af te laai.
Click here to read the extended abstract in English.