F-transformācijas balstītas uz polinomiāliem splainiem
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Latvijas Universitāte
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Abstract
Darbs tiek veltīts F–transformācijām (nestriktām transformācijām), kuras ir cieši saistītas ar nestriktiem sadalījumiem. Ar tiešo F–transformāciju funkcija tiek pārvērsta par vektoru, kura komponentes apraksta transformējamās funkcijas vērtības atbilstoši intervāla [a,b] nestriktajam sadalījumam. Savukārt, ar inverso F–transformāciju vektors tiek pārvērsts atpakaļ par sākotnējās funkcijas aproksimāciju. Sadalījumu konstruēšanai darbā tiek piedāvātas uz polinomiālo splainu pamata konstruētās harakteristiskās funkcijas. Tiek apskatīti divu veidu nestrikti sadalījumi: klasiskais un vispārinātais. Darbā ir aprakstītas arī augstāku kārtu F–transformācijas, iegūti kļūdu novērtējumi. Rezultāti ilustrēti ar testa piemēriem.
Atslēgas vārdi: nestrikta kopa; nestrikts sadalījums; vispārināts nestrikts sadalījums; F–transformācija; augstākas kārtas F–transformācija.
This paper is devoted to F–transform (fuzzy transform). The core idea of F–transform is inwrought with an interval [a,b] fuzzy partitioning into fuzzy subsets, determined by their membership functions. In this work we consider polynomial splines with degree m and defect 1 as membership functions of two types of fuzzy partition: classical and generalized. The idea of F–transform is transformation from a function space to a finite dimensional vector space, the inverse F–transform is transformation back to the function space. We claim that for a sufficient representation of a function defined on [a,b], we may consider its average values over fuzzy subsets from the used partition. We consider also F–transform of higher degree with respect to both types of fuzzy partitions. We investigate approximative properties of the spline based inverse F–transforms and illustrate them with numerical examples. Keywords: fuzzy sets; fuzzy partition; generalized fuzzy partition; fuzzy transform; fuzzy transform of higher degree.
This paper is devoted to F–transform (fuzzy transform). The core idea of F–transform is inwrought with an interval [a,b] fuzzy partitioning into fuzzy subsets, determined by their membership functions. In this work we consider polynomial splines with degree m and defect 1 as membership functions of two types of fuzzy partition: classical and generalized. The idea of F–transform is transformation from a function space to a finite dimensional vector space, the inverse F–transform is transformation back to the function space. We claim that for a sufficient representation of a function defined on [a,b], we may consider its average values over fuzzy subsets from the used partition. We consider also F–transform of higher degree with respect to both types of fuzzy partitions. We investigate approximative properties of the spline based inverse F–transforms and illustrate them with numerical examples. Keywords: fuzzy sets; fuzzy partition; generalized fuzzy partition; fuzzy transform; fuzzy transform of higher degree.