The function sequences and Taylor series with a fuzzy argument


  • I. Ya. Spektorsky


The main consideration subject is functional sequences fn(A) with fuzzy number A for an argument. It is supposed that limn→∞fn(x)=f(x) and limn→∞fn’(x)=f’(x), and these convergences are uniform on each interval within supp A. It is also supposed that the equation f(x)=y with respect to x has finite number of solutions for each y on each interval within supp A. The paper proposes sufficient conditions for fn(A) to converge in the sense that the sequence of membership functions μfn(A)(y): converges point-wise. It is proved that limn→∞ μfn(A)(y)= μf(A)(y) for all y ϵ P, except such y=f(x), that x is a discontinuity point of μA(x), or f‘(x)=0. As a particular case of sequence fn(A), the generalization of Taylor series f(x)=∑i=0(f(i)(x0)(x-x0)i/(i!)) is considered for real analytical function f(x) for the case of fuzzy argument x=A. Convergence of the series is considered in the sense of point-wise convergence of the partial sum μSn(A)(y), where Sn(x)=∑i=0n(f(i)(x0)(x-x0)i/(i!)).


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