Skopje, Makedonija INTERPOLATION POLYNOMIALS VIA AIFS 1 ...

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$Fractional and operational calculus with generalized fractional ...$

$Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...$

64 LJ. KOCIĆ, S. G-ZAJKOVA, V. ANDOVA According to the contraction mapping theorem, W has the unique fixed point, A ∈H(R m ) called the attractor of the IFS, satisfying. n⋃ A = W (A) = w i (A) Definition 1.1. A (non-degenerate) m-dimensional simplex ˆP m (or simplex) is theconvexhullofasetofm+1 affinely independent points (vectors) p 1 , p 2 , ..., p m+1 in Euclidean space of dimension m or higher, ˆP m = conv {p 1 , p 2 , ..., p m+1 }.The vertices of ˆP m will be denoted by P m and represented by the vector P m = [p T 1 p T 2 ... p T m+1 ] T . Let S m+1 =[s i,j ] m+1 i,j=1 be an (m +1)× (m + 1) row-stochastic real matrix (its rows sum up to 1). Definition 1.2. We refer to the linear mapping L : R m+1 → R m+1 , such that L(x)=S T x as the linear mapping associated with S. The corresponding Hutchinson operator is n⋃ W ′ (B) = L i (B), ∀B ∈H ( R m+1) (1.3) i=1 Definition 1.3. Let ˆP m be a non-degenerate simplex and let {S i } n i=1 be a set of real square ) nonsingular row-stochastic } matrices of order m + 1. The system Ω (ˆP m = {ˆP m ;S 1 , S 2 , ..., S n is called (hyperbolic) Affine invariant IFS (AIFS), provided that the linear mappings associated with S i are contractions in (R m ,d E )([4− 6]). Theorem 1. One eigenvalue of the matrix S i is 1, other m eigenvalues coincide with eigenvalues of A i , the matrix that makes the linear part of the affine mapping w i given by (1.1). Inotherwords,sp {S i } =sp{A i }∪{1}. i=1 2. Polynomials and subdivision Although the notion of subdivision is usually attributed to m-dimensional (m ≥ 1) continuous parametric mapping t ↦→ P n (t), t ∈ [a, b] (a
INTERPOLATION POLYNOMIALS VIA AIFS 65 is some functional basis, it may happened that both A k and B n (t) depend on the interval of definition. To stress this fact, in the case when confusion may appear, it is suitable to write A k [a, b] aswellasB n [a, b](t). Then, the subdivision is defined as follows. Definition 2.1. The function P n , defined by (2.1) is said to permit linear subdivision if and only if for each nonempty subinterval [p, q] ⊂ [a, b], there exists a set of coefficients {A k [p, q]} n k=0 such that n∑ n∑ A k [p, q] Bk n [p, q](t) = A k [a, b] Bk n [a, b](ϕ(t)) k=0 k=0 for t ∈ [a, b], where ϕ(t) = 1 ((q − p)t + bp− aq) b − a is the affine contraction that maps [a, b] into[p, q]. As it is shown by Goldman and Heath, linear subdivision is strictly a polynomial phenomenon. Theorem 2. ([3]) The function P n , defined by (2.1) admit linear subdivision if and only if B n (t) is a polynomial basis. The classic, and best known subdivision phenomena is connected with Bernstein polynomial basis {b n k }n k=0 ,ofdegreen ∈ N0 , where the basis functions t ↦→ b n k (t) are usually defined on the interval [0, 1] ( ) b n n k(x) = x k (1 − x) n−k , k =0, ..., n k or, on [a, b] b n k[a, b](x) = 1 (b − a) n ( n k ) (x − a) k (b − x) n−k , k =0, ..., n. It is suitable, for the sake of operational efficacy, to write Bernstein basis in vector form [ T. b [a, b] (x) = b n 0 [a, b](x) b n 1 [a, b](x) ... b n n[a, b](x)] Also, the convention b [0,1] (x) =b(x) will be adopted. Theorem 3. (Subdivision) For all 0 ≤ p
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