Approximation of Functions by Rational Functions on Closed Curves of the Complex Plane

Abstract

Let be a quasi-smooth curve in the complex plane C. In this study, a direct theorem of approximation theory in certain subclasses of the functions which have continuous derivatives through order r on a closed curve is proved. Mathematics Subject Classification Definitions, Some Auxiliary Results and Main ResultLet be an arbitrary restricted Jordan curve with two-component complements = C = 1 ∪ 2 , (0 ∈ 1 , ∞ ∈ 2 ). Let us consider the functions w = φ i (z), (i = 1, 2), that conformally and univalently map, respectively, i onto i , 1 = {w : |w| < 1}, 2 = {w : |w| > 1} , with norm φ 1 (0) =0, φ 1 (0) > 0, φ 2 (∞) = ∞, lim z→∞ 1 z φ 2 (z) > 0. Let us extend each φ i (z), (i = 1, 2) continuously up to the bound = ∂ 1 = ∂ 2 (generally speaking φ 1 (z) = φ 2 (z) for z ∈ ). We preserve the notation φ i , (i = 1, 2) for the extension. Let z = i (w) be the inverse mapping of w = φ i (z), (i = 1, 2) . LetFor arbitrary natural number n we set1+ (−1) i n (z) := inf ζ 1+ (−1) i n |ζ − z| , ρ1 n (z) := min ρ 1+ (−1) i n (z) , i = 1, 2

On approximation of functions byp–Faber-Laurent rational functions

“…[7,[21][22][23][24][25] We remark that problems of approximation functions by p−Faber polynomials and p−Faber-Laurent rational functions were investigated by Çavuş and Israfilov [1], Israfilov [2,3]. In the meantime, investigation of similar problems about the approximation of the functions by rational functions for different spaces was carried out by several researchers, such as, [3,[25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].…”

Approximation of functions by de la Vallée-Poussin sums in weighted Orlicz spaces

“…The problems of approximation theory in weighted and nonweighted Lebesgue spaces, weighted and nonweighted Orlicz spaces have been investigated by several authors (see, e.g., [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][30][31][32]36,37,45,46]). …”

On approximation of functions by rational functions in weighted generalized grand Smirnov classes

“…In this study the approximation problems of the functions by Faber-Laurent rational functions in the weighted generalized grand Smirnov classes E p),θ (G, ω), θ > 0, defined in the doubly connected domains with the regular boundaries are studied. Similar problems in the different spaces were investigated by several authors (see for example, [1][2][3][4][5][14][15][16][17][18][19][20][21][22][23]25,[27][28][29][30][31][32]38,39]).…”