A Note on Approximation by Trigonometric Polynomials

• Published in: Journal of mathematical sciences (New York, N.Y.) N.Y.), 2020-03, Vol.243 (6), p.981-984
• Main Author: Shirokov, N. A.
• Format: Article
• Language: English
• Subjects: Approximation; Mathematical analysis; Mathematics; Mathematics and Statistics; Polynomials
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-019-04598-y

Let E = ∪ k = 1 n a k b k ⊂ ℝ ; if n > 1, then we assume that the segments [ a k ,  b k ] are pairwise disjoint. Assume that the following property holds: E  ∩ ( E  + 2π ν ) = ∅, ν  ∈  ℤ , ν  ≠ 0. Denote by H ω  +  r ( E ) the space of functions f defined on E such that | f ( r ) ( x 2 ) −  f ( r ) ( x 1 )| ≤  c f ω (| x 2  −  x 1 |), x 1 , x 2  ∈  E , f (0)  ≡  f . Assume that a modulus of continuity ω satisfies the condition ∫ 0 x ω t t dt + x ∫ x ∞ ω t t 2 dt ≤ cω x . We find a constructive description of the space H ω  +  r ( E ) in terms of the rate of nonuniform approximation of a function f  ∈  H ω  +  r ( E ) by trigonometric polynomials if E and ω satisfy the above conditions.