Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by finite Fourier-Haar series
- Daghestan Scientific Centre of the Russian Academy of Sciences, Makhachkala (Russian Federation)
The paper deals with the space L{sup p(x)} consisting of classes of real measurable functions f(x) on [0,1] with finite integral ∫{sub 0}{sup 1}|f(x)|{sup p(x)} dx. If 1≤p(x)≤ p-bar <∞, then the space L{sup p(x)} can be made into a Banach space with the norm ∥f∥{sub p(⋅)}=inf(α > 0:∫{sub 0}{sup 1}|f(x)/α|{sup p(x)} dx≤ 1). The inequality ∥f−Q{sub n}(f)∥{sub p(⋅)}≤c(p)Ω(f,1/n){sub p(⋅)}, which is an analogue of the first Jackson theorem, is shown to hold for the finite Fourier-Haar series Q{sub n}(f), provided that the variable exponent p(x) satisfies the condition |p(x)−p(y)|ln (1/|x−y|)≤ c. Here, Ω(f,δ){sub p(⋅)} is the modulus of continuity in L{sup p(x)} defined in terms of Steklov functions. If the function f(x) lies in the Sobolev space W{sub p(⋅)}{sup 1} with variable exponent p(x), it is shown that ∥f−Q{sub n}(f)∥{sub p(⋅)}≤c(p)/n∥f{sup ′}∥{sub p(⋅)}. Methods for estimating the deviation |f(x)−Q{sub n}(f,x)| for f(x)∈W{sub p(⋅)}{sup 1} at a given point x∈[0,1] are also examined. The value of sup{sub f∈W{sub p{sup 1}(1)}}|f(x)−Q{sub n}(f,x)| is calculated in the case when p(x)≡p= const, where W{sub p}{sup 1}(1)=(f∈W{sub p}{sup 1}:∥f{sup ′}∥{sub p(⋅)}≤1). Bibliography: 17 titles.
- OSTI ID:
- 22365661
- Journal Information:
- Sbornik. Mathematics, Vol. 205, Issue 2; Other Information: Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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