Volume 4 • Issue 2 • PP: 01–05 • 2024
On the Hausdorff Method Applications in the Problem of Finding the Degree of Functions Approximation
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© 2024 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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In this research, we study the problem of determining the degree of approximation of functions using the Hausdorff method, and we can do this by proving the following results:
If f∈Lip(α,p)with α>1/p and be a continues aimost everywhere and 2m periodic function,Then the degree of approximation of (f ) ̃using hausdorff means of conjugate fourier series, is given by:
〗|(|H ̃_((n+λ) ) (f,a)-(f ) ̃(a)|)|_p=0((n+λ)^(1/p-α) )
If f be a 2m periodic function, continues almost everywhere on [–m,m] andbelonging to the class Z_(α,p ),p≥1 .then the degree of approximation of function f of fouier series using hausdorff means,is given by:
E_((n+λ) ) (f)= inf_((n+λ) ) ‖H_((n+λ) )-f‖_(α,p)=0(1/((n+λ) ) ∫_(1/(n+λ))^m▒〖t^(α-2)/v(t) dt〗) (5)
where〖 t〗^αand v the zygmund moduli of continuity sunch that t^α/v(t) positive and monotonic function.
Keywords
References
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