In mathematics, the Whitney inequality gives an upper bound for the error of best approximation of a function by polynomials in terms of the moduli of smoothness. It was first proved by Hassler Whitney in 1957, and is an important tool in the field of approximation theory for obtaining upper estimates on the errors of best approximation.
Statement of the theorem Denote the value of the best uniform approximation of a function f ∈ C ( [ a , b ] ) {\displaystyle f\in C([a,b])} by algebraic polynomials P n {\displaystyle P_{n}} of degree ≤ n {\displaystyle \leq n} by
E n ( f ) [ a , b ] := inf P n ‖ f − P n ‖ C ( [ a , b ] ) {\displaystyle E_{n}(f)_{[a,b]}:=\inf _{P_{n}}{\|f-P_{n}\|_{C([a,b])}}}
The moduli of smoothness of order k {\displaystyle k} of a function f ∈ C ( [ a , b ] ) {\displaystyle f\in C([a,b])} are defined as:
ω k ( t ) := ω k ( t ; f ; [ a , b ] ) := sup h ∈ [ 0 , t ] ‖ Δ h k ( f ; ⋅ ) ‖ C ( [ a , b − k h ] ) for t ∈ [ 0 , ( b − a ) / k ] , {\displaystyle \omega _{k}(t):=\omega _{k}(t;f;[a,b]):=\sup _{h\in [0,t]}\|\Delta _{h}^{k}(f;\cdot )\|_{C([a,b-kh])}\quad {\text{ for }}\quad t\in [0,(b-a)/k],}
ω k ( t ) := ω k ( ( b − a ) / k ) for t > ( b − a ) / k , {\displaystyle \omega _{k}(t):=\omega _{k}((b-a)/k)\quad {\text{ for}}\quad t>(b-a)/k,}
where Δ h k {\displaystyle \Delta _{h}^{k}} is the finite difference of order k {\displaystyle k} . Theorem: [Whitney, 1957] If f ∈ C ( [ a , b ] ) {\displaystyle f\in C([a,b])} , then
E k − 1 ( f ) [ a , b ] ≤ W k ω k ( b − a k ; f ; [ a , b ] ) {\displaystyle E_{k-1}(f)_{[a,b]}\leq W_{k}\omega _{k}\left({\frac {b-a}{k}};f;[a,b]\right)}
where W k {\displaystyle W_{k}} is a constant depending only on k {\displaystyle k} . The Whitney constant W ( k ) {\displaystyle W(k)} is the smallest value of W k {\displaystyle W_{k}} for which the above inequality holds. The theorem is particularly useful when applied on intervals of small length, leading to good estimates on the error of spline approximation.
Proof The original proof given by Whitney follows an analytic argument which utilizes the properties of moduli of smoothness. However, it can also be proved in a much shorter way using Peetre's K-functionals. Let:
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