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Constructive function theory

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In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Bernstein.

Example

Let f be a 2π-periodic function. Then f is α-Hölder for some 0 < α < 1 if and only if for every natural n there exists a trigonometric polynomial Pn of degree n such that

max 0 ≤ x ≤ 2 π | f ( x ) − P n ( x ) | ≤ C ( f ) n α , {\displaystyle \max _{0\leq x\leq 2\pi }|f(x)-P_{n}(x)|\leq {\frac {C(f)}{n^{\alpha }}},}

where C(f) is a positive number depending on f. The "only if" is due to Dunham Jackson, see Jackson's inequality; the "if" part is due to Sergei Bernstein, see Bernstein's theorem (approximation theory).

Notes

  1. "Constructive Theory of Functions".
  2. Telyakovskii, S.A. (2001) , "Constructive theory of functions", Encyclopedia of Mathematics, EMS Press

References

  • Achiezer, N. I. (1956). Theory of approximation. Translated by Charles J. Hyman. New York: Frederick Ungar Publishing.
  • Natanson, I. P. (1964). Constructive function theory. Vol. I. Uniform approximation. New York: Frederick Ungar Publishing Co. MR 0196340.
Natanson, I. P. (1965). Constructive function theory. Vol. II. Approximation in mean. New York: Frederick Ungar Publishing Co. MR 0196341.
Natanson, I. P. (1965). Constructive function theory. Vol. III. Interpolation and approximation quadratures. New York: Ungar Publishing Co. MR 0196342.
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