Constructive function theory
In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation.{{cite web|url=http://encyclopedia2.thefreedictionary.com/Constructive+Theory+of+Functions|title=Constructive Theory of Functions}}{{SpringerEOM|id=Constructive_theory_of_functions|title=Constructive theory of functions|first=S.A.|last=Telyakovskii}} 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
:
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
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References
- {{Cite book |first=N. I. |last=Achiezer |author-link=Naum Akhiezer|title=Theory of approximation |translator=Charles J. Hyman |publisher=Frederick Ungar Publishing |location=New York |year=1956 }}
- {{cite book|mr=0196340|last=Natanson|first=I. P.|author-link=Isidor Natanson|title=Constructive function theory. Vol. I. Uniform approximation|publisher=Frederick Ungar Publishing Co.|location=New York|year=1964}}
: {{cite book|mr=0196341|last=Natanson|first=I. P.|author-link=Isidor Natanson|title=Constructive function theory. Vol. II. Approximation in mean|publisher=Frederick Ungar Publishing Co.|location=New York|year=1965}}
: {{cite book|mr=0196342|last=Natanson|first=I. P.|author-link=Isidor Natanson|title=Constructive function theory. Vol. III. Interpolation and approximation quadratures|publisher=Ungar Publishing Co.|location=New York|year=1965}}