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Lethargy theorem

{{Short description|Mathematical theorem}}

{{No footnotes|date=April 2023}}

In mathematics, a lethargy theorem is a statement about the distance of points in a metric space from members of a sequence of subspaces; one application in numerical analysis is to approximation theory, where such theorems quantify the difficulty of approximating general functions by functions of special form, such as polynomials. In more recent work, the convergence of a sequence of operators is studied: these operators generalise the projections of the earlier work.

Bernstein's lethargy theorem

Let V_1 \subset V_2 \subset \ldots be a strictly ascending sequence of finite-dimensional linear subspaces of a Banach space X, and let \epsilon_1 \ge \epsilon_2 \ge \ldots be a decreasing sequence of real numbers tending to zero. Then there exists a point x in X such that the distance of x to Vi is exactly \epsilon_i.

See also

References

  • {{cite journal | author=S.N. Bernstein | authorlink=Sergei Natanovich Bernstein | title=On the inverse problem of the theory of the best approximation of continuous functions | journal=Sochinenya | volume=II | year=1938 | pages=292–294 }}
  • {{cite book | author=Elliott Ward Cheney | title=Introduction to Approximation Theory | publisher=American Mathematical Society | edition=2nd | year=1982 | isbn=978-0-8218-1374-4 }}
  • {{cite book | editor1-first=Heinz H. | editor1-last=Bauschke | editor2-first=Regina S. | editor2-last=Burachik |editor2-link=Regina Burachik | editor3-first=Patrick L. | editor3-last=Combettes | editor4-first=Veit | editor4-last=Elser | editor5-first=D. Russell | editor5-last=Luke | editor6-first=Henry | editor6-last=Wolkowicz | series=Springer Optimization and Its Applications | number=49 | title=Fixed-Point Algorithms for Inverse Problems in Science and Engineering | year=2011 | volume=49 | isbn=9781441995681 | doi=10.1007/978-1-4419-9569-8 }}
  • {{cite journal | title=Slow convergence of sequences of linear operators I: almost arbitrarily slow convergence | author1=Frank Deutsch | author2=Hein Hundal | journal=Journal of Approximation Theory | volume=162 | year=2010 | number=9 | pages=1701–1716 | mr=2718892 | doi=10.1016/j.jat.2010.05.001| doi-access=free }}
  • {{cite journal | title=Slow convergence of sequences of linear operators II: arbitrarily slow convergence | author1=Frank Deutsch | author2=Hein Hundal | journal=Journal of Approximation Theory | volume=162 | year=2010 | number=9 | pages=1717–1738 | mr=2718893 | doi=10.1016/j.jat.2010.05.002| doi-access= }}
  • {{cite journal

| last1 = Badea | first1 = C.

| last2 = Grivaux | first2 = S.

| last3 = Müller | first3 = V.

| doi = 10.1090/S1061-0022-2012-01202-1

| issue = 3

| journal = Algebra i Analiz

| mr = 2896163

| pages = 1–30

| title = The rate of convergence in the method of alternating projections

| volume = 23

| year = 2011| arxiv = 1006.2047

}}

Category:Theorems in approximation theory