Hair room

A Haar space , or Haarscher space (named after Alfréd Haar ) is defined as follows in approximation theory :

If linearly independent functions continuous on an interval have the property that each element has at most zeros , then the set is called hair-space. ${\ displaystyle n}$ ${\ displaystyle [a, b]}$ ${\ displaystyle g_ {1}, \ dots, g_ {n}}$ ${\ displaystyle {f} \ in \ mathrm {span} \ left \ {g_ {1}, \ dots, g_ {n} \ right \}, f \ neq 0}$ ${\ displaystyle [a, b]}$ ${\ displaystyle (n-1)}$ ${\ displaystyle U: = \ mathrm {span} \ left \ {g_ {1}, \ dots, g_ {n} \ right \}}$

A system of such functions that span a hair space is also called the Haar's system or Chebyshev system . If a continuous function is approximated by elements of a hair space, there is always exactly one best approximation with respect to the maximum norm . ${\ displaystyle g_ {1}, \ dots, g_ {n}}$ ${\ displaystyle \ | \ cdot \ | _ {\ infty}}$

Interpolation in hair spaces

If you have different points (support points) and data in pairs , there is exactly one with . This is equivalent to the regularity of the Vandermonde matrix . ${\ displaystyle x_ {1}, \ dots, x_ {n} \ in [a, b]}$ ${\ displaystyle y_ {i}, i = 1, \ dots, n}$ ${\ displaystyle g \ in U}$ ${\ displaystyle g (x_ {i}) = y_ {i}, i = 1, \ dots, n}$

Proof The map is linear . Because each has at most n-1 zeros, the core of the map is just the null function , i.e. H. L is injective . Because of , L is surjective , i.e. bijective as a whole . From this follows the existence and uniqueness of the interpolation function g.

{\ displaystyle L \ colon U \ to [a, b] ^ {n}, f \ to \ left (f (x_ {1}), ... f (x_ {n}) \ right)}

{\ displaystyle f \ in U}

{\ displaystyle dim [a, b] ^ {n} = n}

Examples

The vector space of the polynomials at most nth degree is a hair space. is a Haar system. ${\ displaystyle \ Pi _ {n}}$ ${\ displaystyle \ left \ {1, x, \ dots, x ^ {n} \ right \}}$

However, the system is not a Haarscher room.

{\ displaystyle \ left \ {x, \ dots, x ^ {n} \ right \}}

The trigonometric polynomials form a Haar space with the Haar system (polynomials in ).

{\ displaystyle \ left \ {1, e ^ {ix}, \ dots, e ^ {i (n-1) x} \ right \}}

{\ displaystyle e ^ {ix}}

{\ displaystyle \ left \ {1, \ sin (x), \ cos (x), \ dots, \ sin (nx), \ cos (nx) \ right \}, \ left \ {1, \ sin (x ), \ dots, \ sin (nx) \ right \}, \ left \ {1, \ cos (x), \ dots, \ cos (nx) \ right \}, \; x \ in [0.2 \ pi)}

are both Haar systems.

history

For the first time, Haar formulated and proved the Haar condition in 1918 in: The Minkowski Geometry and the Approach to Continuous Functions , Mathematische Annalen , Volume 78, Pages 294-311. Other proofs formulated by Vlastimil Pták 1958 ( A remark on approximation of continuous functions in Czechoslovak Math. Journal, Volume 8, Pages 251-256) and Singer 1960 ( On best approximation of continuous functions in Mathematische Annalen, Volume 140, Pages 165-168) .

literature

Günther Hämmerlin, Karl-Heinz Hoffmann: Numerical Mathematics. Springer, Berlin 1994, ISBN 3-540-58033-6

Individual evidence

^ Elliot Ward Cheney: Introduction to Approximation Theory , McGraw-Hill Book Company, 1966, Library of Congress Catalog Card Number 65-25916, ISBN 007-010757-2 , pages 227 + 242 + 248 + 251

[Cheney-1] Elliot Ward Cheney: Introduction to Approximation Theory , McGraw-Hill Book Company, 1966, Library of Congress Catalog Card Number 65-25916, ISBN 007-010757-2 , pages 227 + 242 + 248 + 251