Unisolvent functions

<h2 id="examples">Examples</h2>
<ul><li>1, x, x2 is unisolvent on any interval by the unisolvence theorem</li>
<li>1, x2 is unisolvent on [0, 1], but not unisolvent on [−1, 1]</li>
<li>1, cos(x), cos(2x), ..., cos(nx), sin(x), sin(2x), ..., sin(nx) is unisolvent on [−π, π]</li>
<li>Unisolvent functions are used in <a href="/facts/Linear_inverse_problem/Fe01pQ9E">linear inverse problems</a>.</li></ul>
<h2 id="unisolvence-in-the-finite-element-method">Unisolvence in the finite element method</h2>
When using "simple" functions to approximate an unknown function, such as in the <a href="/facts/Finite_element_method/eUcfP1vn">finite element method</a>, it is useful to consider a set of functionals 
 
 
 
 {
 
 f
 
 i
 
 
 
 }
 
 i
 =
 1
 
 
 n
 
 
 
 
 {\displaystyle \{f_{i}\}_{i=1}^{n}}
 
 that act on a finite dimensional <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
 
 
 
 
 V
 
 h
 
 
 
 
 {\displaystyle V_{h}}
 
 of functions, usually polynomials. Often, the functionals are given by evaluation at points in <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> or some subset of it.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
For example, let 
 
 
 
 
 V
 
 h
 
 
 =
 
 
 {
 
 
 p
 (
 x
 )
 =
 
 ∑
 
 k
 =
 0
 
 
 n
 
 
 
 p
 
 k
 
 
 
 x
 
 k
 
 
 
 
 }
 
 
 
 
 {\displaystyle V_{h}={\big \{}p(x)=\sum _{k=0}^{n}p_{k}x^{k}{\big \}}}
 
 be the space of univariate polynomials of degree 
 
 
 
 n
 
 
 {\displaystyle n}
 
 or less, and let 
 
 
 
 
 f
 
 k
 
 
 (
 p
 )
 :=
 f
 
 
 (
 
 
 
 
 k
 n
 
 
 
 
 )
 
 
 
 
 {\displaystyle f_{k}(p):=f{\Big (}{\frac {k}{n}}{\Big )}}
 
 for 
 
 
 
 0
 ≤
 k
 ≤
 n
 
 
 {\displaystyle 0\leq k\leq n}
 
 be defined by evaluation at 
 
 
 
 n
 +
 1
 
 
 {\displaystyle n+1}
 
 equidistant points on the unit interval 
 
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle [0,1]}
 
. In this context, the unisolvence of 
 
 
 
 
 V
 
 h
 
 
 
 
 {\displaystyle V_{h}}
 
 with respect to 
 
 
 
 {
 
 f
 
 k
 
 
 
 }
 
 k
 =
 1
 
 
 n
 
 
 
 
 {\displaystyle \{f_{k}\}_{k=1}^{n}}
 
 means that 
 
 
 
 {
 
 f
 
 k
 
 
 
 }
 
 k
 =
 1
 
 
 n
 
 
 
 
 {\displaystyle \{f_{k}\}_{k=1}^{n}}
 
 is a basis for 
 
 
 
 
 V
 
 h
 
 
 ∗
 
 
 
 
 {\displaystyle V_{h}^{*}}
 
, the dual space of 
 
 
 
 
 V
 
 h
 
 
 
 
 {\displaystyle V_{h}}
 
. Equivalently, and perhaps more intuitively, unisolvence here means that given any set of values 
 
 
 
 {
 
 c
 
 k
 
 
 
 }
 
 k
 =
 1
 
 
 n
 
 
 
 
 {\displaystyle \{c_{k}\}_{k=1}^{n}}
 
, there exists a unique polynomial 
 
 
 
 q
 (
 x
 )
 ∈
 
 V
 
 h
 
 
 
 
 {\displaystyle q(x)\in V_{h}}
 
 such that 
 
 
 
 
 f
 
 k
 
 
 (
 q
 )
 =
 q
 (
 
 
 
 k
 n
 
 
 
 )
 =
 
 c
 
 k
 
 
 
 
 {\displaystyle f_{k}(q)=q({\tfrac {k}{n}})=c_{k}}
 
. Results of this type are widely applied in <a href="/facts/Polynomial_interpolation/DmnuxaKT">polynomial interpolation</a>; given any function on 
 
 
 
 ϕ
 ∈
 C
 (
 [
 0
 ,
 1
 ]
 )
 
 
 {\displaystyle \phi \in C([0,1])}
 
, by letting 
 
 
 
 
 c
 
 k
 
 
 =
 ϕ
 (
 
 
 
 k
 n
 
 
 
 )
 
 
 {\displaystyle c_{k}=\phi ({\tfrac {k}{n}})}
 
, we can find a polynomial 
 
 
 
 q
 ∈
 
 V
 
 h
 
 
 
 
 {\displaystyle q\in V_{h}}
 
 that interpolates 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 at each of the 
 
 
 
 n
 +
 1
 
 
 {\displaystyle n+1}
 
 points: . 
 
 
 
 ϕ
 (
 
 
 
 k
 n
 
 
 
 )
 =
 q
 (
 
 
 
 k
 n
 
 
 
 )
 ,
  
 ∀
 k
 ∈
 {
 0
 ,
 1
 ,
 .
 .
 ,
 n
 }
 
 
 {\displaystyle \phi ({\tfrac {k}{n}})=q({\tfrac {k}{n}}),\ \forall k\in \{0,1,..,n\}}

<h2 id="dimensions">Dimensions</h2>
Systems of unisolvent functions are much more common in 1 dimension than in higher dimensions. In dimension d = 2 and higher (Ω ⊂ Rd), the functions f1, f2, ..., fn cannot be unisolvent on Ω if there exists a single open set on which they are all continuous. To see this, consider moving points x1 and x2 along continuous paths in the open set until they have switched positions, such that x1 and x2 never intersect each other or any of the other xi. The determinant of the resulting system (with x1 and x2 swapped) is the negative of the determinant of the initial system. Since the functions fi are continuous, the <a href="/facts/Intermediate_value_theorem/SaFdYE7W">intermediate value theorem</a> implies that some intermediate configuration has determinant zero, hence the functions cannot be unisolvent.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Inverse_problem/Fe01pQ9E">Inverse problem</a></li></ul>

<ul><li><a href="/facts/Philip_J._Davis/VMXtMMpJ">Davis, Philip J.</a> (2014) [1975]. "2. Interpolation §2.4 Unisolvence". <a href="https://books.google.com/books?id=2PaJAwAAQBAJ&pg=PR11">Interpolation and Approximation</a>. Dover. pp. 31–32. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-62495-2. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/886524559">886524559</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Brenner, Susanne C.; Scott, L. Ridgway (2008). "The Mathematical Theory of Finite Element Methods". Texts in Applied Mathematics. 15. doi:10.1007/978-0-387-75934-0. ISBN 978-0-387-75933-3. ISSN 0939-2475. <a href="978-0-387-75933-3" target="_blank">978-0-387-75933-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Ern, Alexandre; Guermond, Jean-Luc (2004). "Theory and Practice of Finite Elements". Applied Mathematical Sciences. 159. doi:10.1007/978-1-4757-4355-5. ISBN 978-1-4419-1918-2. ISSN 0066-5452. <a href="978-1-4419-1918-2" target="_blank">978-1-4419-1918-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Unisolvent functions open-in-new

Unisolvent functions