Lebesgue's lemma

For other uses, see: Lebesgue constant (disambiguation).

In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree of the polynomials are obviously fixed). The Lebesgue constant for polynomials of degree at most n and for the set of n + 1 nodes T is generally denoted by Λ_n(T). These constants are named after Henri Lebesgue.

Definition

We fix the interpolation nodes x₀, …, x_n and an interval [a, b] containing all the interpolation nodes. The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C([a, b]) of all continuous functions on [a, b] to itself. The map X is linear and it is a projection on the subspace Π_n of polynomials of degree n or less.

The Lebesgue constant Λ_n(T) is defined as the operator norm of X. This definition requires us to specify a norm on C([a, b]). The maximum norm is usually the most convenient.

Properties

The Lebesgue constant bounds the interpolation error:

$\Vert f-X(f)\Vert \le ({\mathrm{\Lambda }}_n(T)+1)\Vert f-p^{*}\Vert .$

We will here prove this statement with the maximum norm. Let p^∗ denote the best approximation of f among the polynomials of degree n or less. In other words, p^∗ minimizes ||p − f|| among all p in Π_n. Then

$\Vert f-X(f)\Vert \le \Vert f-p^{\ast }\Vert +\Vert p^{\ast }-X(f)\Vert$

by the triangle inequality. But X is a projection on Π_n, so p^∗ − X(f) =X(p^∗) − X(f) = X(p^∗−f). This finishes the proof. Note that this relation comes also as a special case of Lebesgue's lemma.

In other words, the interpolation polynomial is at most a factor Λ_n(T) + 1 worse than the best possible approximation. This suggests that we look for a set of interpolation nodes with a small Lebesgue constant.

The Lebesgue constant can be expressed in terms of the Lagrange basis polynomials:

$l_j(x):=\prod _{\begin{array}{c}i=0\\ j\ne i\end{array}}^n\frac{x-x_i}{x_j-x_i}$

In fact, we have the Lebesgue function

${\lambda }_n(x)=\sum _{j=0}^n|l_j(x)|.$

and the Lebesgue constant (or Lebesgue number) for the grid is its maximum value

${\mathrm{\Lambda }}_n(T)={\max }_{x\in [a,b]}{\lambda }_n(x)$

Nevertheless, it is not easy to find an explicit expression for Λ_n(T).

Minimal Lebesgue constants

In the case of equidistant nodes, the Lebesgue constant grows exponentially. More precisely, we have the following asymptotic estimate

${\mathrm{\Lambda }}_n(T)\sim \frac{2^{n+1}}{en\log n}\text{as}n\to \mathrm{\infty }.$

On the other hand, the Lebesgue constant grows only logarithmically if Chebyshev nodes are used, since we have

$\frac{2}{\pi }\log (n+1)+a<{\mathrm{\Lambda }}_n(T)<\frac{2}{\pi }\log (n+1)+1,$

where a = 0.9625….

We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation. However, there is an easy (linear) transformation of Chebyshev nodes that gives a better Lebesgue constant. Let t_i denote the i^{th Chebyshev node. Then, define s_i = t_i}⁄cos(^π⁄_2(n+1)). For such nodes:

${\mathrm{\Lambda }}_n(S)<\frac{2}{\pi }\log (n+1)+b,$

where b = 0.7219….

Those nodes are, however, not optimal (i.e. they do not minimize the Lebesgue constants) and the search for an optimal set of nodes (which has already been proved to be unique under some assumptions) is still an intriguing topic in mathematics today. Using a computer, one can approximate the values of the minimal constants, here for the canonical interval [−1,1]:

n	1	2	3	4	5	6	7	8	9
Λn(T)	1.0000	1.2500	1.4229	1.5595	1.6722	1.7681	1.8516	1.9255	1.9917

There are several sets of nodes that minimize, for fixed n, the Lebesgue constant. Though if we assume that we always take −1 and 1 as nodes for interpolation, then such a set is unique. To illustrate this property, we shall see what happens when n = 2 (i.e. we consider 3 interpolation nodes in which case the property is not trivial). One can check that each set of nodes of type (−a,0,a) is optimal when ^√8⁄₃ ≤ a ≤ 1 (we consider only nodes in [−1,1]). If we force the set of nodes to be of the type (−1,b,1), then b must equal 0 (look at the Lebesgue function, whose maximum is the Lebesgue constant).

If we assume that we take −1 and 1 as nodes for interpolation, then as shown by H.-J. Rack, for the case n = 3, the explicit values of the optimal nodes and the explicit value of the minimal Lebesgue constant are known.

The Padua points provide another set of nodes with slow growth (although not as slow as the Chebyshev nodes) and with the additional property of being a unisolvent point set.

Sensitivity of the values of a polynomial

The Lebesgue constants also arise in another problem. Let $p(x)$ be a polynomial of degree $n$ expressed in the Lagrangian form associated with the points in the vector $t$ (i.e. the vector $u$ of its coefficients is the vector containing the values $p(t_i)$). Let $\hat{p}(x)$ be a polynomial obtained by slightly changing the coefficients $u$ of the original polynomial $p(x)$ to $\hat{u}$. Let us consider the inequality:

$\frac{\Vert p-\hat{p}\Vert }{\Vert p\Vert }\le {\mathrm{\Lambda }}_n(T)\frac{\Vert u-\hat{u}\Vert }{\Vert u\Vert }$

This means that the (relative) error in the values of $\hat{p}(x)$ will not be higher than the appropriate Lebesgue constant times the relative error in the coefficients. In this sense, the Lebesgue constant can be viewed as the relative condition number of the operator mapping each coefficient vector $u$ to the set of the values of the polynomial with coefficients $u$ in the Lagrange form. We can actually define such an operator for each polynomial basis but its condition number is greater than the optimal Lebesgue constant for most convenient bases.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Lebesgue constants on MathWorld.