POLYNOMİALS

Presented by : Nashat Al-ghrairiSubject : MathematicSupervisor : Assoc. prof MURAT AlTEKINDATE : 23, December 2015

POLYNOMİALS

Chebyshev Polynomials

Hermite Polynomials

CHEBYSHEV POLYNOMİALS In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and Chebyshev polynomials of the second kind which are denoted Un. The letter T is used because of the alternative transliterations of the name Chebyshev as T chebycheff, T chebyshev (French) or T schebyschow (German).The Chebyshev polynomials Tn or Un are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.Chebyshev polynomials are polynomials with the largest possible leading coefficient, but subject to the condition that their absolute value on the interval [-1,1] is bounded by 1. They are also the extremal polynomials for many other properties.[2]

Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw–Curtis quadrature.

CHEBYSHEV POLYNOMIALS

Differential Equation and Its Solution

The Chebyshev differential equation is written as(1 − ) If we let x = cos t we obtain + y = 0

whose general solution isy = A cos nt + B sin ntor as

y = A cos(nx) + B sin(n x) |x| < 1

or equivalently

y = ATn (x) + BUn (x) |x| < 1where Tn (x) and Un (x) are defined as Chebyshev polynomials of the first and second kind of degree n, respectively

CHEBYSHEV POLYNOMIALSIf we let x = cosh t we obtain− whose general solution isy = A cosh nt + B sinh ntor asy = A cosh(n cosh−1 x) + B sinh(n cosh−1 x) |x| > 1

or equivalently

y = ATn (x) + BUn (x) |x| > 1

The function Tn (x) is a polynomial. For |x| < 1 we have

(x) + i (x) = (cos t + i sin t = ( x +i (x) i (x) = (cos t i sin t = ( x i

CHEBYSHEV POLYNOMIALSfrom which we obtain (x)=[( x +i ]For |x| > 1 we haveTn (x) + Un (x) = = (x

Tn (x) − Un (x) = = (x The sum of the last two relationships give the same result for Tn (x).

Chebyshev Polynomials of the First Kind of Degree n The Chebyshev polynomials Tn (x) can be obtained by means of Rodrigue’s formula

(x)=(1- n=0,1,2,3,……….

CHEBYSHEV POLYNOMIALSThe first twelve Chebyshev polynomials are listed in Table 1 and then as powers of x in terms of Tn (x) in Table 2.Table 1: Chebyshev Polynomials of the First Kind (x)=1

(x)=2 (x)=4 (x)=8 (x)=16 (x)=32 (x)=6 (x)=1 (x)=2 (x)=5 5 (x)=1 220

CHEBYSHEV POLYNOMIALSTable 2: Powers of x as functions of Tn (x)1= x=()=(3+)=(10+)=(10++)=(3+7+)=(3+28++)=(126+36++)=(126+120+++)=(4+1+++)

CHEBYSHEV POLYNOMIALSGenerating Function for Tn(x)

The Chebyshev polynomials of the first kind can be developed by means of the generating function= Recurrence Formulas for Tn(x)When the first two Chebyshev polynomials T0 (x) and T1 (x) are known, all other polynomials Tn (x), n ≥ 2 can be obtained by means of the recurrence formula (x) = 2x (x) −n (x)The derivative of Tn (x) with respect to x can be obtained from(1 − )(x)=(x)+n (x)

CHEBYSHEV POLYNOMIALSSpecial Values of Tn(x)The following special values and properties of Tn (x) are often useful: (−x) = (−1 (x) (0) = (−1 (1) = 1 +1 (0) = 0 (−1) = (−1

Orthogonality Property of (x)We can determine the orthogonality properties for the Chebyshev polynomials of the first kind from our knowledge of the orthogonality of the cosine functions, namely,

CHEBYSHEV POLYNOMIALSThen substituting

(x) = cos(nθ)cos θ = x

to obtain the orthogonality properties of the Chebyshev polynomials: We observe that the Chebyshev polynomials form an orthogonal set on the interval −1 ≤ x ≤ 1 with the weighting function (1 −

CHEBYSHEV POLYNOMIALSOrthogonal Series of Chebyshev Polynomials

An arbitrary function f (x) which is continuous and single-valued, defined over the interval −1 ≤ x ≤ 1, can be expanded as a series of Chebyshev polynomials

f (x) = (x) + (x) + (x) + . . . = (x)where the coefficients An are given by

= And =

CHEBYSHEV POLYNOMIALSThe following definite integrals are often useful in the series expansion of f (x): 0 0 Chebyshev Polynomials Over a Discrete Set of Points A continuous function over a continuous interval is often replaced by a set of discrete values of the function at discrete points. It can be shown that the Chebyshev polynomials Tn (x) are orthogonal over the following discrete set of N + 1 points xi, equally spaced on θ,=0,, ……(N , where xi = arccos θi

CHEBYSHEV POLYNOMIALS

We have

The Tm(x) are also orthogonal over the following N points ti equally spaced

=0,, , = arccos=

CHEBYSHEV POLYNOMIALSThe set of points ti are clearly the midpoints in θ of the first case. The unequal spacing of the points in xi(N ti) compensates for the weight factor

W(x)= (1 −

in the continuous case.

Additional Identities of Chebyshev Polynomials The Chebyshev polynomials are both orthogonal polynomials and the trigonometric cos nx functions in disguise, therefore they satisfy a large number of useful relationships. The differentiation and integration properties are very important in analytical and numerical work. We begin with

(x) = cos[(n + 1) x] and (x) = cos[(n - 1) x]

CHEBYSHEV POLYNOMIALSDifferentiating both expressions gives

And

Subtracting the last two expressions yields

Or

CHEBYSHEV POLYNOMIALSTherefore  (x)=4 (x)= (x)=We have the formulas for the differentiation of Chebyshev polynomials, therefore these formulas can be used to develop integration for the Chebyshev polynomials 

CHEBYSHEV POLYNOMIALS

The Shifted Chebyshev Polynomials For analytical and numerical work it is often convenient to use the half interval 0 ≤ x ≤ 1instead of the full interval −1 ≤ x ≤ 1. For this purpose the shifted Chebyshev polynomialsare defined: (x)= *(2x-1)Thus we have for the first few polynomials(x)=1 1(x)=8(x)=32(x)=1

CHEBYSHEV POLYNOMIALSand the following powers of x as functions of (x); 1=x()=(3+)=(10++)=(3+2+ +)The recurrence relationship for the shifted polynomials is:Or

where

CHEBYSHEV POLYNOMIALSExpansion of in a Series of (x) A method of expanding in a series of Chebyshev polynomials employes the recurrence relation written as

x (x) = (x)To illustrate the method, consider

This result is consistent with the expansion of x4 given in Table 2

CHEBYSHEV POLYNOMIALSApproximation of Functions by Chebyshev Polynomials Sometimes when a function f (x) is to be approximated by a polynomial of the form

where | (x)| does not exceed an allowed limit, it is possible to reduce the degree of the polynomial by a process called economization of power series. The procedure is to convert the polynomial to a linear combination of Chebyshev polynomials:

It may be possible to drop some of the last terms without permitting the error to exceed the prescribed limit. Since |Tn (x)| ≤ 1, the number of terms which can be omitted is determined by the magnitude of the coefficient b.

CHEBYSHEV POLYNOMIALSThe Chebyshev polynomials are useful in numerical work for the interval −1 ≤ x ≤ 1 because1. |Tn (x)] ≤ 1 within −1 ≤ x ≤ 12. The maxima and minima are of comparable value3. The maxima and minima are spread reasonably uniformly over the interval −1 ≤ x ≤ 14. All Chebyshev polynomials satisfy a three term recurrence relation5. They are easy to compute and to convert to and from a power series form.These properties together produce an approximating polynomial which minimizes error in its application. This is different from the least squares approximation where the sum of the squares of the errors is minimized; the maximum error itself can be quite large. In the Chebyshev approximation, the average error can be large but the maximum error is minimized. Chebyshev approximations of a function are sometimes said to be mini-max approximations of the function.

CHEBYSHEV POLYNOMIALSThe following table gives the Chebyshev polynomial approximation of several power series.

Table 3: Power Series and its Chebyshev Approximation

CHEBYSHEV POLYNOMIALS

CHEBYSHEV POLYNOMIALSTable 4: Formulas for Economization of Power Series

CHEBYSHEV POLYNOMIALSFor easy reference the formulas for economization of power series in terms of Chebyshev are given in Table 4.

Assigned ProblemsProblem Set for Chebyshev Polynomials1-Obtain the first three Chebyshev polynomials (x), (x) and (x) by means of the Rodrigue’s formula2. Show that the Chebyshev polynomial (x) is a solution of Chebyshev’s equation of order 3.3. By means of the recurrence formula obtain Chebyshev polynomials (x) and (x) given (x) and (x).4. Show that (1) = 1 and (−1) = (−15. Show that (0) = 0 if n is odd and (−1 if n is even.6. Setting x = cos θ show that

CHEBYSHEV POLYNOMIALS

Where i=7. Find the general solution of Chebyshev’s equation for n = 08. Obtain a series expansion for f (x) = in terms of Chebyshev polynomials (x),

9. Express x4 as a sum of Chebyshev polynomials of the first kind

HERMITE POLYNOMIALSIn mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator. They are named in honor of Charles Hermite.

DefinitionThe Hermite polynomials are defined either by

(the "probabilists' Hermite polynomials"), or sometimes by

HERMITE POLYNOMIALS(the "physicists' Hermite polynomials"). These two definitions are not exactly equivalent; either is a rescaling of the other, to wit

These are Hermite polynomial sequences of different variances; see the material on variances below. Below, we usually follow the first convention. That convention is often preferred by probabilists because

is the probability density function for the normal distribution with expected value 0 and standard deviation1.

HERMITE POLYNOMIALS

HERMITE POLYNOMIALSThe first eleven probabilists' Hermite polynomials are:

HERMITE POLYNOMIALS

HERMITE POLYNOMIALSthe first ten physicists' Hermite polynomials are:

HERMITE POLYNOMIALSPropertiesHn is a polynomial of degree n. The probabilists' version has leading coefficient 1, while the physicists' version has leading coefficient 2n.Orthogonality Hn(x) is an nth-degree polynomial for n = 0, 1, 2, 3, .... These polynomials are respect to Orthogonality with the weight function (measuer)

W(x)= (probabilist)orW(x)= (physicist) i.e., we have(x) dx=n! (probabilist)

HERMITE POLYNOMIALSOR (x) dx=n! (physicist) The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.

CompletenessThe Hermite polynomials (probabilist or physicist) form an orthogonal basis of the Hilbert space of functions satisfying w(x) dxin which the inner product is given by the integral including the Gaussian weight function w(x) defined in the preceding section,

An orthogonal basis for L2(R, w(x) dx) is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function ƒ ∈ L2(R, w(x) dx) orthogonal to all functions in the system. Since the linear span of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that if ƒ satisfies

HERMITE POLYNOMIALSdx =0for every n ≥ 0, then ƒ = 0. One possible way to do it is to see that the entire function

F(z) = dx =0vanishes identically. The fact that F(it) = 0 for every t real means that the Fourier transform of ƒ(x) exp(−x2) is 0, hence ƒ is 0 almost everywhere. Variants of the above completeness proof apply to other weights with exponential decay. In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see "Completeness relation" below).An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for L2(R, w(x) dx) consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for L2(R).

Hermite's differential equation The probabilists' Hermite polynomials are solutions of the differential equation

ù)`+λ u=0where λ is a constant, with the boundary conditions that u should be polynomially bounded at infinity. With these boundary conditions, the equation has solutions only if λ is a positive integer, and up to an overall scaling, the solution is uniquely given by u(x) = Hλ(x). Rewriting the differential equation as an eigenvalue problem

 

HERMITE POLYNOMIALSL(u)=Ű solutions are the eigenfunctions of the differential operator L. This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation

Ű whose solutions are the physicists' Hermite polynomials. With more general boundary conditions, the Hermite polynomials can be generalized to obtain more general analytic functions Hλ(z) for λ a complex index. An explicit formula can be given in terms of a contour integral (Courant & Hilbert 1953).Recursion relationThe sequence of Hermite polynomials also satisfies the recursion(x)=x(x) (x). (probabilist)

(x)=2x(x) (x). (physicist)

HERMITE POLYNOMIALSThe Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity

(x)=n(x) (probabilist)

(x)=2n(x) (physicist)

or equivalently(x+y)=(y) (probabilist)

(x+y)=(2y (physicist)Generating functionThe Hermite polynomials are given by the exponential generating function

(probabilist)

HERMITE POLYNOMIALS (physicist)

This equality is valid for all x, t complex, and can be obtained by writing the Taylor expansion at x of the entire function z → exp(−z2) (in physicist's case). One can also derive the (physicist's) generating function by using Cauchy's Integral Formula to write the Hermite polynomials as

Using this in the sum , one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function

Expected value

(probabilist)