The Chebyshev polynomials are two sequences of
polynomials related to the
cosine and sine functions, notated as
and
. They can be defined in several equivalent ways, one of which starts with
trigonometric functions:
The Chebyshev polynomials of the first kind
are defined by
:
Similarly, the Chebyshev polynomials of the second kind
are defined by
:
That these expressions define polynomials in
may not be obvious at first sight, but follows by rewriting
and
using
de Moivre's formula or by using the
angle sum formulas for
and
repeatedly. For example, the
double angle formulas
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
, which follow directly from the angle sum formulas, may be used to obtain
and
, which are respectively a polynomial in
and a polynomial in
multiplied by
. Hence
and
.
An important and convenient property of the is that they are
''orthogonal'' with respect to the
inner product:
:
and are orthogonal with respect to another, analogous inner product, given below.
The Chebyshev polynomials are polynomials with the largest possible leading coefficient whose
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
on the
interval is bounded by 1. They are also the "extremal" polynomials for many other properties.
Chebyshev polynomials are important in
approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' ...
because the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of , which are also called ''
Chebyshev nodes'', are used as matching points for optimizing
polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset.
Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no ...
. The resulting interpolation polynomial minimizes the problem of
Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a
continuous function under the
maximum norm, also called the "
minimax" criterion. This approximation leads directly to the method of
Clenshaw–Curtis quadrature
Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables x = \cos ...
.
These polynomials were named after
Pafnuty Chebyshev. The letter is used because of the alternative
transliterations of the name ''Chebyshev'' as , (French) or (German).
Definitions
Recurrence definition
The Chebyshev polynomials of the first kind are obtained from the
recurrence relation
:
The
ordinary generating function for is
:
There are several other
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
s for the Chebyshev polynomials; the
exponential generating function is
:
The generating function relevant for 2-dimensional
potential theory and
multipole expansion is
:
The Chebyshev polynomials of the second kind are defined by the recurrence relation
:
Notice that the two sets of recurrence relations are identical, except for
vs.
The ordinary generating function for is
:
and the exponential generating function is
:
Trigonometric definition
As described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying
:
or, in other words, as the unique polynomials satisfying
:
for which as a technical point is a variant (equivalent transpose) of
Schröder's equation. That is, is functionally conjugate to , codified in the nesting property below.
The polynomials of the second kind satisfy:
:
or
:
which is structurally quite similar to the
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac,
where is any nonn ...
:
:
(The Dirichlet kernel, in fact, coincides with what is now known as the
Chebyshev polynomial of the fourth kind.)
That is an th-
degree polynomial in can be seen by observing that is the
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of one side of
de Moivre's formula. The real part of the other side is a polynomial in and , in which all powers of are
even
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
**Even language, a language spoken by the Evens
* Odd and Even, a solitaire game wh ...
and thus replaceable through the identity .
By the same reasoning, is the
imaginary part of the polynomial, in which all powers of are
odd and thus, if one factor of is factored out, the remaining factors can be replaced to create a st-degree polynomial in .
The identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integer multiple of an
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
solely in terms of the cosine of the base angle.
The first two Chebyshev polynomials of the first kind are computed directly from the definition to be
:
and
:
while the rest may be evaluated using a specialization of the
product-to-sum identity
:
as, for example,
:
Conversely, an arbitrary
integer power of trigonometric functions may be expressed as a linear combination of trigonometric functions using Chebyshev polynomials
:
where the prime at the summation symbol indicates that the contribution of needs to be halved if it appears, and
.
An immediate corollary is the expression of
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
exponentiation in terms of Chebyshev polynomials: given ,
:
Commuting polynomials definition
Chebyshev polynomials can also be characterized by the following theorem:
If
is a family of monic polynomials with coefficients in a field of characteristic
such that
and
for all
and
, then, up to a simple change of variables, either
for all
or
for all
.
Pell equation definition
The Chebyshev polynomials can also be defined as the solutions to the
Pell equation
:
in a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
. Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:
:
Relations between the two kinds of Chebyshev polynomials
The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of
Lucas sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation
: x_n = P \cdot x_ - Q \cdot x_
where P and Q are fixed integers. Any sequence satisfying this r ...
s and with parameters and :
:
It follows that they also satisfy a pair of mutual recurrence equations:
https://resolver.caltech.edu/CaltechAUTHORS:20140123-104529738] (xvii+1 errata page+396 pages, red cloth hardcover) (NB. Copyright was renewed by California Institute of Technology
The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...
in 1981.); Reprint: Robert E. Krieger Publishing Co., Inc., Melbourne, Florida, USA. 1981. ; Planned Dover reprint: .
:
The second of these may be rearranged using the
recurrence definition for the Chebyshev polynomials of the second kind to give
:
Using this formula iteratively gives the sum formula
:
while replacing
and
using the
derivative formula for
gives the recurrence relationship for the derivative of
:
:
This relationship is used in the
Chebyshev spectral method of solving differential equations.
Turán's inequalities for the Chebyshev polynomials are
:
The
integral relations are
:
where integrals are considered as principal value.
Explicit expressions
Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:
:
with inverse
:
where the prime at the summation symbol indicates that the contribution of needs to be halved if it appears.
:
where is a
hypergeometric function.
Properties
Symmetry
:
That is, Chebyshev polynomials of even order have
even symmetry and therefore contain only even powers of . Chebyshev polynomials of odd order have
odd symmetry and therefore contain only odd powers of .
Roots and extrema
A Chebyshev polynomial of either kind with degree has different
simple roots, called Chebyshev roots, in the interval . The roots of the Chebyshev polynomial of the first kind are sometimes called
Chebyshev nodes because they are used as ''nodes'' in polynomial interpolation. Using the trigonometric definition and the fact that
:
one can show that the roots of are
:
Similarly, the roots of are
:
The
extrema of on the interval are located at
:
One unique property of the Chebyshev polynomials of the first kind is that on the interval all of the
extrema have values that are either −1 or 1. Thus these polynomials have only two finite
critical value
Critical value may refer to:
*In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) Æ’(''x'') in ''N'' of a critical point ''x'' in ''M''.
*In statistical hypothesis ...
s, the defining property of
Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:
:
:
:
:
The
extrema of
on the interval
where
are located at
values of
. They are
, or
where
,
,
and
, i.e.,
and
are relatively prime numbers.
Specifically,
when
is even,
*
if
, or
and
is even. There are
such values of
.
*
if
and
is odd. There are
such values of
.
When
is odd,
*
if
, or
and
is even. There are
such values of
.
*
if
, or
and
is odd. There are
such values of
.
This result has been generalized to solutions of
,
and to
and
for Chebyshev polynomials of the third and fourth kinds, respectively.
Differentiation and integration
The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it can be shown that:
:
The last two formulas can be numerically troublesome due to the division by zero (
indeterminate form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...
, specifically) at and . It can be shown that:
:
More general formula states:
:
which is of great use in the numerical solution of
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
problems.
Also, we have
:
where the prime at the summation symbols means that the term contributed by is to be halved, if it appears.
Concerning integration, the first derivative of the implies that
:
and the recurrence relation for the first kind polynomials involving derivatives establishes that for
:
The last formula can be further manipulated to express the integral of as a function of Chebyshev polynomials of the first kind only:
:
Furthermore, we have
:
Products of Chebyshev polynomials
The Chebyshev polynomials of the first kind satisfy the relation
:
which is easily proved from the
product-to-sum formula for the cosine,
:
For this results in the already known recurrence formula, just arranged differently, and with it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest ) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:
:
The polynomials of the second kind satisfy the similar relation
:
(with the definition by convention ).
They also satisfy
:
for .
For this recurrence reduces to
:
which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether starts with 2 or 3.
Composition and divisibility properties
The trigonometric definitions of and imply the composition or nesting properties
:
For the order of composition may be reversed, making the family of polynomial functions a
commutative semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'' ...
under composition.
Since is divisible by if is odd, it follows that is divisible by if is odd. Furthermore, is divisible by , and in the case that is even, divisible by .
Orthogonality
Both and form a sequence of
orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight
:
on the interval , i.e. we have:
:
This can be proven by letting and using the defining identity .
Similarly, the polynomials of the second kind are orthogonal with respect to the weight
:
on the interval , i.e. we have:
:
(The measure is, to within a normalizing constant, the
Wigner semicircle distribution
The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0):
:f(x)=\sq ...
.)
These orthogonality properties follow from the fact that the Chebyshev polynomials solve the
Chebyshev differential equations
:
:
which are
Sturm–Liouville differential equations. It is a general feature of such
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to
those equations.)
The also satisfy a discrete orthogonality condition:
:
where is any integer greater than , and the are the
Chebyshev nodes (see above) of :
:
For the polynomials of the second kind and any integer with the same Chebyshev nodes , there are similar sums:
:
and without the weight function:
:
For any integer , based on the zeros of :
:
one can get the sum:
:
and again without the weight function:
:
Minimal -norm
For any given , among the polynomials of degree with leading coefficient 1 (
monic polynomials),
:
is the one of which the maximal absolute value on the interval
��1, 1is minimal.
This maximal absolute value is
:
and reaches this maximum exactly times at
:
Remark
By the
equioscillation theorem
In mathematics, the equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attributed to Chebyshev.
Statement
Let f be a ...
, among all the polynomials of degree , the polynomial minimizes on
if and only if there are points such that .
Of course, the null polynomial on the interval can be approximated by itself and minimizes the -norm.
Above, however, reaches its maximum only times because we are searching for the best polynomial of degree (therefore the theorem evoked previously cannot be used).
Chebyshev polynomials as special cases of more general polynomial families
The Chebyshev polynomials are a special case of the ultraspherical or
Gegenbauer polynomials , which themselves are a special case of the
Jacobi polynomials :
:
Chebyshev polynomials are also a special case of
Dickson polynomials:
::
::
In particular, when
, they are related by
and
.
Other properties
The curves given by , or equivalently, by the parametric equations , , are a special case of
Lissajous curve
A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations
: x=A\sin(at+\delta),\quad y=B\sin(bt),
which describe the superposition of two perpendicular oscillations in x and y dire ...
s with frequency ratio equal to .
Similar to the formula
:
we have the analogous formula
:
For ,
:
and
:
which follows from the fact that this holds by definition for .
Examples
First kind
The first few Chebyshev polynomials of the first kind are
:
Second kind
The first few Chebyshev polynomials of the second kind are
:
As a basis set
In the appropriate
Sobolev space, the set of Chebyshev polynomials form an
orthonormal basis, so that a function in the same space can, on , be expressed via the expansion:
:
Furthermore, as mentioned previously, the Chebyshev polynomials form an
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
basis which (among other things) implies that the coefficients can be determined easily through the application of an
inner product. This sum is called a Chebyshev series or a Chebyshev expansion.
Since a Chebyshev series is related to a
Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
have a Chebyshev counterpart.
[ These attributes include:
* The Chebyshev polynomials form a ]complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
orthogonal system.
* The Chebyshev series converges to if the function is piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. ...
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
and continuous. The smoothness requirement can be relaxed in most cases as long as there are a finite number of discontinuities in and its derivatives.
* At a discontinuity, the series will converge to the average of the right and left limits.
The abundance of the theorems and identities inherited from Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis function ...
,[ often in favor of trigonometric series due to generally faster convergence for continuous functions ( Gibbs' phenomenon is still a problem).
]
Example 1
Consider the Chebyshev expansion of . One can express
:
One can find the coefficients either through the application of an inner product or by the discrete orthogonality condition. For the inner product,
:
which gives
:
Alternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for ''approximate'' coefficients,
:
where is the Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
function and the are the Gauss–Chebyshev zeros of :
:
For any , these approximate coefficients provide an exact approximation to the function at with a controlled error between those points. The exact coefficients are obtained with , thus representing the function exactly at all points in . The rate of convergence depends on the function and its smoothness.
This allows us to compute the approximate coefficients very efficiently through the discrete cosine transform
:
Example 2
To provide another example:
:
Partial sums
The partial sums of
:
are very useful in the approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix '' ...
of various functions and in the solution of differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s (see spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis function ...
). Two common methods for determining the coefficients are through the use of the inner product as in Galerkin's method and through the use of collocation
In corpus linguistics, a collocation is a series of words or terms that co-occur more often than would be expected by chance. In phraseology, a collocation is a type of compositional phraseme, meaning that it can be understood from the words ...
which is related to interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one often has a ...
.
As an interpolant, the coefficients of the st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:
:
Polynomial in Chebyshev form
An arbitrary polynomial of degree can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial is of the form
:
Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.
Families of polynomials related to Chebyshev polynomials
Polynomials denoted and closely related to Chebyshev polynomials are sometimes used. They are defined by
:
and satisfy
:
A. F. Horadam called the polynomials Vieta–Lucas polynomials and denoted them . He called the polynomials Vieta–Fibonacci polynomials and denoted them . Lists of both sets of polynomials are given in Viète's ''Opera Mathematica'', Chapter IX, Theorems VI and VII. The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials and of imaginary argument.
Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by
:
When the argument of the Chebyshev polynomial satisfies the argument of the shifted Chebyshev polynomial satisfies . Similarly, one can define shifted polynomials for generic intervals .
Around 1990 the terms "third-kind" and "fourth-kind" came into use in connection with Chebyshev polynomials, although the polynomials denoted by these terms had an earlier development under the name airfoil polynomials. According to J. C. Mason and G. H. Elliott, the terminology "third-kind" and "fourth-kind" is due to Walter Gautschi
Walter Gautschi (born December 11, 1927) is a Swiss-American mathematician, known for his contributions to numerical analysis. He has authored over 200 papers in his
area and published four books.
Born in Basel, he has a Ph.D. in mathematics from ...
, "in consultation with colleagues in the field of orthogonal polynomials." The Chebyshev polynomials of the third kind are defined as
:
and the Chebyshev polynomials of the fourth kind are defined as
:
where . In the airfoil literature and are denoted and . The polynomial families , , , and are orthogonal with respect to the weights
:
and are proportional to Jacobi polynomials with
:[
All four families satisfy the recurrence with , where , , , or , but they differ according to whether equals , , , or .][
]
See also
*Chebyshev filter
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error be ...
* Chebyshev cube root
* Dickson polynomials
* Legendre polynomials
* Hermite polynomials
*Minimal polynomial of 2cos(2pi/n)
For an integer n \geq 1, the minimal polynomial \Psi_n(x) of 2\cos\left(\frac\right) is the non-zero monic polynomial of degree 1 for n=1,2 and degree \frac\varphi(n) for n\geq 3 with integer coefficients, such that \Psi_n\!\left(2\cos\left(\frac ...
*Romanovski polynomials In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription) within the context of probability distribution functions in statistics ...
* Chebyshev rational functions
*Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' ...
* The Chebfun system
*Discrete Chebyshev transform
In applied mathematics, the discrete Chebyshev transform (DCT), named after Pafnuty Chebyshev, is either of two main varieties of DCTs: the discrete Chebyshev transform on the 'roots' grid of the Chebyshev polynomials of the first kind T_n (x) an ...
* Markov brothers' inequality
* Clenshaw algorithm
References
Sources
*
*
*
*
*
*
*
*
*
*
*
*
*
*
External links
*
*
*
* – includes illustrative Java applet
Java applets were small applications written in the Java programming language, or another programming language that compiles to Java bytecode, and delivered to users in the form of Java bytecode. The user launched the Java applet from a ...
.
*
*
*
{{Authority control
Special hypergeometric functions
Orthogonal polynomials
Polynomials
Approximation theory