They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the following inner product:
[1] In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems;[2] the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation.
[3] The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
(The Dirichlet kernel, in fact, coincides with what is now known as the Chebyshev polynomial of the fourth kind.)
[6] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:
The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences Ṽn(P, Q) and Ũn(P, Q) with parameters P = 2x and Q = 1:
From this trigonometric form, the recurrence definition can be recovered by computing directly that the bases cases hold:
Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression:
An explicit form of the Chebyshev polynomial in terms of monomials xk follows from de Moivre's formula:
The real part of the expression is obtained from summands corresponding to even indices.
A related expression for Tn as a sum of monomials with binomial coefficients and powers of two is
One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ x ≤ 1 all of the extrema have values that are either −1 or 1.
The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at x = 1 and x = −1.
where the prime at the summation symbols means that the term contributed by k = 0 is to be halved, if it appears.
The last formula can be further manipulated to express the integral of Tn as a function of Chebyshev polynomials of the first kind only:
For n = 1 this results in the already known recurrence formula, just arranged differently, and with n = 2 it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest m) which implies the evenness or oddness of these polynomials.
It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions.
For the polynomials of the second kind and any integer N > i + j with the same Chebyshev nodes xk, there are similar sums:
Let's assume that wn(x) is a polynomial of degree n with leading coefficient 1 with maximal absolute value on the interval [−1, 1] less than 1 / 2n − 1.
However, this is impossible, as fn(x) is a polynomial of degree n − 1, so the fundamental theorem of algebra implies it has at most n − 1 roots.
Above, however, | f | reaches its maximum only n + 1 times because we are searching for the best polynomial of degree n ≥ 1 (therefore the theorem evoked previously cannot be used).
These identities can be proven using generating functions and discrete convolution The first few Chebyshev polynomials of the first kind are OEIS: A028297
Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients an can be determined easily through the application of an inner product.
[16] These attributes include: The abundance of the theorems and identities inherited from Fourier series 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,[16] often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).
This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:
[19] Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII.
and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials Ln and Fn of imaginary argument.
According to J. C. Mason and G. H. Elliott, the terminology "third-kind" and "fourth-kind" is due to Walter Gautschi, "in consultation with colleagues in the field of orthogonal polynomials.
Even order Chebyshev filter designs using equally terminated passive networks are an example of this.