Equality is attained for Chebyshev polynomials of the first kind.

The same is true if the expansion is in terms of Chebyshev polynomials.

Is there an intuitive explanation for an extremal property of Chebyshev polynomials?

The Chebyshev polynomials form a complete orthogonal system.

Chebyshev polynomials of the first kind closely approximate the minimax polynomial.

For , one obtains the Chebyshev polynomials (of the second and first kind, respectively).

A different formula for M involving Chebyshev polynomials was given by .

The elliptic rational functions are related to the Chebyshev polynomials of the first kind by:

Many properties can be derived from the properties of the Chebyshev polynomials of the first kind.

For further details, including the expressions for the first few polynomials, see Chebyshev polynomials.

Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:

These disadvantages can be reduced by using spline interpolation or restricting attention to Chebyshev polynomials.

The ephemeris data are distributed in the form of a file of numerical coefficients for Chebyshev polynomials.

Evaluating the first two Chebyshev polynomials:

Where is the nth cardinal function of the chebyshev polynomials of the first kind with input argument y.

The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution.

Chebyshev polynomials and the "Chebyshev form"

Evaluation of the Chebyshev polynomials can recover planetary and lunar coordinates to high precision relative to the original numerical integration.

We recognize in p(x) and q(x) respectively the Chebyshev polynomials of degree 3 and 5.

The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind.

In PS optimal control, Legendre and Chebyshev polynomials are commonly used.

That is, Fejér only used the interior extrema of the Chebyshev polynomials, i.e. the true stationary points.

Demeyer (2007) mentions a connection between Pell's equation and the Chebyshev polynomials:

An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind.

This includes Legendre polynomials and Chebyshev polynomials.