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Dec 29, 2020 · The way to use Nest [] for computing the Chebyshev polynomial of the first kind is to recognize that a three-term recurrence is equivalent to repeated multiplication by a certain 2 × 2 matrix. To wit, With [ {n = 5}, First [Nest [ { {2 x, -1}, {1, 0}}.# &, {x, 1}, n - 1]]] // Expand 5 x - 20 x^3 + 16 x^5 ChebyshevT [5, x] 5 x - 20 x^3 + 16 x^5 ...

This section concerns about Chebyshev equations and its solutions, known as Chebyshev functions.

The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted ...

Is there functionality in Mathematica to expand a function into a series with Chebyshev polynomials? The Series function only approximates with Taylor ...

Cell[BoxData[RowBox[List[RowBox[List["ChebyshevU", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], FractionBox ...

For certain special arguments, ChebyshevU automatically evaluates to exact values. ChebyshevU can be evaluated to arbitrary numerical precision. ChebyshevU automatically threads over lists. ChebyshevU [ n, z] has a branch cut discontinuity in the complex z …

05.01.2016 · Of course, not every function will admit a closed form Chebyshev series representation, since the Fourier integrals involved won't necessarily …

Chebyshev2FilterModel[n] creates a lowpass Chebyshev type 2 filter of order n. Chebyshev2FilterModel[{n, \[Omega]c}] uses the cutoff frequency \[Omega]c.

01.11.2021 · There are two kinds of Chebyshev expansions for a function on the finite interval [-1, 1] depending which kind of Chebyshev function is used. The Chebyshev polynomials of first kind. T n ( x) are solutions of the differential …

ChebyshevT can be evaluated to arbitrary numerical precision. ChebyshevT automatically threads over lists. ChebyshevT [ n, z] has a branch cut discontinuity in the complex z plane running from to . ChebyshevT can be used with CenteredInterval objects.

For certain special arguments, ChebyshevU automatically evaluates to exact values. ChebyshevU can be evaluated to arbitrary numerical precision. ChebyshevU automatically threads over lists. ChebyshevU [ n, z] has a branch cut discontinuity in the complex z plane running from to for noninteger n.

A number of spellings of "Chebyshev" (which is the spelling used exclusively in this work) are commonly found in the ... DOWNLOAD Mathematica Notebook ...

The Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients can be determined easily through the application of ...

Jan 06, 2016 · One slick way to derive the analytic Chebyshev series of a function is to use the relationship between the Chebyshev polynomials and the cosine, and then use the built-in FourierCosSeries []. As an example: f [x_] := Exp [x]; n = 5; (* degree of approximation *) approx [x_] = FourierCosSeries [f [Cos [t]], t, n] /.

Nov 01, 2021 · There are two kinds of Chebyshev expansions for a function on the finite interval [-1, 1] depending which kind of Chebyshev function is used. The Chebyshev polynomials of first kind. T n ( x) are solutions of the differential equation. ( 1 − x 2) y ″ − x y ′ + n 2 y = 0, and they are orthogonal on the interval [-1, 1] with weight function.

As of Version 7.0, ChebyshevDistance is superseded by ChessboardDistance. ... gives the Chebyshev or sup norm distance between vectors u and v.

ChebyshevU[n, x] gives the Chebyshev polynomial of the second kind Un (x).

ChebyshevT[n, x] gives the Chebyshev polynomial of the first kind Tn (x).

ChebyshevT can be evaluated to arbitrary numerical precision. ChebyshevT automatically threads over lists. ChebyshevT [ n, z] has a branch cut discontinuity in the complex z plane running from to . ChebyshevT can be used with CenteredInterval objects.

Chebyshev1FilterModel[n] creates a lowpass Chebyshev type 1 filter of order n. Chebyshev1FilterModel[{n, \[Omega]c}] uses the cutoff frequency \[Omega]c.