Введение
Полиномиальная последовательность
Polynomial sequence
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Полиномы Чебышева — это две последовательности полиномов, связанных с косинусом и синусом, обозначаемые как и . Они могут быть определены несколькими эквивалентными способами, один из которых начинается с тригонометрических функций:
Polynomial sequence
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Полиномы Чебышева первого рода определяются следующим образом:
Polynomial sequence
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Аналогично, полиномы Чебышева второго рода определяются следующим образом:
Polynomial sequence
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Тот факт, что эти выражения определяют полиномы, может быть не сразу очевиден, но следует из переписывания и с использованием формулы Муавра или повторного применения формул сложения углов для и . Например, формулы двойного угла, непосредственно вытекающие из формул сложения углов, могут быть использованы для получения и , которые являются соответственно полиномом от и полиномом от , умноженным на . Следовательно, и
Polynomial sequence
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Важное и удобное свойство полиномов Tn(x) заключается в их ортогональности относительно внутреннего произведения:
Polynomial sequence
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
а полиномы Un(x) ортогональны относительно другого, аналогичного внутреннего произведения, приведенного ниже. Полиномы Чебышева Tn — это полиномы с максимально возможным старшим коэффициентом, абсолютная величина которых на интервале ограничена 1. Они также являются "экстремальными" полиномами для многих других свойств. В 1952 году Корнелиус Ланчос показал, что полиномы Чебышева важны в теории аппроксимаций для решения линейных систем; корни Tn(x), также называемые узлами Чебышева, используются в качестве точек сопоставления для оптимизации полиномиальной интерполяции. Полученный интерполяционный полином минимизирует проблему феномена Рунге и обеспечивает приближение, близкое к наилучшей полиномиальной аппроксимации непрерывной функции в максимальной норме, также называемой критерием "мини-макс". Это приближение непосредственно приводит к методу квадратур Кленшоу — Куртиса. Эти полиномы названы в честь Пафнутия Чебышева. Буква T используется из-за альтернативных транслитераций имени Чебышев как Tchebycheff, Tchebyshev (французский) или Tschebyschow (немецкий).
Polynomial sequence
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
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, 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 Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. 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. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Симметрия
То есть, полиномы Чебышева четной степени обладают четной симметрией и, следовательно, содержат только четные степени x. Полиномы Чебышева нечетной степени обладают нечетной симметрией и, следовательно, содержат только нечетные степени x.
That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of x.
Замечание
По теореме эквиосцилляции, среди всех многочленов степени ≤ n, многочлен f минимизирует ошибку на [-1, 1] тогда и только тогда, когда существуют n + 2 точки −1 ≤ x0 < x1 < ⋯ < xn + 1 ≤ 1 такие, что |f(x)| = ε.
Конечно, нулевой многочлен на интервале [-1, 1] может быть приближен самим собой и минимизирует ∞-норму. Однако вышеуказанная функция достигает своего максимума только n + 1 раз, поскольку мы ищем наилучший многочлен степени n ≥ 1 (следовательно, ранее упомянутая теорема неприменима).
By the equioscillation theorem, among all the polynomials of degree ≤ n, the polynomial f minimizes on if and only if there are n + 2 points −1 ≤ x0 < x1 < ⋯ < xn + 1 ≤ 1 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 n + 1 times because we are searching for the best polynomial of degree n ≥ 1 (therefore the theorem evoked previously cannot be used).
Of course, the null polynomial on the interval can be approximated by itself and minimizes the ∞ norm. Above, however, 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).
Полином в форме Чебышева
Произвольный многочлен степени N может быть представлен через многочлены Чебышева первого рода. Такой многочлен p(x) имеет вид:
An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial p(x) is of the form:
Многочлены, представленные в форме Чебышева, могут быть вычислены с использованием алгоритма Кленшоу.
Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.