Sentence examples for by a trigonometric polynomial from inspiring English sources

First a simple practical procedure for approximating a stationary Gaussian process over a finite interval by a trigonometric polynomial with predetermined error is described.

The degree of approximation of a function f: R → R by a trigonometric polynomial t n of degree n under sup norm || ||∞ is defined by.

The degree of approximation of a function f : R → R by a trigonometric polynomial t n of order n under sup norm ∥ ∥ ∞ is defined by ∥ t n − f ∥ ∞ = sup { | t n ( x ) − f ( x ) | : x ∈ R }.

The error (degree) of approximation of a function (f: R to R) by a trigonometric polynomial (t_{n}) of order n under sup norm (Vert Vert _{ infty}) is given by [25] Vert t_{n} - f Vert _{infty} = sup_{x in R} bigl{ biglvert t_{n} (x) - f ( x ) bigrvert bigr} and (E_{n}(f)) of a signal (f in L_{ r}) is defined by (E_{n}(f) = min_{n} Vert t_{n} - f Vert _{ r}).

Let ∥ ∥ ∞ of a function f : [ 0, 2 π ] → R be defined by ∥ f ∥ ∞ = ess sup 0 ≤ x ≤ 2 π | f ( x ) |. and the degree of approximation of a function f : [ 0, 2 π ] → R by a trigonometric polynomial of order n, t n = 1 2 a 0 + ∑ ν = 1 n ( a ν cos ν x + b ν sin ν x ), under esssup norm ∥ ∥ ∞ be defined by (Zygmund [12], p.114) ∥ t n − f ∥ ∞ = ess sup 0 ≤ x ≤ 2 π | t n ( x ) − f ( x ) |.

By (2.3), we have a trigonometric polynomial (P_{n}) of degree at most (( frac{log_{2} L}{p} +4) n) such that P_{n}^{p} sim W_{n}.

A trigonometric polynomial is a truncated Fourier series of the form fN t)≡∑j= 0Najcos jt)+∑j= 1Nbjsin jt). It has been previously shown by the author that zeros of such a polynomial can be computed as the eigenvalues of a companion matrix with elements which are complex valued combinations of the Fourier coefficients, the "CCM" method.

A signal (function) f is approximated by trigonometric polynomials t n of order n and the degree of approximation E n ( f ) is given by Zygmund [13] E n ( f ) = min n ∥ f ( x ) − t n ( f ; x ) ∥ r (1.6). in terms of n, where t n ( f ; x ) is a trigonometric polynomial of degree n.

(ii) there is a trigonometric polynomial where such that.

Hence ((B_{psi} f)(b,a)) is a trigonometric polynomial in v.

Let (f(t)=sum_{k=1}^{n} a_{k} e^{ilambda_{k} t}) be a trigonometric polynomial.