Trigonometric polynomials

W. Gautschi. Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math., 3 381-397, 1961. 

Any function in L — 1, 1] can be approximated by trigonometric polynomials (of period 2). A trigonometric polynomial is a finite (complex) linear combination of the functions... 

Notice that we have approximated a discontinuous function by a continuous one. It turns out that any function in L —1, 1] can be approximated by trigonometric polynomials — this is one of the important results of the theory of Fourier series. ... 

Exercise 3.32 Show that the set Tf of trigonometric polynomials of period... 

However, the expansion above is still impractical for our purposes, because the functions As(q), Bs(q),. .. still need to be expanded in an infinite Fourier series of the angles e.g., we should write A q) = J2kez akexp(i(k,qj). It is more convenient to work with trigonometric polynomials, so that every part of the expansion contains only a finite number of terms. To this end, we introduce a Fourier cutoff by splitting every function of the angles in an infinite number of slices that contain only a finite number of Fourier modes. This may be done in many arbitrary ways, so let us illustrate just one method. We choose an arbitray integer K, ad write, e.g.,... 

H(° Here we must pay a little attention to the action of the Lie derivative L (i) on a function fi° m p,q)- Since xi is independent of p, the Poisson bracket decrements by one degree on p on the other hand, since Xi is a trigonometric polynomial of degree K it increments by K the trigonometric degree. This is illustrated in the following diagram ... 

Let us add a few remarks in this case too. The generating function is linear in p, and is a trigonometric polynomial of degree K. Hence,... 

There is only one point to be noticed the generating functions x and %2 are trigonometric polynomials of degree sK thus the Lie derivatives L (s) and L (s) increase the trigonometric degree by sK. Moreover, they... 

The first operation (i) contains derivatives of polynomials of degree at most s + 1 and trigonometric polynomials of degree at most sK. Hence, the size of is multiplied by a factor < s + 1, besides other possible... 

These scheme is quite convenient for calculation, because for all xo, only trigonometric polynomials are to be integrated in order to find x] (t). 

We want to find an approximate periodic solution of the system (1.1) in the form of a trigonometric polynomial... 

The trigonometric polynomial x (r) satisfying (1.4) is called Bubnov-Galerkin s approximation of the mth order. The system (1.4) can be rewritten as follows... 

This implies that — — and, hence, z (t) are trigonometric polynomials of the mth dt... 

In the uniform metric, any quasiperiodic function can be approximated by trigonometric polynomials as accurately as desired. Therefore, one can try to construct a quasiperiodic solution of the system (1.19) (or a periodic solution of the equation (1.24)) in the form of a sequence of trigonometric polynomials... 

Assume that the trigonometric polynomial (O 9) satisfies the following inequalities... 

Giesbrecht, F. G, W. F. McClure, and A. Hamid. 1981. The use of trigonometric polynomials to approximate visible and near infrared spectra of agricultural products. Appl. Spectrosc., 35 210-214. 

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