trigonometric polynomial

For approximation by trigonometric polynomials, the result is as follows:

In the complex case the trigonometric polynomials are spanned by the positive and negative powers of e.

In mathematics, a K-finite function is a type of generalized trigonometric polynomial.

The partial sums for f are trigonometric polynomials.

This approach relies on the fact that trigonometric polynomials are an orthonormal basis for .

Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.

For trigonometric polynomials, the following was proved by Dunham Jackson:

This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.

In 1916, he introduced the Riesz interpolation formula for trigonometric polynomials, which allowed him to give a new proof of Bernstein's inequality.

In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials.

This can be shown to be a trigonometric polynomial by employing the multiple-angle formula and other identities for sin (x x).

Approximation theory - part of analysis that studies how well functions can be approximated by simpler ones (such as polynomials or trigonometric polynomials)

Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions.

A trigonometric polynomial can be considered a periodic function on the real line, with period some multiple of 2π, or as a function on the unit circle.

"Mean Number of Real Zeroes of a Random Trigonometric Polynomial.

Similar results exist for trigonometric polynomials, matrix polynomials, polynomials in free variables, various quantum polynomials, etc.

For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods.

This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial.

This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials.

For instance, rational interpolation is interpolation by rational functions, and trigonometric interpolation is interpolation by trigonometric polynomials.

The term trigonometric polynomial for the real-valued case can be seen as using the analogy: the functions sin(nx) and cos(nx) are similar to the monomial basis for polynomials.

Toeplitz matrices are also closely connected with Fourier series, because the multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix.

According to DTFT definition, is a sum of trigonometric functions, and since f(t) is time-limited, this sum will be finite, so will be actually a trigonometric polynomial.

All trigonometric polynomials are holomorphic on a whole complex plane, and there is a simple theorem in complex analysis that says that all zeros of non-constant holomorphic function are isolated.

A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm ; this is a special case of the Stone-Weierstrass theorem.