Trigonometric Polynomials
Trigonometric Polynomials:
A Trigonometric Polynomial is a polynomial expression involving cos x and sin x:
cos5x 6 cos3x sin2x 3 sin4x 2 cos2x 5
Because of the identity cos2x sin2x = 1, most trigonometric polynomials can be written in several dierent ways. For example, the above polynomial can be rewritten as:
5 cos3x sin2x 3 sin4x cos3x - 2 sin2x 7
Fourier Sums:
A Fourier Sum is a Fourier Series with nitely many terms:
5 3 sin 2x 4 cos 5x - 3 sin 5x 2 cos 8x:
Every Fourier sum is actually a trigonometric polynomial, and any trigonometric polynomial can be expressed as a Fourier sum. Converting a Fourier sum to a trigonometric polynomial is fairly straight forward: simply substitute the appropriate multiple-angle identity for each cos nx and sin nx (see table 1). It is less obvious that every trigonometric polynomial can be expressed as a Fouriersum. This depends on the three product-to-sum formulas:
These identities allow us to transform any product of trigonometric functions into a sum. By applying them repeatedly, we can remove all of the multiplications from a trigonometric polynomial, resulting in a Fourier sum. Alternatively, one can use these identities to derive power-reduction formulas for cosjx sinkx, the first few of which are listed below:
These formulas tell us how to convert each term of a trigonometric polynomials directly into Fourier Sum.