Trigonometric polynomial

Summary

In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series.

Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform.

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. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of , i.e., Laurent polynomials in under the change of variables .

Definition

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Any function T of the form

 

with coefficients   and at least one of the highest-degree coefficients   and   non-zero, is called a complex trigonometric polynomial of degree N.[1] Using Euler's formula the polynomial can be rewritten as

  with  .

Analogously, letting coefficients  , and at least one of   and   non-zero or, equivalently,   and   for all  , then

 

is called a real trigonometric polynomial of degree N.[2][3]

Properties

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

Trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm;[4] this is a special case of the Stone–Weierstrass theorem. More concretely, for every continuous function   and every   there exists a trigonometric polynomial   such that   for all  . Fejér's theorem states that the arithmetic means of the partial sums of the Fourier series of   converge uniformly to   provided   is continuous on the circle; these partial sums can be used to approximate  .

A trigonometric polynomial of degree   has a maximum of   roots in a real interval   unless it is the zero function.[5]

Fejér-Riesz theorem

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The Fejér-Riesz theorem states that every positive real trigonometric polynomial   satisfying   for all  , can be represented as the square of the modulus of another (usually complex) trigonometric polynomial   such that:[6]   Or, equivalently, every Laurent polynomial   with   that satisfies   for all   can be written as:   for some polynomial  .[7]

Notes

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References

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  • Dritschel, Michael A.; Rovnyak, James (2010). "The Operator Fejér-Riesz Theorem". A Glimpse at Hilbert Space Operators. Basel: Springer Basel. doi:10.1007/978-3-0346-0347-8_14. ISBN 978-3-0346-0346-1.
  • Hussen, Abdulmtalb; Zeyani, Abdelbaset (2021). "Fejer-Riesz Theorem and Its Generalization". International Journal of Scientific and Research Publications (IJSRP). 11 (6): 286–292. doi:10.29322/IJSRP.11.06.2021.p11437.
  • Powell, Michael J. D. (1981), Approximation Theory and Methods, Cambridge University Press, ISBN 978-0-521-29514-7
  • Riesz, Frigyes; Szőkefalvi-Nagy, Béla (1990). Functional analysis. New York: Dover Publications. ISBN 978-0-486-66289-3.
  • Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157.