If P is a polynomial in CC^n and if int_(T_n) |P|d sigma_n=0, then P is identically zero.

Chasity Kane

Chasity Kane

Answered question


If P is a polynomial in C n and if
T n | P | d σ n = 0 ,
then P is identically zero.

Answer & Explanation

Emilia Boyle

Emilia Boyle

Beginner2022-10-09Added 10 answers

If you restrict P as a function T n , then the Fourier series for P is simply the polynomial itself (up to constant maybe). Therefore, by Plancherel's theorem, we have
T n | P | = 0 T n | P | 2 = j | a j | 2 = 0
again with possibly a constant C > 0 in front, and the a j 's are the Fourier coefficients of P, which are the coefficients of the polynomial! Therefore a j = 0 for all j, hence P 0

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