state and prove the localization theorem for Fourier series
Answers
Answer:
Basic texts in Fourier analysis typically state the Riemann localization principle as follows. Suppose f is integrable on T1 = R/(2πZ), i.e., f ∈ L1(T1). Assume f = g on an open set O ⊂ T1, and that g ∈ C(T1) has a Fourier series that converges uniformly on T1.
Answer:
The localization theorem under certain conditions allows to infer the nullity of a function given only information about its continuity and the value of its integral.
Step-by-step explanation:
A Fourier transform refers to a mathematical transform which decomposes functions depending on space or time into functions depending on spatial frequency or temporal frequency.
The localization theorem under certain conditions allows to infer the nullity of a function given only information about its continuity and the value of its integral.
If there were a point x0 within Ω for which F(x0) ≠ 0,
then the continuity of F will require the existence of a neighborhood of x0 where the value of F was nonzero, and in particular of the same sign than in x0.
Hence proved.