Question

According to the theorem, which of the Dirichlet conditions for the existence of the Fourier transform violates the following function? x(t)=sin1/x Theorem: (Dirichlet conditions) If f:(−[infinity],[infinity])→C is absolutely convergent in R, that is the integral ∫ₐᵇ∣f(t)∣dt exists, is finite, and in every closed and bounded interval of the form [a,b] occurs: 1. f(t) is continuous on [a,b] except for a finite number of discontinuities, all of them by infinite jump. 24. f(t) have in each [a,b] a finite number of maxima and minima, then there exists F[f(t)](ω).

Answer 1

The function x(t) = sin(1/x) violates the second condition of the Dirichlet conditions for the existence of the Fourier transform.

Step-by-step explanation:

The function x(t) = sin(1/x) violates the second condition of the Dirichlet conditions for the existence of the Fourier transform. According to the second condition, f(t) should have a finite number of maxima and minima in each closed and bounded interval [a, b]. However, the function x(t) = sin(1/x) oscillates infinitely with decreasing amplitude as x approaches 0. Therefore, it violates the second condition and the Fourier transform of this function may not exist.