Sage-Tips

What is are the conditions for the convergence of Fourier series?

If f satisfies a Holder condition, then its Fourier series converges uniformly. If f is of bounded variation, then its Fourier series converges everywhere. If f is continuous and its Fourier coefficients are absolutely summable, then the Fourier series converges uniformly.

How do you know if a function can be represented by a Fourier series?

Well there are 3 conditions for a Fourier Series of a function to be exist: 1. It has to be periodic. 2. It must be single valued, continuous.it can have finite number of finite discontinuities.

Can the Fourier transform of a function be zero?

What is the Fourier transform of 0? – Quora. Only functions have fourier transforms. 0 is a number not a function so it does not have a fourier transform. Now if you defined a function f(t)=0 for all t then you could calculate ther first transform of that function.

How can we find the convergence of Fourier series at a point of discontinuity?

Fourier series representation of such function has been studied, and it has been pointed out that, at the point of discontinuity, this series converges to the average value between the two limits of the function about the jump point. So for a step function, this convergence occurs at the exact value of one half.

Does the Fourier series converge?

The Fourier series of f(x) will be continuous and will converge to f(x) on −L≤x≤L − L ≤ x ≤ L provided f(x) is continuous on −L≤x≤L − L ≤ x ≤ L and f(−L)=f(L) f ( − L ) = f ( L ) .

For which of the following Fourier series Cannot be defined Mcq?

→ e sin 25t Due to decaying exponential decaying function it is not periodic. So Fourier series cannot be defined for it.

What are the conditions for the Fourier series to converge?

There are many known sufficient conditions for the Fourier series of a function to converge at a given point x, for example if the function is differentiable at x.

What are the symmetry properties of Fourier series?

There are two symmetry properties of functions that will be useful in the study of Fourier series. Even and Odd Function A function f (x) is said to be even if f (−x) = f (x). The function f (x) is said to be odd if f (−x) = −f (x).

What is an example of summation in Fourier series?

As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving the weights that describe a given function is a form of Fourier analysis.

What are the Fourier series formulas in calculus?

The above Fourier series formulas help in solving different types of problems easily. Example: Determine the fourier series of the function f (x) = 1 – x2 in the interval [-1, 1]. We know that, the fourier series of the function f (x) in the interval [-L, L], i.e. -L ≤ x ≤ L is written as: