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After calculating the fourier series of $$f(x) = \sqrt {\left| x \right|} $$ which is: $$\frac{2}{3}\sqrt \pi - \frac{2}{\pi }\sum\limits_{n = 1}^\infty {\frac{{S(\sqrt {n\pi } )}}{{{n^{1.5}}}}} \cos (nx)$$ where $$ S(x) $$ is Fresnel instegral. I showed that the series converges at $$x = 0$$ by bounding it.

Now I am trying to calculate the series: $$\sum\limits_{n = 1}^\infty {\frac{{S(n\pi )}}{{{n^{1.5}}}}} $$ which is the above fourier series at $$x = 0$$ However ,since the derivatives at $$x = 0$$ aren't finite I am unable to use Dirichlet, and cannot say that it converges to $$f(0)$$

In what way is it possible to find the value of convergenge in that case ?

breeze
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1 Answers1

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Derivatives aren't necessary. If $f$ satisfies a Hölder condition, its Fourier series converges uniformly to $f$. If $f$ is of bounded variation, the series converges everywhere. If $f$ is continuous and the Fourier series is absolutely summable, it converges uniformly. All of that is satisfied in your case.
https://en.wikipedia.org/wiki/Convergence_of_Fourier_series#Pointwise_convergence