NPTEL IIT Guwahati

Dirichlet was the first to prove pointwise convergence of Fourier series. In 1829 and 1847, he showed that the Fourier series of piecewise smooth function on $[-\pi, \; \pi]$ converges pointwise at every to $f^{*}(x)$ .

Definition: A function is piecewise continuous on $[a, \; b]$ if all of the following hold.
(1) is defined and continuous at all but a finite number of points in $[a, \; b]$ .
(2) For all , the limits $f(x_{0}^{+})$ and $f(x_{0}^{-})$ exist.
(3) $f(a^{+})$ and $f(b^{-})$ exists.

Definition: A function is piecewise smooth (or piecewise $C^{1}$ ) on $[a, \; b]$ if and $f^{\prime}(x)$ are piecewise continuous on $[a, \; b]$ .

Theorem: Let be a piecewise smooth function on $[-p, \; p]$ and let $f^{*}$ be the adjusted function of . Then the Fourier series of converges pointwise to $f^{*}(x)$ for all $x \in [-p, \; p]$ .

Note: It is easy to observe that a Fourier series of a function cannot converge uniformly on an interval that contains a point at which is discontinuous.


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Dirichlet was the first to prove pointwise convergence of Fourier series. In 1829 and 1847, he showed that the Fourier series of piecewise smooth function on $[-\pi, \; \pi]$ converges pointwise at every to $f^{}(x)$ . Definition:* A function is piecewise continuous on $[a, \; b]$ if all of the following hold. (1) is defined and continuous at all but a finite number of points in $[a, \; b]$ . (2) For all , the limits $f(x_{0}^{+})$ and $f(x_{0}^{-})$ exist. (3) $f(a^{+})$ and $f(b^{-})$ exists. Definition: A function is piecewise smooth (or piecewise $C^{1}$ ) on $[a, \; b]$ if and $f^{\prime}(x)$ are piecewise continuous on $[a, \; b]$ . Theorem: Let be a piecewise smooth function on $[-p, \; p]$ and let $f^{}$ be the adjusted function of . Then the Fourier series of converges pointwise to $f^{}(x)$ for all $x \in [-p, \; p]$ . Note: It is easy to observe that a Fourier series of a function cannot converge uniformly on an interval that contains a point at which is discontinuous.

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