They converge uniformly for continuously differentiable functions, and are uniformly Cesaro summable for continuous functions, as was shown by Fejer, but uniform convergence need not always hold, as is shown by Gibbs Phenomenon. They do converge in the mean square, however, which gives rise to Parseval's identity regarding their coefficients. In actuality, they can be defined for any complete orthonormal set of functions, an important result in Harmonic Analysis. FTP, name this series useful in analyzing waves and other
They converge uniformly for continuously differentiable functions, and are uniformly Cesaro summable for continuous functions, as was shown by Fejer, but uniform convergence need not always hold, as is shown by Gibbs Phenomenon. They do converge in the mean square, however, which gives rise to Parseval's identity regarding their coefficients. In actuality, they can be defined for any complete orthonormal set of functions, an important result in Harmonic Analysis. FTP, name this series useful in analyzing waves and other periodic functions which expands functions as an infinite sum of sines and cosines.
They converge uniformly for continuously differentiable functions, and are uniformly Cesaro summable for continuous functions, as was shown by Fejer, but uniform convergence need not always hold, as is shown by Gibbs Phenomenon.