The summation of power series and Fourier series

Summation of Fourier Series

A general summability method of different orthogonal series is given with the help of an integrable function θ. As special cases the trigonometric Fourier, Walsh-, Walsh-Kaczmarz-, Vilenkinand Ciesielski-Fourier series and the Fourier transforms are considered. For each orthonormal system a different Hardy space is introduced and the atomic decomposition of these Hardy spaces are presented. A s...

Determination of a jump by Fourier and Fourier-Chebyshev series

‎By observing the equivalence of assertions on determining the jump of a‎ ‎function by its differentiated or integrated Fourier series‎, ‎we generalize a‎ ‎previous result of Kvernadze‎, ‎Hagstrom and Shapiro to the whole class of‎ ‎functions of harmonic bounded variation‎. ‎This is achieved without the finiteness assumption on‎ ‎the number of discontinuities‎. ‎Two results on determination of ...

Power and RMS Values of Fourier Series

1 . Average power Let us consider the transmission of energy from a source to a load, through a given surface as in Fig. 1. In the network of Fig. 1, the voltage waveform v(t) (not necessarily sinusoidal) is given by the source, and the current waveform is determined by the response of the load. In the more general case in which the source output impedance is significant, then v(t) and i(t) bot...

Summation of Multiple Fourier Series by Spherical Means.

Summation Notation and Series

Corresponding material in the book: Section 12.1, 12.2, 12.3. What students should definitely get: The summation notation and how it works, series, concepts of convergence. The use of telescoping and forward difference operator ideas to sum up series. The use of the integral test and other tests to determine whether a series converges and obtain numerical estimates. Convergence rules for ration...