## A First Course in Fourier AnalysisThis book provides a meaningful resource for applied mathematics through Fourier analysis. It develops a unified theory of discrete and continuous (univariate) Fourier analysis, the fast Fourier transform, and a powerful elementary theory of generalized functions and shows how these mathematical ideas can be used to study sampling theory, PDEs, probability, diffraction, musical tones, and wavelets. The book contains an unusually complete presentation of the Fourier transform calculus. It uses concepts from calculus to present an elementary theory of generalized functions. FT calculus and generalized functions are then used to study the wave equation, diffusion equation, and diffraction equation. Real-world applications of Fourier analysis are described in the chapter on musical tones. A valuable reference on Fourier analysis for a variety of students and scientific professionals, including mathematicians, physicists, chemists, geologists, electrical engineers, mechanical engineers, and others. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

1 | |

A14 | 14 |

A23 | 23 |

A33 | 33 |

Sampling | 483 |

Partial diﬀerential equations | 523 |

Wavelets | 593 |

Musical tones | 693 |

1 | xi |

### Other editions - View all

### Common terms and phrases

absolutely integrable algorithm analogous analysis equation approximation argument calculus characteristic function coeﬃcients components compute continuous function converges convolution product corresponding deﬁned derivative diﬀerential equation diﬀraction diﬀusion dilation discrete Fourier transform Example Find Example Let EXERCISE factorization Figure ﬁlter ﬁnd Find the Fourier ﬁnite ﬁrst formula Fourier analysis Fourier representation Fourier series frequency function f function on Tp graph Haar wavelet Hartley transform Hilbert transform Hint identity interval Let f linear mathematical matrix multiplication nonnegative obtain oo oo operator p-periodic function parameter piecewise continuous piecewise smooth piecewise smooth function points probability density random variables rule samples satisﬁes Schwartz function Show that f shown in Fig smooth function Solution spectrogram string suitably regular functions symmetry synthesis equation theorem transform F translation trigonometric polynomial vector verify wavelet