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Fourier Series and Fourier Transforms
2003In Chapter 3, we touched upon the analogy between the diffraction of x-rays and that of visible light. Here, we extend that discussion and consider some aspects of Fourier series and Fourier transforms.
Mark Ladd, Rex A. Palmer
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On the Fourier series and Fourier transforms
Journal of Mathematical Sciences, 2019This survey article is addresses to classical harmonic analysis. In particular, a number of classical theorems are presented with the simplest, in our opinion, proofs (see also [1] and references therein). Some results of the present article are new and are published for the first time.
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2013
Publisher Summary This chapter analyzes trigonometric Fourier series, which are series of trigonometric functions. This set is chosen because the mathematics is the best developed and is the simplest to demonstrate. It will serve to illustrate the basic questions that need to be addressed for each system.
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Publisher Summary This chapter analyzes trigonometric Fourier series, which are series of trigonometric functions. This set is chosen because the mathematics is the best developed and is the simplest to demonstrate. It will serve to illustrate the basic questions that need to be addressed for each system.
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1966
Publisher Summary This chapter focuses on “function space” as opposed to a three-dimensional “vector space.” This function space is infinite dimensional, in the sense that an infinite sequence of mutually orthogonal functions is needed to represent an arbitrary function.
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Publisher Summary This chapter focuses on “function space” as opposed to a three-dimensional “vector space.” This function space is infinite dimensional, in the sense that an infinite sequence of mutually orthogonal functions is needed to represent an arbitrary function.
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, 1977 In the subsequent sections use will be made of the expansion of given functions in Fourier series and it will be more convenient to represent them in complex form; some remarks will now be made about this.
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Periodic Hyperfunctions and Fourier Series Fourier Series
1992We have now almost finished the general discussion of hyperfunctions. From now on we shall attempt to apply this general theory to various cases. In this chapter we study periodic hyperfunctions. Then we shall see that the theory of Fourier series is naturally absorbed into the theory of Fourier transformations.
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2014
Fourier Series are a powerful tool in Applied Mathematics; indeed, their importance is twofold since Fourier Series are used to represent both periodic real functions as well as solutions admitted by linear partial differential equations with assigned initial and boundary conditions.
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Fourier Series are a powerful tool in Applied Mathematics; indeed, their importance is twofold since Fourier Series are used to represent both periodic real functions as well as solutions admitted by linear partial differential equations with assigned initial and boundary conditions.
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The Fourier Series and Fourier Transform
2020We encountered the Fourier series in passing in Chap. 5. Then it was just to illustrate the importance of sine waves as a fundamental waveform from which more complex ones such as a square wave could be constructed by adding them with different frequencies and amplitudes.
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