Question: In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after which French mathematician and physicist, born in Auxerre?

Context articles: Fourier series
Discrete-time Fourier transform
Fourier sine and cosine series
Walsh function
Madhava series
Joseph Fourier
Fourier analysis
Sine and cosine transforms
List of harmonic analysis topics
Non-sinusoidal waveform

In mathematics, a Fourier series ( ) is a way to represent a function as the sum of simple sine waves.
 More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials).
 The discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series.
 The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1.
 Fourier series are also central to the original proof of the Nyquist–Shannon sampling theorem.
 The study of Fourier series is a branch of Fourier analysis.
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to the uniformly-spaced samples of a continuous function.
 The term "discrete-time" refers to the fact that the transform operates on discrete data (samples) whose interval often has units of time.
 From only the samples, it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.
 Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples.
 The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see Sampling the DTFT), which is by far the most common method of modern Fourier analysis.
In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.
In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous function in Fourier analysis.
 They can thus be viewed as a discrete, digital counterpart of the continuous, analog system of trigonometric functions on the unit interval.
 But unlike the sine and cosine functions, which are continuous, Walsh functions are piecewise constant.
 They take the values −1 and +1 only, on sub-intervals defined by dyadic fractions.
In mathematics, a Leibniz or Madhava series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics and later by Gottfried Wilhelm Leibniz, among others.
 These expressions are the Maclaurin series expansions of the trigonometric sine, cosine and arctangent functions, and the special case of the power series expansion of the arctangent function yielding a formula for computing π.
 The power series expansions of sine and cosine functions are respectively called "Madhava's sine series" and "Madhava's cosine series".
 The power series expansion of the arctangent function is sometimes called "Madhava–Gregory series" or "Gregory–Madhava series".
 These power series are also collectively called "Taylor–Madhava series".
 The formula for π is referred to as "Madhava–Newton series" or "Madhava–Leibniz series" or Leibniz formula for pi or Leibnitz–Gregory–Madhava series.
 These further names for the various series are reflective of the names of the Western discoverers or popularizers of the respective series.
Jean-Baptiste Joseph Fourier - ( ; ] ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.
 The Fourier transform and Fourier's law are also named in his honour.
 Fourier is also generally credited with the discovery of the greenhouse effect.
In mathematics, Fourier analysis ( ) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.
 Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.
In mathematics, the Fourier sine and cosine transforms are forms of the Fourier integral transform that do not use complex numbers.
 They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics.
This is a list of harmonic analysis topics.
 See also list of Fourier analysis topics and list of Fourier-related transforms, which are more directed towards the classical Fourier series and Fourier transform of mathematical analysis, mathematical physics and engineering.
Non-sinusoidal waveforms are waveforms that are not pure sine waves.
 They are usually derived from simple math functions.
 While a pure sine consists of a single frequency, non-sinusoidal waveforms can be described as containing multiple sine waves of different frequencies.
 These "component" sine waves will be whole number multiples of a fundamental or "lowest" frequency.
 The frequency and amplitude of each component can be found using a mathematical technique known as Fourier analysis.