Fourier series

Fourier series   (Redirected from Fourier theorem) Jump to navigation Jump to search Fourier transforms Continuous Fourier transform Fourier series Discrete-time Fourier transform Discrete Fourier transform Discrete Fourier transform over a ring Fourier transform on finite groups Fourier analysis Related transforms This section may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (November 2021) (Learn how and when to remove this template message) A Fourier series (/ˈfʊrieɪ, -iər/[1]) is a sum that represents a periodic function as a sum of sine and cosine waves. The frequency of each wave in the sum, or harmonic, is an integer multiple of the periodic function's fundamental frequency. Each harmonic's phase and amplitude can be determined using harmonic analysis. A Fourier series may potentially contain an infinite number of harmonics. Summing part of but not all the harmonics in a function's Fourier series produces an approximation to that function. For example, using the first few harmonics of the Fourier series for a square wave yields an approximation of a square wave.

A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum).

The first four partial sums of the Fourier series for a square wave. As more harmonics are added, the partial sums converge to (become more and more like) the square wave.

Function {displaystyle s_{6}(x)} (in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform {displaystyle S(f)} is a frequency-domain representation that reveals the amplitudes of the summed sine waves.

Almost any[A] periodic function can be represented by a Fourier series that converges.[B] Convergence of Fourier series means that as more and more harmonics from the series are summed, each successive partial Fourier series sum will better approximate the function, and will equal the function with a potentially infinite number of harmonics. The mathematical proofs for this may be collectively referred to as the Fourier Theorem (see § Convergence).

Fourier series can only represent functions that are periodic. However, non-periodic functions can be handled using an extension of the Fourier Series called the Fourier transform which treats non-periodic functions as periodic with infinite period. This transform thus can generate frequency domain representations of non-periodic functions as well as periodic functions, allowing a waveform to be converted between its time domain representation and its frequency domain representation.

Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions as the basis set for the decomposition. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis.

Contents 1 Definition 1.1 Common forms 1.1.1 Amplitude-phase form 1.1.2 Sine-cosine form 1.1.3 Exponential form 1.2 Example 1.3 Convergence 1.4 Complex-valued functions 1.5 Other common notations 2 History 2.1 Beginnings 2.2 Fourier's motivation 2.3 Complex Fourier series animation 2.4 Other applications 3 Table of common Fourier series 4 Table of basic properties 5 Symmetry properties 6 Other properties 6.1 Riemann–Lebesgue lemma 6.2 Parseval's theorem 6.3 Plancherel's theorem 6.4 Convolution theorems 6.5 Derivative property 6.6 Compact groups 6.7 Riemannian manifolds 6.8 Locally compact Abelian groups 7 Extensions 7.1 Fourier series on a square 7.2 Fourier series of Bravais-lattice-periodic-function 7.3 Hilbert space interpretation 8 Fourier theorem proving convergence of Fourier series 8.1 Least squares property 8.2 Convergence theorems 8.3 Divergence 9 See also 10 Notes 11 References 11.1 Further reading 12 External links Definition The Fourier series {displaystyle s_{scriptscriptstyle N}(x)} represents a synthesis of a periodic function {displaystyle s(x)} by summing harmonically related sinusoids (called harmonics) whose coefficients are determined by harmonic analysis.

Common forms The Fourier series can be represented in different forms. The amplitude-phase form, sine-cosine form, and exponential form are commonly used and are expressed here for a real-valued function {displaystyle s(x)} . (See § Complex-valued functions and § Other common notations for alternative forms).

The number of terms summed, {displaystyle N} , is a potentially infinite integer. Even so, the series might not converge or exactly equate to {displaystyle s(x)} at all values of {displaystyle x} (such as a single-point discontinuity) in the analysis interval. For the well-behaved functions typical of physical processes, equality is customarily assumed, and the Dirichlet conditions provide sufficient conditions.

The integer index, {displaystyle n} , is also the number of cycles the {displaystyle n^{text{th}}} harmonic makes in the function's period {displaystyle P} .[C] Therefore: The {displaystyle n^{text{th}}} harmonic's wavelength is {displaystyle {tfrac {P}{n}}} and in units of {displaystyle x} . The {displaystyle n^{text{th}}} harmonic's frequency is {displaystyle {tfrac {n}{P}}} and in reciprocal units of {displaystyle x} . Fig 1. The top graph shows a non-periodic function s(x) in blue defined only over the red interval from 0 to P. The function can be analyzed over this interval to produce the Fourier series in the bottom graph. The Fourier series is always a periodic function, even if original function s(x) wasn't. Amplitude-phase form The Fourier series in amplitude-phase form is: Fourier series, amplitude-phase form {displaystyle s_{scriptscriptstyle N}(x)={frac {A_{0}}{2}}+sum _{n=1}^{N}A_{n}cdot cos left({tfrac {2pi }{P}}nx-varphi _{n}right)}         (Eq.1) Its {displaystyle n^{text{th}}} harmonic is {displaystyle A_{n}cdot cos left({tfrac {2pi }{P}}nx-varphi _{n}right)} . {displaystyle A_{n}} is the {displaystyle n^{text{th}}} harmonic's amplitude and {displaystyle varphi _{n}} is its phase shift. The fundamental frequency of {displaystyle s_{scriptscriptstyle N}(x)} is the term for when {displaystyle n} equals 1, and can be referred to as the {displaystyle 1^{text{st}}} harmonic. {displaystyle {tfrac {A_{o}}{2}}} is sometimes called the {displaystyle 0^{text{th}}} harmonic or DC component. It is the mean value of {displaystyle s(x)} .

Clearly Eq.1 can represent functions that are just a sum of one or more of the harmonic frequencies. The remarkable thing, for those not yet familiar with this concept, is that it can also represent the intermediate frequencies and/or non-sinusoidal functions because of the potentially infinite number of terms ( {displaystyle N} ).

Fig 2. The blue curve is the cross-correlation of a square wave and a cosine function, as the phase lag of the cosine varies over one cycle. The amplitude and phase lag at the maximum value are the polar coordinates of one harmonic in the Fourier series expansion of the square wave. The corresponding Cartesian coordinates can be determined by evaluating the cross-correlation at just two phase lags separated by 90º.

The coefficients {displaystyle A_{n}} and {displaystyle varphi _{n}} can be determined by a harmonic analysis process. Consider a real-valued function {displaystyle s(x)} that is integrable on an interval that starts at any {displaystyle x_{0}} and has length {displaystyle P} . The cross-correlation function: {displaystyle mathrm {X} _{f}(tau )={tfrac {2}{P}}int _{x_{0}}^{x_{0}+P}s(x)cdot cos left(2pi f(x-tau )right),dx;quad tau in left[0,{tfrac {2pi }{f}}right]}         (Eq.2) is essentially a matched filter, with template {displaystyle cos(2pi fx)} .[D] The maximum of {displaystyle mathrm {X} _{f}(tau )} is a measure of the amplitude {displaystyle (A)} of frequency {displaystyle f} in the function {displaystyle s(x)} , and the value of {displaystyle tau } at the maximum determines the phase {displaystyle (varphi )} of that frequency. Figure 2 is an example, where {displaystyle s(x)} is a square wave (not shown), and frequency {displaystyle f} is the {displaystyle 4^{text{th}}} harmonic.

Rather than computationally intensive cross-correlation which requires evaluating every phase, Fourier analysis exploits a trigonometric identity: Equivalence of polar and Cartesian forms {displaystyle cos left({tfrac {2pi }{P}}nx-varphi _{n}right) equiv cos(varphi _{n})cdot cos left({tfrac {2pi }{P}}nxright)+sin(varphi _{n})cdot sin left({tfrac {2pi }{P}}nxright)}         (Eq.3) Substituting this into Eq.2 gives: {displaystyle {begin{aligned}mathrm {X} _{n}(varphi )&={tfrac {2}{P}}int _{P}s(x)cdot cos left({tfrac {2pi }{P}}nx-varphi right),dx;quad varphi in [0,2pi ]\&=cos(varphi )cdot underbrace {{tfrac {2}{P}}int _{P}s(x)cdot cos left({tfrac {2pi }{P}}nxright),dx} _{triangleq a_{n}}+sin(varphi )cdot underbrace {{tfrac {2}{P}}int _{P}s(x)cdot sin left({tfrac {2pi }{P}}nxright),dx} _{triangleq b_{n}}\&=cos(varphi )cdot a_{n}+sin(varphi )cdot b_{n}end{aligned}}} Note the definitions of {displaystyle a_{n}} and {displaystyle b_{n}} ,[2] and that {displaystyle a_{0}} and {displaystyle b_{0}} can be simplified: {displaystyle a_{0}={tfrac {2}{P}}int _{P}s(x)dx~,quad b_{0}=0~.} The derivative of {displaystyle mathrm {X} _{n}(varphi )} is zero at the phase of maximum correlation. {displaystyle mathrm {X} '_{n}(varphi _{n})=sin(varphi _{n})cdot a_{n}-cos(varphi _{n})cdot b_{n}=0quad longrightarrow quad tan(varphi _{n})={frac {b_{n}}{a_{n}}}quad longrightarrow quad varphi _{n}=arctan(b_{n},a_{n})} And the correlation peak value is: {displaystyle {begin{aligned}A_{n}triangleq mathrm {X} _{n}(varphi _{n}) &=cos(varphi _{n})cdot a_{n}+sin(varphi _{n})cdot b_{n}\&={frac {a_{n}}{sqrt {a_{n}^{2}+b_{n}^{2}}}}cdot a_{n}+{frac {b_{n}}{sqrt {a_{n}^{2}+b_{n}^{2}}}}cdot b_{n}={frac {a_{n}^{2}+b_{n}^{2}}{sqrt {a_{n}^{2}+b_{n}^{2}}}}&={sqrt {a_{n}^{2}+b_{n}^{2}}}.end{aligned}}} {displaystyle a_{n}} and {displaystyle b_{n}} are the Cartesian coordinates of a vector with polar coordinates {displaystyle A_{n}} and {displaystyle varphi _{n}.}   Figure 2 is an example of these relationships.

Sine-cosine form Substituting Eq.3 into Eq.1 gives: {displaystyle {displaystyle s_{scriptscriptstyle N}(x)={frac {A_{0}}{2}}+sum _{n=1}^{N}left[A_{n}cos(varphi _{n})cdot cos left({tfrac {2pi }{P}}nxright)+A_{n}sin(varphi _{n})cdot sin left({tfrac {2pi }{P}}nxright)right]}} In terms of the readily computed quantities, {displaystyle a_{n}} and {displaystyle b_{n}} , recall that: {displaystyle cos(varphi _{n})=a_{n}/A_{n}} {displaystyle sin(varphi _{n})=b_{n}/A_{n}} {displaystyle A_{0}={sqrt {a_{0}^{2}+b_{0}^{2}}}={sqrt {a_{0}^{2}}}=a_{0}} Therefore an alternative form of the Fourier series, using the Cartesian coordinates, is the sine-cosine form:[E] Fourier series, sine-cosine form {displaystyle s_{scriptscriptstyle N}(x)={frac {a_{0}}{2}}+sum _{n=1}^{N}left(a_{n}cos left({tfrac {2pi }{P}}nxright)+b_{n}sin left({tfrac {2pi }{P}}nxright)right)}         (Eq.4) Exponential form Another applicable identity is Euler's formula: {displaystyle {begin{aligned}cos left({tfrac {2pi }{P}}nx-varphi _{n}right)&{}equiv {tfrac {1}{2}}e^{ileft(2pi nx/P-varphi _{n}right)}+{tfrac {1}{2}}e^{-ileft(2pi nx/P-varphi _{n}right)}\[6pt]&=left({tfrac {1}{2}}e^{-ivarphi _{n}}right)cdot e^{i2pi (+n)x/P}+left({tfrac {1}{2}}e^{-ivarphi _{n}}right)^{*}cdot e^{i2pi (-n)x/P}end{aligned}}} (Note: the ∗ denotes complex conjugation.) Therefore, with definitions: {displaystyle c_{n}triangleq left{{begin{array}{lll}A_{0}/2&=a_{0}/2,quad &n=0\{tfrac {A_{n}}{2}}e^{-ivarphi _{n}}&={tfrac {1}{2}}(a_{n}-ib_{n}),quad &n>0\c_{|n|}^{*},quad &&n<0end{array}}right}={tfrac {1}{P}}int _{P}s(x)cdot e^{-i2pi nx/P},dx,} the final result is: Fourier series, exponential form {displaystyle s_{_{N}}(x)=sum _{n=-N}^{N}c_{n}cdot e^{i2pi nx/P}}         (Eq.5) This is the customary form for generalizing to § Complex-valued functions. Negative values of {displaystyle n} correspond to negative frequency (explained in Fourier transform § Use of complex sinusoids to represent real sinusoids). Example Plot of the sawtooth wave, a periodic continuation of the linear function {displaystyle s(x)=x/pi } on the interval {displaystyle (-pi ,pi ]} Animated plot of the first five successive partial Fourier series Consider a sawtooth function: {displaystyle s(x)={frac {x}{pi }},quad mathrm {for} -pi 1end{cases}}\end{aligned}}} Half-wave rectified sine [17]: p. 193  {displaystyle s(x)={begin{cases}A&quad {text{for }}0leq x1/2} . In the absolutely summable case, the inequality: {displaystyle sup _{x}|s(x)-s_{_{N}}(x)|leq sum _{|n|>N}|S[n]|} proves uniform convergence.

Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at {displaystyle x} if {displaystyle s} is differentiable at {displaystyle x} , to Lennart Carleson's much more sophisticated result that the Fourier series of an {displaystyle L^{2}} function actually converges almost everywhere.

Divergence Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T-periodic function need not converge pointwise.[citation needed] The uniform boundedness principle yields a simple non-constructive proof of this fact.

In 1922, Andrey Kolmogorov published an article titled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere (Katznelson 1976).

See also ATS theorem Dirichlet kernel Discrete Fourier transform Fast Fourier transform Fejér's theorem Fourier analysis Fourier sine and cosine series Fourier transform Gibbs phenomenon Half range Fourier series Laurent series – the substitution q = eix transforms a Fourier series into a Laurent series, or conversely. This is used in the q-series expansion of the j-invariant. Least-squares spectral analysis Multidimensional transform Spectral theory Sturm–Liouville theory Residue theorem integrals of f(z), singularities, poles Notes ^ except for pathological functions that don't satisfy the Dirichlet conditions ^ Convergence is possible only where the function is continuous. Jump discontinuities result in the Gibbs phenomenon. The infinite series will converge almost everywhere except the point of discontinuity. ^ Some texts define P=2π to simplify the sinusoid's argument at the expense of generality. ^ The scale factor {displaystyle {tfrac {2}{P}},} which could be inserted later, results in a series that converges to {displaystyle s(x)} instead of {displaystyle {tfrac {P}{2}}s(x).} ^ Some authors define {displaystyle a_{0}} differently than {displaystyle left.a_{n}right|_{n=0}.}   Rather their scale factor is just {displaystyle {tfrac {1}{P}},} and that of course changes Eq.4 accordingly. ^ Since the integral defining the Fourier transform of a periodic function is not convergent, it is necessary to view the periodic function and its transform as distributions. In this sense {displaystyle {mathcal {F}}{e^{i{frac {2pi nx}{P}}}}} is a Dirac delta function, which is an example of a distribution. ^ These three did some important early work on the wave equation, especially D'Alembert. Euler's work in this area was mostly comtemporaneous/ in collaboration with Bernoulli, although the latter made some independent contributions to the theory of waves and vibrations. (See Fetter & Walecka 2003, pp. 209–210). ^ These words are not strictly Fourier's. Whilst the cited article does list the author as Fourier, a footnote indicates that the article was actually written by Poisson (that it was not written by Fourier is also clear from the consistent use of the third person to refer to him) and that it is, "for reasons of historical interest", presented as though it were Fourier's original memoire. References ^ "Fourier". Unabridged (Online). n.d. ^ Dorf, Richard C.; Tallarida, Ronald J. (1993). Pocket Book of Electrical Engineering Formulas (1st ed.). Boca Raton,FL: CRC Press. pp. 171–174. ISBN 0849344735. ^ Tolstov, Georgi P. (1976). Fourier Series. Courier-Dover. ISBN 0-486-63317-9. ^ Wolfram, Eric W. "Fourier Series (eq.30)". MathWorld--A Wolfram Web Resource. Retrieved 3 November 2021. ^ Cheever, Erik. "Derivation of Fourier Series". Retrieved 3 November 2021. ^ Stillwell, John (2013). "Logic and the philosophy of mathematics in the nineteenth century". In Ten, C. L. (ed.). Routledge History of Philosophy. Vol. VII: The Nineteenth Century. Routledge. p. 204. ISBN 978-1-134-92880-4. ^ Fasshauer, Greg (2015). "Fourier Series and Boundary Value Problems" (PDF). Math 461 Course Notes, Ch 3. Department of Applied Mathematics, Illinois Institute of Technology. Retrieved 6 November 2020. ^ Cajori, Florian (1893). A History of Mathematics. Macmillan. p. 283. ^ Lejeune-Dirichlet, Peter Gustav (1829). "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données" [On the convergence of trigonometric series which serve to represent an arbitrary function between two given limits]. Journal für die reine und angewandte Mathematik (in French). 4: 157–169. arXiv:0806.1294. ^ "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" [About the representability of a function by a trigonometric series]. Habilitationsschrift, Göttingen; 1854. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 13, 1867. Published posthumously for Riemann by Richard Dedekind (in German). Archived from the original on 20 May 2008. Retrieved 19 May 2008. ^ Mascre, D.; Riemann, Bernhard (1867), "Posthumous Thesis on the Representation of Functions by Trigonometric Series", in Grattan-Guinness, Ivor (ed.), Landmark Writings in Western Mathematics 1640–1940, Elsevier (published 2005), p. 49, ISBN 9780080457444 ^ Remmert, Reinhold (1991). Theory of Complex Functions: Readings in Mathematics. Springer. p. 29. ISBN 9780387971957. ^ Nerlove, Marc; Grether, David M.; Carvalho, Jose L. (1995). Analysis of Economic Time Series. Economic Theory, Econometrics, and Mathematical Economics. Elsevier. ISBN 0-12-515751-7. ^ Wilhelm Flügge, Stresses in Shells (1973) 2nd edition. ISBN 978-3-642-88291-3. Originally published in German as Statik und Dynamik der Schalen (1937). ^ Fourier, Jean-Baptiste-Joseph (1888). Gaston Darboux (ed.). Oeuvres de Fourier [The Works of Fourier] (in French). Paris: Gauthier-Villars et Fils. pp. 218–219 – via Gallica. ^ Sepesi, G (13 February 2022). "Zeno's Enduring Example". Towards Data Science. pp. Appendix B. ^ Jump up to: a b c d e Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler [Mathematical Functions for Engineers and Physicists] (in German). Vieweg+Teubner Verlag. ISBN 978-3834807571. ^ Jump up to: a b c d Shmaliy, Y.S. (2007). Continuous-Time Signals. Springer. ISBN 978-1402062711. ^ Proakis, John G.; Manolakis, Dimitris G. (1996). Digital Signal Processing: Principles, Algorithms, and Applications (3rd ed.). Prentice Hall. p. 291. ISBN 978-0-13-373762-2. ^ "Characterizations of a linear subspace associated with Fourier series". MathOverflow. 2010-11-19. Retrieved 2014-08-08. ^ Vanishing of Half the Fourier Coefficients in Staggered Arrays ^ Siebert, William McC. (1985). Circuits, signals, and systems. MIT Press. p. 402. ISBN 978-0-262-19229-3. ^ Marton, L.; Marton, Claire (1990). Advances in Electronics and Electron Physics. Academic Press. p. 369. ISBN 978-0-12-014650-5. ^ Kuzmany, Hans (1998). Solid-state spectroscopy. Springer. p. 14. ISBN 978-3-540-63913-8. ^ Pribram, Karl H.; Yasue, Kunio; Jibu, Mari (1991). Brain and perception. Lawrence Erlbaum Associates. p. 26. ISBN 978-0-89859-995-4. Further reading William E. Boyce; Richard C. DiPrima (2005). Elementary Differential Equations and Boundary Value Problems (8th ed.). New Jersey: John Wiley & Sons, Inc. ISBN 0-471-43338-1. Joseph Fourier, translated by Alexander Freeman (2003). The Analytical Theory of Heat. Dover Publications. ISBN 0-486-49531-0. 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work Théorie Analytique de la Chaleur, originally published in 1822. Enrique A. Gonzalez-Velasco (1992). "Connections in Mathematical Analysis: The Case of Fourier Series". American Mathematical Monthly. 99 (5): 427–441. doi:10.2307/2325087. JSTOR 2325087. Fetter, Alexander L.; Walecka, John Dirk (2003). Theoretical Mechanics of Particles and Continua. Courier. ISBN 978-0-486-43261-8. Katznelson, Yitzhak (1976). An introduction to harmonic analysis (Second corrected ed.). New York: Dover Publications, Inc. ISBN 0-486-63331-4. Felix Klein, Development of mathematics in the 19th century. Mathsci Press Brookline, Mass, 1979. Translated by M. Ackerman from Vorlesungen über die Entwicklung der Mathematik im 19 Jahrhundert, Springer, Berlin, 1928. Walter Rudin (1976). Principles of mathematical analysis (3rd ed.). New York: McGraw-Hill, Inc. ISBN 0-07-054235-X. A. Zygmund (2002). Trigonometric Series (third ed.). Cambridge: Cambridge University Press. ISBN 0-521-89053-5. The first edition was published in 1935. External links "Fourier series", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Hobson, Ernest (1911). "Fourier's Series" . Encyclopædia Britannica. Vol. 10 (11th ed.). pp. 753–758. Weisstein, Eric W. "Fourier Series". MathWorld. Joseph Fourier – A site on Fourier's life which was used for the historical section of this article at the Wayback Machine (archived December 5, 2001) This article incorporates material from example of Fourier series on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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