Fourier Analysis
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Mr. V. V. Chandavale
Asst Prof
Department of mathematics
Raje Ramrao Mahavidyalaya, Jath
Fourier Analysis
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The central starting point of Fourier analysis is Fourier series. They are infinite series designed to represent general periodic functions in terms of simple ones, namely, cosines and sines. This trigonometric system is orthogonal, allowing the computation of the coefficients of the Fourier series by use of the well-known Euler formulas. Fourier series are very important to the engineer and physicist because they allow the solution of differential equations in connection with forced oscillations and the approximation of periodic functions.
Fourier Analysis
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The underlying idea of the Fourier series can be extended in two important ways. We can replace the trigonometric system by other families of orthogonal functions, e.g., Bessel functions and obtain the Sturm–Liouville expansions. The second expansion is applying Fourier series to nonperiodic phenomena and obtaining Fourier integrals and Fourier transforms.
Both extensions have important applications to solving differential equations. In a digital age, the discrete Fourier transform plays an important role. Signals, such as voice or music, are sampled and analyzed for frequencies. An important algorithm, in this context, is the fast Fourier transform.
Fourier Series
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A function f(x) is called a periodic function if f(x) is defined for all real x, except possibly at some points, and if there is some positive number p, called a period of f(x), such that
f(x+p) = f(x) (and also f (x + np) = f (x))
The graph of a periodic function has the characteristic that it can be obtained by periodic repetition of its graph in any interval of length p. The smallest positive period is often called the fundamental period.
Fourier Series
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Fourier Series
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Fourier series (examples)
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From Period 2π to Any Period p = 2L
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From Period 2π to Any Period p = 2L
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Sum and Scalar Multiple�T H E O R E M 1
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The Fourier coefficients of a sum f1 + f2 are the sums of the corresponding Fourier coefficients of f1 and f2.
The Fourier coefficients of cf are c times the corresponding Fourier coefficients of f.
Orthogonal systems
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The idea of the Fourier series was to represent general periodic functions in terms of cosines and sines. The latter formed a trigonometric system. This trigonometric system has the desirable property of orthogonality which allows us to compute the coefficient of the Fourier series by the Euler formulas. The question then arises, can this approach be generalized? That is, can we replace the trigonometric system by other orthogonal systems (sets of other orthogonal functions)? The answer is “yes” and leads to generalized Fourier series, including the Fourier-Legendre series and the Fourier-Bessel series.
Orthogonal functions
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Orthogonal series
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Orthogonal systems
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Fourier Integral
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Fourier series are powerful tools for problems involving functions that are periodic or are of interest on a finite interval only. Since, of course, many problems involve functions that are nonperiodic and are of interest on the whole x-axis, we ask what can be done to extend the method of Fourier series to such functions. This idea will lead to Fourier integrals.
The main application of Fourier integrals is in solving ODEs (Ordinary differential equations) and PDEs (Partial differential equations).
Fourier Integral
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Fourier Integral
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Fourier Integral
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Fourier Integral
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Fourier Integral
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���Thank you
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