Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals.

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Lecture 2: Fourier Series Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Jean Baptiste Joseph Fourier Development of a function defined in an interval into Fourier Series

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics REPRESENTING A FUNCTION BY NUMBERS function f coefficients α 1, α 2,... of the function f = a 1 φ 1 + a 2 φ 2 +... known base functions φ 1, φ 2,... 0Τ t f (t)f (t)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0 +1 –1–1 0Τ 0 –1–1 0Τ 0 –1–1 0Τ 0 –1–1 0Τ 0 –1–1 0Τ 0 –1–1 0Τ 0 –1–1 0 Τ 0 –1–1 0Τ 0 –1–1 0Τ The base functions of Fourier series

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Every base function has: Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: period:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Every base function has: Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: period: frequency:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Every base function has: Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: period: frequency: angular frequency:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Every base function has: Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: period: frequency: angular frequency: fundamental periodfundamental angular frequencyfundamental frequency

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Every base function has: Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: period: frequency: angular frequency: fundamental periodfundamental angular frequencyfundamental frequency term periodsterm frequenciesterm angular frequencies

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics simplest form: Development of a real function f(t) defined in the interval [0,T ] into Fourier series

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier basis (base functions): simplest form: Development of a real function f(t) defined in the interval [0,T ] into Fourier series

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics simplest form: Development of a real function f(t) defined in the interval [0,T ] into Fourier series Fourier basis (base functions):

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics An example for the development of a function in Fourier series Separate analysis of each term for k = 0, 1, 2, 3, 4, … 0 +1 –1–1 f (x)f (x)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0 +1 –1–1 base function k = 0 0 +1 –1–1 f (x)f (x)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0 +1 –1–1 contribution of term k = 0 0 +1 –1–1 f (x)f (x)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0 +1 –1–1 0 –1–1 k = 1 0 +1 –1–1 f (x)f (x) base functions

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics k = 1 0 +1 –1–1 0 –1–1 0 –1–1 f (x)f (x) contributions of term

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0 +1 –1–1 f (t)f (t)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Exploiting the idea of function othogonality vector: orthogonal vector basis inner product:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Exploiting the idea of function othogonality Computation of vector components: vector: orthogonal vector basis inner product:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0 0 00 00 Exploiting the idea of function othogonality Computation of vector components: vector: orthogonal vector basis inner product:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Computation of vector components: 0 0 00 00 Exploiting the idea of function othogonality vector: orthogonal vector basis inner product:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Orthogonality of Fourier base functions Inner product of two functions: Fourier basis: Norm (length) of a function: Orthogonality relations (k  m):

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ortjhogonality relations (k  m): Computation of Fourier series coefficients

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0 0 for k  m 0 00 00 Ortjhogonality relations (k  m): Computation of Fourier series coefficients

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0 00 00 0 Ortjhogonality relations (k  m): Computation of Fourier series coefficients 0 for k  m

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0 00 00 Ortjhogonality relations (k  m): 0 Computation of Fourier series coefficients 0 for k  m

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Computation of Fourier series coefficients of a known function: Computation of Fourier series coefficients

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Computation of Fourier series coefficients

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics change of notation Computation of Fourier series coefficients

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Computation of Fourier series coefficients change of notation

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Computation of Fourier series coefficients

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics ρ k = «length» θ k = «azimuth» φ k = «direction angle» φ k + θ k = 90  Alternative forms of Fourier series (polar forms) Polar coordinates ρ k, θ k or ρ k, φ k, from the Cartesian a k, b k !

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Polar coordinates ρ k, θ k or ρ k, φ k, from the Cartesian a k, b k ! ρ k = «length» θ k = «azimuth» φ k = «direction angle» φ k + θ k = 90  Alternative forms of Fourier series (polar forms)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Alternative forms of Fourier series (polar forms)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics θ k = phase (sin) Alternative forms of Fourier series (polar forms)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics θ k = phase (sin) φ k = phase (cosine) Alternative forms of Fourier series (polar forms)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series of real functions: «real» part «imaginary» part Fourier series for a complex function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series of real functions: «real» part «imaginary» part setting Fourier series for a complex function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series for a complex function Implementation of complex symbolism:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Complex form of Fourier series Development of a complex function into a Fourier series with complex base functions and complex coefficients Computation of complex coefficients for a known complex function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ortjhogonality of the complex basis Conjugateς z* of a complex number z : inner product:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Implementation of complex symbolism: Fourier series of a real function using complex notation

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series of a real function using complex notation

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics f (t) = real function Fourier series of a real function using complex notation

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics f (t) = real function The imaginary part disappears ! Fourier series of a real function using complex notation

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series of a real function: Real and complex form

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Extension of the function f (t) outside the interval [0, T ] for every integer n The extension is a periodic function, with period Τ CAUSE OF USUAL MISCONCEPTION: “Fourier series expansion deals with periodic functions» CAUSE OF USUAL MISCONCEPTION: “Fourier series expansion deals with periodic functions» T02T2T3T3T–T–T–2T–2T

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series on the circle (naturally periodic domain) θ (angle)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  T x, 0  y  T y ) Base functions: 0 TyTy TxTx 0

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  T x, 0  y  T y ) Base functions: (angular frequencies along x and y ) 0 TyTy TxTx 0

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  T x, 0  y  T y )

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Equivalent to double Fourier series: First along x and then along y (or vice-versa) Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  T x, 0  y  T y )

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Inner product: Orthogonal Fourier basis ! ή for every Α = a,b,c,d and B = a,b,c,d Fourier series on the plane

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Inner product: Orthogonal Fourier basis ! ή for every Α = a,b,c,d and B = a,b,c,d Computation of coefficients: Fourier series on the plane

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Complex form: Fourier series on the plane

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Complex form: Fourier series in n dimensions Fourier series on the plane

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series in n dimensions

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics In matrix notation: (orthogonal hyper-parallelepiped) domain of definition: ( parallelepiped volume ) Fourier series in n dimensions

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series on any interval [Α, Β]

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Question :What is the meaning of the symbol in the Fourier series expansion? Approximating a function by a finite Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Question :What is the meaning of the symbol in the Fourier series expansion? Certainly not that infinite terms must be summed! This is impossible! Approximating a function by a finite Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Question :What is the meaning of the symbol in the Fourier series expansion? Certainly not that infinite terms must be summed! This is impossible! In practice we can use only a finite sum with a «sufficiently large» integer Ν Approximating a function by a finite Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Sufficiently large Ν means: For whatever small ε > 0 there exists an integer Ν such that || f(t) – f N (t)|| < ε Approximating a function by a finite Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Attention: || f(t) – f N (t)|| < ε does not necessarily mean that the difference | f(t) – f N (t)| is small for every t !!! Sufficiently large Ν means: For whatever small ε > 0 there exists an integer Ν such that || f(t) – f N (t)|| < ε Approximating a function by a finite Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Attention: || f(t) – f N (t)|| < ε does not necessarily mean that the difference | f(t) – f N (t)| is small for every t !!! It would be desirable (though not plausible) that max | f(t) – fN(t)| < ε in the interval [0,Τ] Sufficiently large Ν means: For whatever small ε > 0 there exists an integer Ν such that || f(t) – f N (t)|| < ε Approximating a function by a finite Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics  The coefficients a k, b k become generally smaller as k increases Characteristics of the Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics  The base functions cosω k t, sinω k t have larger frequency ω k = kω T and smaller period Τ k = T/k (i.e. more detail) as k increases  The coefficients a k, b k become generally smaller as k increases Characteristics of the Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics  The base functions cosω k t, sinω k t have larger frequency ω k = kω T and smaller period Τ k = T/k (i.e. more detail) as k increases  The terms [a k cosω k t + b k sinω k t] have a more detailed contribution to f N (t) a k increases  The coefficients a k, b k become generally smaller as k increases Characteristics of the Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics  The base functions cosω k t, sinω k t have larger frequency ω k = kω T and smaller period Τ k = T/k (i.e. more detail) as k increases  The terms [a k cosω k t + b k sinω k t] have a more detailed contribution to f N (t) a k increases  As Ν increases more details are added to the Fourier series expansion  The coefficients a k, b k become generally smaller as k increases Characteristics of the Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics  The base functions cosω k t, sinω k t have larger frequency ω k = kω T and smaller period Τ k = T/k (i.e. more detail) as k increases  The terms [a k cosω k t + b k sinω k t] have a more detailed contribution to f N (t) a k increases  As Ν increases more details are added to the Fourier series expansion  For a sufficient large Ν (which?) f N (t) ia a satisfactory approximation to f(t) within a particular application  The coefficients a k, b k become generally smaller as k increases Characteristics of the Fourier series expansion

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Question :In an expansion with finite number of terms Ν, of the form which are the values of the coefficients Α 0, A k, B k for which the sum f N (t) best approximates f(t), in the sense that The finite sum of the Fourier series expansion as the best approximation of a function within an interval

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Question :In an expansion with finite number of terms Ν, of the form which are the values of the coefficients Α 0, A k, B k for which the sum f N (t) best approximates f(t), in the sense that Answer : The Fourier coefficients a 0, a k, b k The finite sum of the Fourier series expansion as the best approximation of a function within an interval

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The finite sum of the Fourier series expansion as the best approximation of a function within an interval Question :What is the meaning of the symbol in the Fourier series expansion?

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics ANSWER :The symbol means that we can choose a sufficiently large Ν, so that we can make satisfactorily small the error δf(t) = f(t) – f N (t) The finite sum of the Fourier series expansion as the best approximation of a function within an interval Question :What is the meaning of the symbol in the Fourier series expansion?

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Specifically: For every small ε there exists a corresponding integer Ν = Ν(ε) such that small mean square error ! ANSWER :The symbol means that we can choose a sufficiently large Ν, so that we can make satisfactorily small the error δf(t) = f(t) – f N (t) The finite sum of the Fourier series expansion as the best approximation of a function within an interval Question :What is the meaning of the symbol in the Fourier series expansion?

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics END

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals.

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Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals.

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