Fourier Series and SturmLiouville Eigenvalue Problems


 Dale Gordon
 2 years ago
 Views:
Transcription
1 Fourier Series and SturmLiouville Eigenvalue Problems 2009
2 Outline Functions Fourier Series Representation Halfrange Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex Form of Fourier Series Inner Products Orthogonal Functions Selfadjoint Operators SturmLiouville Eigenvalue Probelms
3 Periodic Functions Definition (Periodic Function) A 2Lperiodic function f : R R {± } is a function such that there exists a constant L > 0 such that f(x) = f(x + 2L), x R. (1) Here 2L is called the fundamental period or just period. For example, f(x) = sin(x) is a 2πperiodic funtion with period 2π, since sin(x + 2π) = sin(x).
4 Even and Odd Functions Definition (Even and Odd Functions) A function f is even if and only if f( x) = f(x), x. A function f is odd if and only if f( x) = f(x), x. For example sin x is an odd function since sin( x) = sin x. cos x is an even function since cos( x) = cos x. Note that even function is symmetric about the yaxis. On the other hand, odd function is symmetric about the origin.
5 Examples of Even and Odd Functions
6 Piecewise Continuous Functions Definition (Piecewise Continuous Functions) A function f is said to be piecewise continuous on the interval [a, b] if 1. f(a+) and f(b ) exist, and 2. f is defined and continous on (a, b) except at a finite number of points in (a, b) where the left and right limits exits
7 Piecewise Smooth Functions Definition (Piecewise Smooth Functions) A function f, defined on the interval [a, b], is said to be piecewise smooth if f and f are piecewise continous on [a, b]. Thus f is piecewise smooth if 1. f is piecewise continous on [a, b], 2. f exists and is continous in (a, b) except possibly at finitely many points c where the onesided limits lim x c f (x) and lim x c + f (x) exist. Furthermore, lim x a + f (x) and lim x b f (x) exist.
8 Examples of Piecewise Functions Figure: A piecewise smooth function. Figure: Another piecewise smooth function.
9 Some useful integrations If f is even. Then, for any a R a f(x) dx = 2 a a 0 If f is odd. Then, for any a R a a f(x) dx = 0. f(x) dx. If f is piecewise continuous and 2Lperiodic. Then, for any a R 2L f(x) dx = 0 a a+2l f(x) dx.
10 Orthogonal Functions Definition (Orthogonal Functions) Two functions f and g are said to be orthogonal in the interval [a, b] if b a f(x)g(x) dx = 0. (2) We will come back to orthogonal functions again later.
11 Orthogonal Properties of Trigonometric Functions The orthogonal properties of sine and cosine functions are summarised as follow: π π π π π π where δ mn is the Kronecker s delta. cos mx sin nx dx = 0, (3) cos mx cos nx dx = πδ mn, (4) sin mx sin nx dx = πδ mn (5)
12 Kronecker s Delta Definition (Kronecker s Delta) δ mn = { 1, m = n 0, m n. (6)
13 Fourier Series Theorem (Fourier Series Representation) Suppose f is a 2Lperiodic piecewise smooth function, then Fourier series of f is given by f(x) = a (a n cos nωx + b n sin nωx) (7) n=1 and the Fourier series converges to f(x) if f is continuous at x and to 1 [f(x+) + f(x )] otherwise. 2 Here ω = 2π is called the fundamental frequency, while the 2L amplitudes a 0, a n, and b n are called Fourier coefficients of f and they are given by the Euler formula.
14 Euler Formula Definition (Euler Formula) The Fourier coefficients for a 2Lperiodic function f are given by a 0 = 1 L a n = 1 L b n = 1 L L L L f(x) dx, L f(x) cos nωx dx = 1 L L for n = 1, 2,.... L f(x) sin nωx dx = 1 L L L L L f(x) cos nπx L f(x) sin nπx L dx, dx.
15 Collorary If f is even and 2Lperiodic, then the Fourier series representation is f(x) = a a n cos nωx. n=1 If f is odd and 2Lperiodic, then the Fourier series representation is f(x) = b n sin nωx. n=1 Here, a 0, a n, and b n are given by the Euler formula.
16 Examples of Fourier Series (Odd function, digital impulses) Find the Fourier representation of the periodic function f(x) with period 2π, where { 1, π < x < 0, f(x) = 1, 0 < x < π. [Answer:] f(x) = n odd 4 sin nx nπ = k=1 4 sin(2k 1)x (2k 1)π.
17 Graphs for Digital Pulse Train and its Fourier Series
18 Examples of Fourier Series (Even function) Find the Fourier series for f(x) = x if 1 < x < 1 and f(x + 2) = f(x). [Answer:] f(x) = 1 2 n=1 4 cos(2n 1)x (2n 1) 2 π 2.
19 Graphs for f(x) = x, f(x + 2) = f(x)
20 Examples of Fourier Series Find the Fourier series of the 2periodic function f(x) = x 3 + π if 1 < x < 1. [Answer:] f(x) = π + n=1 ( 1) n 12 2n2 π 2 n 3 π 3 sin nπx.
21 Graphs for f(x) = x 3 + π, f(x + 2) = f(x)
22 Fullrange Extensions Consider a function f that is only defined in the interval [0, p). We could always extension the function outside the range to produce a new function. Of course, we have infinite many ways to extend the function, but here we will focus only on three specific extensions. Definition (Fullrange Periodic Extension) The fullrange periodic extension g of a function f defined in [0, p) is a pperiodic function given by g(x) = f(x) if 0 x < p, g(x) = g(x + p) if x < 0, g(x) = g(x p) if x p.
23 Halfrange Periodic Extensions Definition (Halfrange Even Periodic Extension) The halfrange even periodic extension f e of a function f defined in [0, p) is a 2pperiodic even function given by { f(x) 0 x < p, f e (x) = f( x) p x < 0.
24 Halfrange Periodic Extensions Definition (Halfrange Odd Periodic Extension) The halfrange odd periodic extension f o of a function f defined in [0, p) is a 2pperiodic odd function given by { f(x) 0 x < p, f e (x) = f( x) p x < 0.
25 Examples Periodic Extensions
26 Fullrange Fourier Series for f defined on [0, p) Theorem If f(x) is a piecewise smooth function defined on an interval [0, p), then f has a fullrange Fourier series expansion f(x) = a (a n cos nωx + b n sin nωx), 0 x < p, (8) n=1 where ω = 2π and the Fourier coefficients p a 0 = 1 p f(x) dx, a n = 1 p f(x) cos nωx dx, and p/2 0 p/2 0 b n = 1 p f(x) sin nωx dx. p/2 0
27 Fourier Cosine Series for f defined on [0, p) Theorem If f(x) is a piecewise smooth function defined on an interval [0, p), then f has a halfrange Fourier cosine series expansion f(x) = a a n cos nωx, 0 x < p, (9) n=1 where ω = π p and the Fourier coefficients a 0 = 2 p and a n = 2 p p 0 f(x) cos nωx dx. p 0 f(x) dx,
28 Fourier Since Series for f defined on [0, p) Theorem If f(x) is a piecewise smooth function defined on an interval [0, p), then f has a halfrange Fourier sine series expansion where ω = π p b n = 2 p p 0 f(x) = b n sin nωx, 0 x < p, (10) n=1 and the Fourier coefficients f(x) sin nωx dx.
29 Examples Consider a signal f(t) = t measured from an experiment over the duration given by 0 <= t < 4. Sketch the fullrange periodic extension of f(t). Find the corresponding Fourier expansion of f(t). Sketch the halfrange even extension of f(t) and find the corresponding Fourier cosine expansion of f(t). Sketch the halfrange odd extension of f(t) and find the corresponding Fourier sine expansion of f(t).
30 Convergence of Fourier Series In the Fourier Series Representation Theorem, we were saying that for every 2Lperiodic piecewise smooth function f, we could construct a partial sum s N (x) = a N 2 + (a n cos nωx + b n sin nωx). n=1 And, when N, the partial sum s N (x) converges to f(x), if f(x) is continuous for all x; 1 [f(x+) + f(x )], at the discontinuous points, or jumps. 2
31 Figure: s N (x) converges to f(x), except at the jumps. Figure: s N (x) converges uniformly on the interval [1, 1]
32 Pointwise Convergence Definition (Pointwise Convergence) A sequence of functions {s n } is said to converge pointwise to the function f on the set E, if the sequence of numbers {s n (x)} converges to the number f(x), for each x in E. Or, if x E, s n (x) f(x), then {s n } converge pointwise to f.
33 Uniform Convergence Definition (Uniform Convergence) We say that s n converges to f uniformly on a set E, and we write s n f uniformly on E if, given ɛ > 0, we can find a positive integer N such that for all n N s n (x) f(x) < ɛ, x E. Definition (Uniform Convergence Series) A series s(x) = k=0 u k(x) is said to converge uniformly to f(x) on a set E if the sequence of partial sums s n (x) = n k=0 u k(x) converges uniformly to f(x).
34 Note that if a sequence of partial sums s n converges uniformly to f, then s n is also pointwise convergence. However, the converse is not always true. In order to determine is a s n is uniformly convergence, we use the WeierstraßMtest.
35 WeierstraßMTest Theorem (WeierstraßMTest) Let {u k } k=0 be a sequence of real or complexvalued functions on E. If there exists a sequence {M k } k=0 of nonnegative real numbers such that the following two conditions hold: u k (x) M k, x E, and Then k=0 M k <. u k (x) converges uniformly on E. k=0
36 Gibbs Phenomena Here is an example of nonuniform convergence. The peaks remain same height but the width of the peaks changes. N=10 N=30 N= sqr(x) S10(x) 1.5 sqr(x) S30(x) 1.5 sqr(x) S50(x) N=10 N=30 N= err(x) 0.8 err(x) 0.8 err(x)
37 Mean Square Error Since s N converges to f only when N, for most practical purposes, we need N to be large but finite. Thus, we are approximating f with s N, and it is important for us to keep track of the error of the approximation. Definition (Mean Square Error) The mean square error of the partial sum s N relative to f is E N = 1 2L = 1 2L L L L [f(x) s N (x)] 2 dx L[f(x)] 2 dx 1 4 a N n=1 ( ) a 2 n + b 2 n.
38 Mean Square Approximation Theorem (Mean Square Approximation) Suppose that f is square integrable, i.e. L L f(x) 2 dx on [ L, L]. Then s N, the Nth partial sum of the Fourier series of f, approximates f in the mean square sense with an error E N that decreases to zero as N. lim E N = 1 L N 2L L[f(x)] 2 dx 1 4 a N n=1 ( ) a 2 n + b 2 n = 0.
39 Bessel s Inequality and Parseval s Identity Since E N > 0, from the definition of E N, we get Definition (Bessel s Inequality) 1 4 a N n=1 (a 2 n + b 2 n) 1 2L L L [f(x)] 2 dx. A stronger result is when taking the limit N Definition (Parseval s Identity) 1 4 a n=1 (a 2 n + b 2 n) = 1 2L L L [f(x)] 2 dx.
40 Multiplication Theorem A generalisation of the Parseval s identity is the multiplication theorem. Theorem (Multiplication (Inner Product) Theorem) If f and g are two 2Lperiodic piecewise smooth functions 1 L f(x)g(x) dx = 2L L n= c n d n where c n and d n are the Fourier coefficients for the complex Fourier series of f and g respectively.
41 Complex Form of Fourier Series Theorem (Complex Form of Fourier Series) Let f be a 2Lperiodic piecewise smooth function. The complex form of the Fourier series of f is n= c n e inωx, where the frequency ω = 2π/2L and the Fourier coefficients c n = 1 L 2L L f(x)e inωx dx, n = 0, ±1, ±2,.... For all x, the complex Fourier series converges to f(x) if f is continuous at x, and to 1 [f(x+) + f(x )] otherwise. 2
42 Relations of Complex and Real Fourier Coefficients c n = c n; c 0 = 1 2 a 0 c n = 1 2 (a n ib n ); c n = 1 2 (a n + ib n ). a 0 = 2c 0 ; a n = c n + c n ; b n = i(c n c n ).
43 Complex Form of Parseval s Identity Theorem (Complex Form of Parseval s Identity) Suppose f is a square integrable 2Lperiodic piecewise smooth function on [ L, L]. Then 1 2L L L [f(x)] 2 dx = n= c n c n = n= where c n is the complex Fourier coefficients of f. c n 2
44 Example Find the complex Fourier series for the 2πperiodic function f(x) = e x defined in ( π, π).
45 Frequency Spectra The distribution of the magnitude of complex Fourier coefficients c n in frequency domain is called the amplitude spectrum of f. The distribution of p 0 = c 0 2 and p n = c n 2 in frequency domain is called the power spectrum of f.
46 Inner Products Definition (Inner Products) Let ψ and φ be (possibly complex) functions of x on the interval (a, b). Then the inner product of ψ and φ is ψ φ = b a ψ (x)φ(x) dx. Note that the notation of inner product in some books is (φ, ψ) = b a ψ (x)φ(x) dx. Please take note on the order of φ and ψ in the brackets.
47 Norm Definition (Norm) Let f be (possibly complex) function of x on the interval (a, b). Then the norm of f is ψ = ( b 1/2 f f = f(x) dx) 2. a
48 Orthogonal Functions Definition (Orthogonal Functions) The functions f and g are called orthogonal on the interval (a, b) if their inner product is zero, f g = b a f (x)g(x) dx = 0. Definition (Orthogonal Set of Functions) A set of functions {F 1, F 2, F 3,... } defined on the interval (a, b) is called an orthogonal set if F n 0 for all n; and F m F n = 0, for m n.
49 Normalisation and Orthonormal Set Definition (Normalisation) A normalised function f n for a function F n, with F n 0, is defined as f n (x) = F n(x) F n. Definition (Orthonormal Set of Functions) A set of functions {f 1, f 2, f 3,... } defined on the interval (a, b) is called an orthonormal set if f n = 1 for all n; and f m f n = 0, for m n; or simply, f m f n = δ mn, where δ mn is the Kronecker s delta.
50 Generalized Fourier Series Theorem (Generalized Fourier Series) If {f 1, f 2, f 3,... } is a complete set of orthogonal functions on (a, b) and if f can be represented as a linear combination of f n, then the generalised Fourier series of f is given by f(x) = where a n = fn f f n 2 a n f n (x) = n=1 n=1 f n f f n 2 f n(x), is the generalised Fourier coefficient.
51 Parseval s Identity Theorem (Generalized Parseval s Identity) If {f 1, f 2, f 3,... } is a complete set of orthogonal functions on (a, b) and let f be such that f is finite. Then b a f(x) 2 dx = n=1 f n f 2 f n 2.
52 Orthogonality with respect to a Weight, w(x) (Inner product) f g = b a (Orthogonality) f m f n = f (x)g(x)w(x) dx b a (Generalised Fourier Series) f n f f(x) = f n f n(x) 2 n=1 f m(x)f n (x)w(x) dx = f m 2 δ mn (Generalised Parseval s Identity) b f(x) 2 f n f 2 w(x) dx = f n 2 a n=1
53 Adjoint and Selfadjoint Operators Definition (Adjoints of Differential Operators) Suppose u and v are (possible complex) functions of x in (a, b) and let L be a linear differential operator. Then, the formal adjoint M of L is another operator such that for all u and v f L[g] = M[f] g. Definition (Selfadjoint Operators) Suppose M is the formal adjoint operator for a linear operator L in space S. If M = L, then the operator L is said to be formally selfadjoint or formally Hermitian.
54 Example: Adjoint for L[φ] p(x)dφ/dx Suppose L[φ] p(x)dφ/dx, then u L[v] = b [ u p d a dx = M[u] v ] v dx = [u pv] ba b a [ ] d v dx (p u) dx In the last step, we set the boundary term to zero. The the adjoint for the operator L consists of Formal adjoint M[φ] d dx (p φ); and Boundary conditions [u pv] b a = 0. Furhermore, if p(x) is a pure imaginary constant, then M = L. i.e. L is selfadjoint.
55 Adjoint for 2 nd Order Linear Differential Operators Suppose L[φ] a 0 (x)φ + a 1 (x)φ + a 2 (x)φ. Then, u L[v] = [u a 0 v + u a 1 v v(a 0 u ) ] b a + b a v [(a 0 u ) (a 1 u ) + a 2 u ] dx = M[u] v with appropriate choice of boundary conditions. Note that M[φ] a 0 φ + (2a 0 a 1 )φ + (a 2 a 1 + a 0)φ. M can be made selfadjoint if a 1 = a 0. The selfadjoint operator S is S[φ] = d ( a 0 (x) dφ ) + a 2 (x)φ. dx dx The neccessary boundary condition is [a 0 u v a 0 v(u ) ] b a = 0.
56 Eigenvalue Problems & SturmLiouville Equation Definition (Eigenvalue Problem) The eigenvalue problem associated to a differential operator L is the equation Ly + λy = 0, where λ is called the eigenvalue, and y is called the eigenfunction. It is possible to find a weight factor w(x) > 0 for L such that S[y] w(x)l[y] is selfadjoint. The resulting eigenvalue equation is called the SturmLiouville Equation Definition (SturmLiouville Equation) [S + λw(x)] y = d dx a < x < b. ( a 0 (x) dy dx ) + a 2 (x)y + λw(x)y = 0 for
57 Regular SturmLiouville Problems Definition (Regular SturmLiouville Problem) A regular SL problem is a boundary value problem on a closed finite interval [a, b] of the form ( d a 0 (x) dy ) + a 2 (x)y + λw(x)y = 0, a < x < b, dx dx satisfying regularity conditions and boundary conditions c 1 y(a) + c 2 y (a) = 0, d 1 y(b) + d 2 y (b) = 0, where at least one of c 1 and c 2 and at least one of d 1 and d 2 are nonzero.
58 Singular SturmLiouville Problems Definition (Regularity Conditions) The regularity conditions of a regular SL problem are a 0 (x), a 0(x), a 2 (x) and w(x) are continuous in [a, b]; a 0 (x) > 0 and w(x) > 0. Definition (Singular SturmLiouville Problem) A singular SL problem is a boundary value problem consists of SturmLiouville equation, but either fails the regularity conditions; or infinite boundary conditions; or one or more of the coefficients become singular.
59 Solutions of SL Problems A trivial (not useful) solution to the SL problem is y = 0. Other nontrivial solutions would be the eigenfunctions y m, and for each of these eigenfunctions there is a corresponding eigenvalue λ m. There are infinite many of these eigenfunctions, and the set of eigenfunctions {y 1, y 2,..., y m,... } forms a complete orthogonal set of functions that span the infinite dimensional Hilbert space. Any function f in the Hilbert space can be expressed as a linear combination of the eigenfunctions, f(x) = a n y n (x). n=1
60 Eigenvalues of SturmLiouville Problems Theorem (SturmLiouville Problem) The eigenvalues and eigenfunctions of a SL problem has the properties of All eigenvalues are real and compose a countably infinite collections satisfying λ 1 < λ 2 < λ 3 <... where λ j as j. To each eigenvalue λ j there corresponds only to one independent eigenfunction y j (x). The eigenfunctions y j (x), j = 1, 2,..., compose a complete orthogonal set with appropriate to the weight functions w(x) in doublyintegrable functions space L 2 (a, b).
61 Eigenfunction Expansions Theorem (Eigenfunction Expansions) If f L 2 (a, b) then eigenfunction expansion of f on {y 1, y 2,... } is where A n = y n f y n 2 = f(x) = b a A n y n, a < x < b, n=1 y n(x)f(x)w(x) dx/ b a y n (x) 2 w(x) dx.
62 Example SL Problem: Harmonic Equation An example of SL equation is the Harmonic Equation with boundary condition. ODE : y + λy = 0, 0 < x < L. Dirichlet Boundary condition: y(0) = y(l) = 0. Eigenvalues: λ n = kn 2 = n2 π 2, n = 1, 2,... ; L2 ( nπ ) Eigenfunctions: y n (x) = sin L x for n = 1, 2,... ; Eigenfunction expansion: f(x) = b n sin nπ L x. n=1
63 Example SL Problem: Harmonic Equation Again, the Harmonic Equation with another boundary condition. ODE : y + λy = 0, 0 < x < L. Neumann Boundary condition: y (0) = y (L) = 0. Eigenvalues: λ n = kn 2 = n2 π 2 L ; 2 ( nπ ) Eigenfunctions: y n (x) = cos L x for n = 0, 1, 2,... ; Eigenfunction expansion: f(x) = a a n cos nπ L x. n=1
64 Example SL Problem: Harmonic Equation The Harmonic Equation with periodic boundary condition. ODE: y + λy = 0, 0 < x < 2π. Periodic boundary condition: y(0) = y(2π). Eigenvalues: λ n = kn 2 = n 2 ; Eigenfunctions: y n (x) = e inx for n = 0, ±1, ±2,... ; Eigenfunction expansion: f(x) = c n e inx. n=
65 Example SL Problem: (Parametric) Bessel Equation Bessel Equation. x 2 y + xy + (λ 2 x 2 µ 2 )y = 0 or [x 2 y ] + (λ 2 x µ2 )y = 0 in the interval 0 < x < L. x Since a 0 (x) = x 2 is zero at x = 0, = singular SL problem. ODE: x 2 y + xy + (λ 2 x 2 µ 2 )y = 0; Boundary conditions: y(x) is bounded, and y(l) = 0; Eigenvalues: λ 2 = λ 2 n = αn, n = 1, 2,..., where α L n is the nthroot of J µ, i.e. J µ (α n ) = 0; Eigenfunctions: y n (x) = J µ (λ n x), n = 1, 2,... ; Weight: w(x) = x; Eigenfunction expansion: f(x) = n=1 a nj µ (λ n x).
66 Example SL Problem: Spherical Bessel Equation Bessel Equation. x 2 y + xy + (λ 2 x 2 n(n + 1))y = 0 in the interval 0 < x <. Since a 0 (x) = x 2 is zero at x = 0, = singular SL problem. ODE: x 2 y + xy + (λ 2 x 2 µ(µ + 1))y = 0; Boundary conditions: y(x) is bounded, and y(l) = 0; Eigenvalues: λ 2 = λ 2 n = αn L, n = 1, 2,..., where α n is the nthroot of j µ, i.e. j µ (α n ) = 0; Eigenfunctions: y n (x) = j µ (λ n x), n = 1, 2,... ; Weight: w(x) = x; Eigenfunction expansion: f(x) = n=1 a nj µ (λ n x).
67 Example SL Problem: Legendre Equation Legendre Equation. (1 x 2 )y 2xy + n(n + 1)y = 0 or [(1 x 2 )y ] + n(n + 1)y = 0, in the interval 1 < x < 1. a 0 (x) = (1 x 2 ) and a 0 (±1) = 0 = singular SL problem. ODE: (1 x 2 )y 2xy + λy = 0; Boundary conditions: y(x) is bounded at x = ±1; Eigenvalues: λ = λ n = n(n + 1), n = 0, 1, 2,... ; Eigenfunctions: y n (x) = P n (x), n = 0, 1, 2,... ; Eigenfunction expansion: f(x) = n=0 a np n (x).
Fourier Series Chapter 3 of Coleman
Fourier Series Chapter 3 of Coleman Dr. Doreen De eon Math 18, Spring 14 1 Introduction Section 3.1 of Coleman The Fourier series takes its name from Joseph Fourier (1768183), who made important contributions
More informationFourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012
Fourier series Jan Philip Solovej English summary of notes for Analysis 1 May 8, 2012 1 JPS, Fourier series 2 Contents 1 Introduction 2 2 Fourier series 3 2.1 Periodic functions, trigonometric polynomials
More informationIntroduction to SturmLiouville Theory
Introduction to Ryan C. Trinity University Partial Differential Equations April 10, 2012 Inner products with weight functions Suppose that w(x) is a nonnegative function on [a,b]. If f (x) and g(x) are
More informationMATH 461: Fourier Series and Boundary Value Problems
MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter
More informationMaths 361 Fourier Series Notes 2
Today s topics: Even and odd functions Real trigonometric Fourier series Section 1. : Odd and even functions Consider a function f : [, ] R. Maths 361 Fourier Series Notes f is odd if f( x) = f(x) for
More information1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is:
CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx
More informationLecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)
ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process
More informationChapter 11 Fourier Analysis
Chapter 11 Fourier Analysis Advanced Engineering Mathematics WeiTa Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2 11.1 Fourier Series Fourier Series Fourier series are infinite series that
More informationCHAPTER 2 FOURIER SERIES
CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =
More informationThe Fourier Transform
The Fourier Transorm Fourier Series Fourier Transorm The Basic Theorems and Applications Sampling Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGrawHill, 2. Eric W. Weisstein.
More informationRecap on Fourier series
Civil Engineering Mathematics Autumn 11 J. Mestel, M. Ottobre, A. Walton Recap on Fourier series A function f(x is called periodic if f(x = f(x + for all x. A continuous periodic function can be represented
More informationWeek 15. Vladimir Dobrushkin
Week 5 Vladimir Dobrushkin 5. Fourier Series In this section, we present a remarkable result that gives the matehmatical explanation for how cellular phones work: Fourier series. It was discovered in the
More information16 Convergence of Fourier Series
16 Convergence of Fourier Series 16.1 Pointwise convergence of Fourier series Definition: Piecewise smooth functions For f defined on interval [a, b], f is piecewise smooth on [a, b] if there is a partition
More informationChapt.12: Orthogonal Functions and Fourier series
Chat.12: Orthogonal Functions and Fourier series J.P. Gabardo gabardo@mcmaster.ca Deartment of Mathematics & Statistics McMaster University Hamilton, ON, Canada Lecture: January 10, 2011. 1/3 12.1:Orthogonal
More informationFourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks
Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results
More informationFourier Series. Some Properties of Functions. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Fourier Series Today 1 / 19
Fourier Series Some Properties of Functions Philippe B. Laval KSU Today Philippe B. Laval (KSU) Fourier Series Today 1 / 19 Introduction We review some results about functions which play an important role
More informationLEGENDRE POLYNOMIALS AND APPLICATIONS. We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.
LEGENDRE POLYNOMIALS AND APPLICATIONS We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.. Legendre equation: series solutions The Legendre equation is
More information1. the function must be periodic; 3. it must have only a finite number of maxima and minima within one periodic;
Fourier Series 1 Dirichlet conditions The particular conditions that a function f(x must fulfil in order that it may be expanded as a Fourier series are known as the Dirichlet conditions, and may be summarized
More informationFourier Series. 1. Fullrange Fourier Series. ) + b n sin L. [ a n cos L )
Fourier Series These summary notes should be used in conjunction with, and should not be a replacement for, your lecture notes. You should be familiar with the following definitions. A function f is periodic
More informationFourier Series Representations
Fourier Series Representations Introduction Before we discuss the technical aspects of Fourier series representations, it might be well to discuss the broader question of why they are needed We ll begin
More informationSine and Cosine Series; Odd and Even Functions
Sine and Cosine Series; Odd and Even Functions A sine series on the interval [, ] is a trigonometric series of the form k = 1 b k sin πkx. All of the terms in a series of this type have values vanishing
More informationIntroduction to Fourier Series
Introduction to Fourier Series A function f(x) is called periodic with period T if f(x+t)=f(x) for all numbers x. The most familiar examples of periodic functions are the trigonometric functions sin and
More informationAn introduction to generalized vector spaces and Fourier analysis. by M. Croft
1 An introduction to generalized vector spaces and Fourier analysis. by M. Croft FOURIER ANALYSIS : Introduction Reading: Brophy p. 5863 This lab is u lab on Fourier analysis and consists of VI parts.
More informationTHE DIFFUSION EQUATION
THE DIFFUSION EQUATION R. E. SHOWALTER 1. Heat Conduction in an Interval We shall describe the diffusion of heat energy through a long thin rod G with uniform cross section S. As before, we identify G
More informationEngineering Mathematics II
PSUT Engineering Mathematics II Fourier Series and Transforms Dr. Mohammad Sababheh 4/14/2009 11.1 Fourier Series 2 Fourier Series and Transforms Contents 11.1 Fourier Series... 3 Periodic Functions...
More informationApplications of Fourier series
Chapter Applications of Fourier series One of the applications of Fourier series is the evaluation of certain infinite sums. For example, n= n,, are computed in Chapter (see for example, Remark.4.). n=
More informationFourier Series and Fejér s Theorem
wwu@ocf.berkeley.edu June 4 Introduction : Background and Motivation A Fourier series can be understood as the decomposition of a periodic function into its projections onto an orthonormal basis. More
More informationMATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π
MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationOutline of Complex Fourier Analysis. 1 Complex Fourier Series
Outline of Complex Fourier Analysis This handout is a summary of three types of Fourier analysis that use complex numbers: Complex Fourier Series, the Discrete Fourier Transform, and the (continuous) Fourier
More informationFourier Series, Integrals, and Transforms
Chap. Sec.. Fourier Series, Integrals, and Transforms Fourier Series Content: Fourier series (5) and their coefficients (6) Calculation of Fourier coefficients by integration (Example ) Reason hy (6) gives
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationCHAPTER 3. Fourier Series
`A SERIES OF CLASS NOTES FOR 20052006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL
More informationM344  ADVANCED ENGINEERING MATHEMATICS Lecture 9: Orthogonal Functions and Trigonometric Fourier Series
M344  ADVANCED ENGINEERING MATHEMATICS ecture 9: Orthogonal Functions and Trigonometric Fourier Series Before learning to solve partial differential equations, it is necessary to know how to approximate
More informationFrom Fourier Series to Fourier Integral
From Fourier Series to Fourier Integral Fourier series for periodic functions Consider the space of doubly differentiable functions of one variable x defined within the interval x [ L/2, L/2]. In this
More informationIntroduction to Fourier Series
Introduction to Fourier Series MA 16021 October 15, 2014 Even and odd functions Definition A function f(x) is said to be even if f( x) = f(x). The function f(x) is said to be odd if f( x) = f(x). Graphically,
More informationFourier Analysis. A cosine wave is just a sine wave shifted in phase by 90 o (φ=90 o ). degrees
Fourier Analysis Fourier analysis follows from Fourier s theorem, which states that every function can be completely expressed as a sum of sines and cosines of various amplitudes and frequencies. This
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationFourier Series and Integrals
Fourier Series and Integrals Fourier Series et f(x) be a piecewise linear function on [,] (This means that f(x) may possess a finite number of finite discontinuities on the interval). Then f(x) can be
More information) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.
Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on Fourier
More informationFourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +
Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.
More informationLecture 3: Fourier Series: pointwise and uniform convergence.
Lecture 3: Fourier Series: pointwise and uniform convergence. 1. Introduction. At the end of the second lecture we saw that we had for each function f L ([, π]) a Fourier series f a + (a k cos kx + b k
More informationSolutions to Linear Algebra Practice Problems
Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the
More information1 The Fourier Transform
Physics 326: Quantum Mechanics I Prof. Michael S. Vogeley The Fourier Transform and Free Particle Wave Functions 1 The Fourier Transform 1.1 Fourier transform of a periodic function A function f(x) that
More informationFourier cosine and sine series even odd
Fourier cosine and sine series Evaluation of the coefficients in the Fourier series of a function f is considerably simler is the function is even or odd. A function is even if f ( x) = f (x) Examle: x
More informationFourier Series Expansion
Fourier Series Expansion Deepesh K P There are many types of series expansions for functions. The Maclaurin series, Taylor series, Laurent series are some such expansions. But these expansions become valid
More informationbe a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that
Problem 1A. Let... X 2 X 1 be a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that i=1 X i is nonempty and connected. Since X i is closed in X, it is compact.
More informationf x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y
Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions:
More informationThe Dirichlet Problem on a Rectangle
The Dirichlet Problem on a Rectangle R. C. Trinity University Partial Differential Equations March 18, 014 The D heat equation Steady state solutions to the D heat equation Laplace s equation Recall:
More informationCHAPTER IV NORMED LINEAR SPACES AND BANACH SPACES
CHAPTER IV NORMED LINEAR SPACES AND BANACH SPACES DEFINITION A Banach space is a real normed linear space that is a complete metric space in the metric defined by its norm. A complex Banach space is a
More informationnπt a n cos y(x) satisfies d 2 y dx 2 + P EI y = 0 If the column is constrained at both ends then we can apply the boundary conditions
Chapter 6: Page 6 Fourier Series 6. INTRODUCTION Fourier Anaysis breaks down functions into their frequency components. In particuar, a function can be written as a series invoving trigonometric functions.
More informationDifferential Operators and their Adjoint Operators
Differential Operators and their Adjoint Operators Differential Operators inear functions from E n to E m may be described, once bases have been selected in both spaces ordinarily one uses the standard
More information, < x < Using separation of variables, u(x, t) = Φ(x)h(t) (4) we obtain the differential equations. d 2 Φ = λφ (6) Φ(± ) < (7)
Chapter1: Fourier Transform Solutions of PDEs In this chapter we show how the method of separation of variables may be extended to solve PDEs defined on an infinite or semiinfinite spatial domain. Several
More informationDISTRIBUTIONS AND FOURIER TRANSFORM
DISTRIBUTIONS AND FOURIER TRANSFORM MIKKO SALO Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth
More informationChapter 2. Fourier Analysis
Chapter 2. Fourier Analysis Reading: Kreyszig, Advanced Engineering Mathematics, 0th Ed., 20 Selection from chapter Prerequisites: Kreyszig, Advanced Engineering Mathematics, 0th Ed., 20 Complex numbers:
More informationMath Spring Review Material  Exam 3
Math 85  Spring 01  Review Material  Exam 3 Section 9.  Fourier Series and Convergence State the definition of a Piecewise Continuous function. Answer: f is Piecewise Continuous if the following to
More informationLecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if
Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close
More informationProblem 1 (10 pts) Find the radius of convergence and interval of convergence of the series
1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,
More informationR U S S E L L L. H E R M A N
R U S S E L L L. H E R M A N A N I N T R O D U C T I O N T O F O U R I E R A N D C O M P L E X A N A LY S I S W I T H A P P L I C AT I O N S T O T H E S P E C T R A L A N A LY S I S O F S I G N A L S R.
More informationINTRODUCTION TO FOURIER ANALYSIS
NTRODUCTON TO FOURER ANALYSS SAGAR TKOO Abstract. n this paper we introduce basic concepts of Fourier analysis, develop basic theory of it, and provide the solution of Basel problem as its application.
More informationCourse MA2C02, Hilary Term 2012 Section 8: Periodic Functions and Fourier Series
Course MAC, Hiary Term Section 8: Periodic Functions and Fourier Series David R. Wikins Copyright c David R. Wikins Contents 8 Periodic Functions and Fourier Series 37 8. Fourier Series of Even and Odd
More informationBounded Linear Operators on a Hilbert Space
Chapter 8 Bounded Linear Operators on a Hilbert Space In this chapter we describe some important classes of bounded linear operators on Hilbert spaces, including projections, unitary operators, and selfadjoint
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More information1 Mathematical Induction
Extra Credit Homework Problems Note: these problems are of varying difficulty, so you might want to assign different point values for the different problems. I have suggested the point values each problem
More informationSECONDORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS
L SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A secondorder linear differential equation is one of the form d
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationIn this chapter, we define continuous functions and study their properties.
Chapter 7 Continuous Functions In this chapter, we define continuous functions and study their properties. 7.1. Continuity Continuous functions are functions that take nearby values at nearby points. Definition
More informationChapter 5: Application: Fourier Series
321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1
More informationProblems for Quiz 14
Problems for Quiz 14 Math 3. Spring, 7. 1. Consider the initial value problem (IVP defined by the partial differential equation (PDE u t = u xx u x + u, < x < 1, t > (1 with boundary conditions and initial
More informationLecture5. Fourier Series
Lecture5. Fourier Series In 1807 the French mathematician Joseph Fourier (17681830) submitted a paper to the Academy of Sciences in Paris. In it he presented a mathematical treatment of problems involving
More informationQuick Reference Guide to Linear Algebra in Quantum Mechanics
Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................
More informationFourier Series & Fourier Transforms
Fourier Series & Fourier Transforms nicholas.harrison@imperial.ac.uk 19th October 003 Synopsis ecture 1 : Review of trigonometric identities Fourier Series Analysing the square wave ecture : The Fourier
More informationSolutions to Sample Midterm 2 Math 121, Fall 2004
Solutions to Sample Midterm Math, Fall 4. Use Fourier series to find the solution u(x, y) of the following boundary value problem for Laplace s equation in the semiinfinite strip < x : u x + u
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationAn Introduction to Frame Theory
An Introduction to Frame Theory Taylor Hines April 4, 2010 Abstract In this talk, I give a very brief introduction to orthonormal bases for Hilbert spaces, and how they can be naturally generalized to
More informationCAN WE INTEGRATE x 2 e x2 /2? 1. THE PROBLEM. xe x2 /2 dx
CAN WE INTEGRATE x 2 e x2 /2? MICHAEL ANSHELEVICH ABSTRACT. Not every nice function has an integral given by a formula. The standard example is e x 2 /2 dx which is not an elementary function. On the other
More informationLinear Algebra Concepts
University of Central Florida School of Electrical Engineering Computer Science EGN3420  Engineering Analysis. Fall 2009  dcm Linear Algebra Concepts Vector Spaces To define the concept of a vector
More informationDistributions, the Fourier Transform and Applications. Teodor Alfson, Martin Andersson, Lars Moberg
Distributions, the Fourier Transform and Applications Teodor Alfson, Martin Andersson, Lars Moberg 1 Contents 1 Distributions 2 1.1 Basic Definitions......................... 2 1.2 Derivatives of Distributions...................
More informationChemistry 431. NC State University. Lecture 3. The Schrödinger Equation The Particle in a Box (part 1) Orthogonality Postulates of Quantum Mechanics
Chemistry 431 Lecture 3 The Schrödinger Equation The Particle in a Box (part 1) Orthogonality Postulates of Quantum Mechanics NC State University Derivation of the Schrödinger Equation The Schrödinger
More informationNotes on Fourier Series
Notes on Fourier Series Alberto Candel This notes on Fourier series complement the textbook. Besides the textbook, other introductions to Fourier series (deeper but still elementary) are Chapter 8 of CourantJohn
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationExamination paper for Solutions to Matematikk 4M and 4N
Department of Mathematical Sciences Examination paper for Solutions to Matematikk 4M and 4N Academic contact during examination: Trygve K. Karper Phone: 99 63 9 5 Examination date:. mai 04 Examination
More informationFourier Analysis. Nikki Truss. Abstract:
Fourier Analysis Nikki Truss 09369481 Abstract: The aim of this experiment was to investigate the Fourier transforms of periodic waveforms, and using harmonic analysis of Fourier transforms to gain information
More informationSolutions for homework assignment #4
Solutions for homework assignment #4 Problem 1. Solve aplace s equation inside a rectangle x, y, with the following boundary conditions:, y) =,, y) =, ux, ) =, ux, ) = fx). y ux, y) = b + b n sinh nπ )
More informationSolutions to Problems for 2D & 3D Heat and Wave Equations
Solutions to Problems for D & 3D Heat and Wave Equations 18.303 Linear Partial Differential Equations Matthew J. Hancock Fall 006 1 Problem 1 A rectangular metal plate with sides of lengths L, H and insulated
More informationFourier Transforms The Fourier Transform Properties of the Fourier Transform Some Special Fourier Transform Pairs 27
24 Contents Fourier Transforms 24.1 The Fourier Transform 2 24.2 Properties of the Fourier Transform 14 24.3 Some Special Fourier Transform Pairs 27 Learning outcomes In this Workbook you will learn about
More informationPeriodic signals and representations
ECE 3640 Lecture 4 Fourier series: expansions of periodic functions. Objective: To build upon the ideas from the previous lecture to learn about Fourier series, which are series representations of periodic
More informationAnalysis for Simple Fourier Series in Sine and Cosine Form
Analysis for Simple Fourier Series in Sine and Cosine Form Chol Yoon August 19, 13 Abstract From the history of Fourier series to the application of Fourier series will be introduced and analyzed to understand
More informationPURE MATHEMATICS AM 27
AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)
More informationLecture Notes for Math 251: ODE and PDE. Lecture 33: 10.4 Even and Odd Functions
Lecture Notes for Math 51: ODE and PDE. Lecture : 1.4 Even and Odd Functions Shawn D. Ryan Spring 1 Last Time: We studied what a given Fourier Series converges to if at a. 1 Even and Odd Functions Before
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and
More informationMATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,
More informationAdvanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 28 Fourier Series (Contd.) Welcome back to the lecture on Fourier
More informationSolvability of Laplace s Equation Kurt Bryan MA 436
1 Introduction Solvability of Laplace s Equation Kurt Bryan MA 436 Let D be a bounded region in lr n, with x = (x 1,..., x n ). We see a function u(x) which satisfies u = in D, (1) u n u = h on D (2) OR
More informationPDE methods for Pricing Derivative Securities
PDE methods for Pricing Derivative Securities Diane Wilcox University of Cape Town. 2005 Contents I Mathematical Theory: Parabolic PDE 4 Introduction to Partial Differential Equations and Diffusion processes
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationCALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent.
CALCULUS tutorials 5/6 Repetition. Find limits of the following sequences or prove that they are divergent. a n = n( ) n, a n = n 3 7 n 5 n +, a n = ( n n 4n + 7 ), a n = n3 5n + 3 4n 7 3n, 3 ( ) 3n 6n
More informationMath 56 Homework 4 Michael Downs
Fourier series theory. As in lecture, let be the L 2 norm on (, 2π). (a) Find a unitnorm function orthogonal to f(x) = x on x < 2π. [Hint: do what you d do in vectorland, eg pick some simple new function
More informationTMA4213/4215 Matematikk 4M/N Vår 2013
Norges teknisk naturvitenskapelige universitet Institutt for matematiske fag TMA43/45 Matematikk 4M/N Vår 3 Løsningsforslag Øving a) The Fourier series of the signal is f(x) =.4 cos ( 4 L x) +cos ( 5 L
More information