Orthonormal Systems. Fourier Series

Transcription

1 Yuliya Gorb Orthonormal Systems. Fourier Series October 31 November 3, 2017

2 Yuliya Gorb Orthonormal Systems (cont.) Let {e α} α A be an orthonormal set of points in an inner product space X. Then {e α} are linearly independent Definition (complete system) System of elements {e α} α A is complete in H iff L{e α} = H In any separable Hilbert space H there exists a complete orthonormal system {e i} Definition (maximal system) System of elements {e α} α A in an inner product space X is called maximal iff (x e α, α x = 0) Let {e α} be any orthonormal set in an inner product space X. Then there is a maximal orthonormal set B in X with {e α} B

3 Yuliya Gorb Orthonormal Systems (cont.) An orthonormal set {e α} in H is complete {e α} is maximal Let X be an inner product space, M = {e α} α A X be any orthonormal set. Then for each x X there exists at most countably many elements y M such that (x,y) 0 Definition (closed system) System of elements {e α} α A in an inner product space X is called closed iff x X : x 2 = x i 2 i=1 An orthonormal set {e α} in H is complete {e α} is closed Definition (orthonormal basis) A complete orthonormal set B in a Hilbert space H is called an orthonormal basis for H

4 Yuliya Gorb Orthonormal Systems (cont.) Lemma Let {e i } i=1 be a countable infinite orthonormal set in a Hilbert space H. Then the following statements hold: 1 The series α k e k (where α k s are scalars) converges iff series of real numbers α k 2 converges 2 Assume α ke k converges and let x = α k e k = β k e k Then α k = β k for all k 1 and 2 α k e k = α k 2

5 Yuliya Gorb Orthonormal Systems (cont.) Let H be a Hilbert space, and M = {e α } α A H be an orthonormal set. Then 1 for every x H the sum y M(x,y)y converges in H, where the sum is taken over all y M with (x,y) 0 2 x = y M(x,y)y if x M 3 x = y M(x,y)y if x span M

6 Fourier Series Yuliya Gorb Let {e i } i=1 be an orthonormal set in a Hilbert space H Definition (Fourier coefficients) Numbers x k := (x,e k ) are called Fourier coefficients of x H Definition (Fourier series) Series x k e k is called a Fourier series for x Lemma (Bessel inequality) Let {e i } i=1 be an orthonormal set in an inner product space X. Then x X: (x,e k ) 2 x 2 k

7 Fourier Series (cont.) Yuliya Gorb Let {e i } i=1 be an orthonormal set in a Hilbert space H. Then the following statements are equivalent: 1 {e i } is an orthonormal basis for H 2 For any x H one has x = (x,e k )e k 3 (Parseval Identity): x,y H: (x,y) = 4 x H: x 2 = (x,e k )(y,e k ) (x,e k ) 2, i.e. {e i } is a closed system 5 Let M be any linear set of H that contains {e i }, then M = H 6 {e i } is a maximal system Every Hilbert space H has an orthonormal basis

8 Fourier Series (cont.) Yuliya Gorb Consider finite-dimensional subspaces H n = L{e 1,...,e n} n Partial sums of Fourier series s n = (x,e k )e k is a projection of x on H n Corollary 1: For every z = n c k e k H n, where c k are some numbers (not necessarily Fourier coefficients): x s n x z This is called a minimal property of Fourier coefficients Corollary 2: Let {e i} be any orthonormal set in a Hilbert space H. Then 1 For x H: the Fourier series (x,e k )e k converges, and its sum s is a projection of x on subspace H n 2 Series (x,e k )e k converges to x {e i} is a basis of H

9 Fourier Series (cont.) Yuliya Gorb (Riesz-Fisher) Let {e i } be any orthonormal set in a Hilbert space H (not necessarily complete). Then for every c 1,...,c n,... s.t. c k 2 < there exists x H: x = (x,e k )e k. Moreover, if c k = (x,e k ) are Fourier coefficients then x 2 = c k 2

10 References Yuliya Gorb Hunter/Nachtergaele Applied Analysis pp , Naylor/Sell Lineat Operator Theory... pp ,