Function Space and Convergence Types

Transcription

1 Function Space and Convergence Types PHYS Southern Illinois University November 1, 2016 PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

2 Recall Theorem Let {φ 1, φ 2, } be an orthonormal set for a Hilbert space H. The following statements are equivalent: 1 The set {φ 1, φ 2, } is complete. 2 Every vector x H can be expressed as x. = k=1 (φ k, x)φ k. 3 Every vector x H satisfies x 2 = k=1 (φ k, x) 2. 4 Every pair of vectors x, y H satisfies (x, y) = k=1 (x, φ k)(φ k, y). PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

3 Recall Theorem Let {φ 1, φ 2, } be an orthonormal set for a Hilbert space H. The following statements are equivalent: 1 The set {φ 1, φ 2, } is complete. 2 Every vector x H can be expressed as x. = k=1 (φ k, x)φ k. 3 Every vector x H satisfies x 2 = k=1 (φ k, x) 2. 4 Every pair of vectors x, y H satisfies (x, y) = k=1 (x, φ k)(φ k, y). Since x. = k=1 a kφ k for some constants a k, we say that the {φ 1, φ 2, } is a basis for the vector space. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

4 Recall Theorem Let {φ 1, φ 2, } be an orthonormal set for a Hilbert space H. The following statements are equivalent: 1 The set {φ 1, φ 2, } is complete. 2 Every vector x H can be expressed as x. = k=1 (φ k, x)φ k. 3 Every vector x H satisfies x 2 = k=1 (φ k, x) 2. 4 Every pair of vectors x, y H satisfies (x, y) = k=1 (x, φ k)(φ k, y). Since x. = k=1 a kφ k for some constants a k, we say that the {φ 1, φ 2, } is a basis for the vector space. But what is the meaning of x. = k=1 (φ k, x)φ k? PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

5 Convergence in the Mean Definition Let H be a Hilbert space and let {φ 1, φ 2, } be an orthonormal set. A sequence of vectors (x 1, x 2, ) converges to a vector x H if x n x 0. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

6 Convergence in the Mean Definition Let H be a Hilbert space and let {φ 1, φ 2, } be an orthonormal set. A sequence of vectors (x 1, x 2, ) converges to a vector x H if x n x 0. Definition Now let {φ 1, φ 2, } be an orthonormal set. We write x. = k=1 (φ k, x)φ k? if x n (φ k, x)φ k 0. k=1 PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

7 Function Space L 2 Definition The set L 2 ([a, b]) consists of all complex-valued functions of a real variable x, defined on a closed interval [a, b] which are square-integrable. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

8 Function Space L 2 Definition The set L 2 ([a, b]) consists of all complex-valued functions of a real variable x, defined on a closed interval [a, b] which are square-integrable. The set L 2 ([a, b]) is a vector space! PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

9 Function Space L 2 Definition The set L 2 ([a, b]) consists of all complex-valued functions of a real variable x, defined on a closed interval [a, b] which are square-integrable. The set L 2 ([a, b]) is a vector space! An inner product can be defined on L 2 ([a, b]) according to (f, g) = where ω(x) > 0 for x [a, b]. b a f (x)ω(x)g(x)dx, PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

10 Function Space L 2 Recall that a Hilbert space is any inner product space that is both separable and complete. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

11 Function Space L 2 Recall that a Hilbert space is any inner product space that is both separable and complete. Completeness Riesz-Fischer Theorem. Let the functions f 1 (x), f 2 (x), be elements of L 2 ([a, b]). If the the sequence is Cauchy, then it converges to some function f L 2 ([a, b]). Note this convergence is in the mean. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

12 Function Space L 2 Recall that a Hilbert space is any inner product space that is both separable and complete. Completeness Riesz-Fischer Theorem. Let the functions f 1 (x), f 2 (x), be elements of L 2 ([a, b]). If the the sequence is Cauchy, then it converges to some function f L 2 ([a, b]). Note this convergence is in the mean. Separability What about separability? We need to show that L 2 ([a, b]) has a countable basis. That is, every function f (x) can be expressed as f (x) =. k=1 a kφ k for some orthonormal set {φ 1, φ 2, }. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

13 Function Space L 2 Recall that a Hilbert space is any inner product space that is both separable and complete. Completeness Riesz-Fischer Theorem. Let the functions f 1 (x), f 2 (x), be elements of L 2 ([a, b]). If the the sequence is Cauchy, then it converges to some function f L 2 ([a, b]). Note this convergence is in the mean. Separability What about separability? We need to show that L 2 ([a, b]) has a countable basis. That is, every function f (x) can be expressed as f (x) =. k=1 a kφ k for some orthonormal set {φ 1, φ 2, }. Below we will show that L 2 ([a, b]) has a countable basis. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

14 Convergence Types in L 2 Often we are interested in different types of convergence in L 2 besides converge in the mean. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

15 Convergence Types in L 2 Often we are interested in different types of convergence in L 2 besides converge in the mean. Definition A sequence of functions (f 1, f 2, ) is said to converge pointwise to f (x) on [a, b] if for every x [a, b] and for every ɛ > 0, there exists an integer N(x, ɛ) such that n > N implies f (x) f n (x) < ɛ. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

16 Convergence Types in L 2 Often we are interested in different types of convergence in L 2 besides converge in the mean. Definition A sequence of functions (f 1, f 2, ) is said to converge pointwise to f (x) on [a, b] if for every x [a, b] and for every ɛ > 0, there exists an integer N(x, ɛ) such that n > N implies f (x) f n (x) < ɛ. Example Consider the sequence f n (x) = x2 +nx n on some interval [a, b]. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

17 Convergence Types in L 2 Definition A sequence of functions (f 1, f 2, ) is said to converge uniformly to f (x) on [a, b] if for every ɛ > 0 there exists an integer N(ɛ) such that n > N implies f (x) f n (x) < ɛ x [a, b]. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

18 Convergence Types in L 2 Definition A sequence of functions (f 1, f 2, ) is said to converge uniformly to f (x) on [a, b] if for every ɛ > 0 there exists an integer N(ɛ) such that n > N implies f (x) f n (x) < ɛ x [a, b]. Example Consider the sequence f n (x) = x n on the interval [0, 1/2]. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

19 Convergence Types in L 2 Definition A sequence of functions (f 1, f 2, ) is said to converge uniformly to f (x) on [a, b] if for every ɛ > 0 there exists an integer N(ɛ) such that n > N implies f (x) f n (x) < ɛ x [a, b]. Example Consider the sequence f n (x) = x n on the interval [0, 1/2]. Example Consider the sequence f n (x) = x n on the interval [0, 1]. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

20 Convergence Types in L 2 Uniform convergence implies convergence in the mean. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

21 Convergence Types in L 2 Uniform convergence implies convergence in the mean. Theorem Weierstrass Approximation Theorem. If the function f (x) is continuous on the closed interval [a, b], then there is a sequence p n (x) of polynomials that converges uniformly to f (x) on [a, b]. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

22 Convergence Types in L 2 Uniform convergence implies convergence in the mean. Theorem Weierstrass Approximation Theorem. If the function f (x) is continuous on the closed interval [a, b], then there is a sequence p n (x) of polynomials that converges uniformly to f (x) on [a, b]. In fact there are many different sequences of polynomials p n (x) with this property. The Legendre Polynomials are one example! PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7

23 Convergence Types in L 2 Uniform convergence implies convergence in the mean. Theorem Weierstrass Approximation Theorem. If the function f (x) is continuous on the closed interval [a, b], then there is a sequence p n (x) of polynomials that converges uniformly to f (x) on [a, b]. In fact there are many different sequences of polynomials p n (x) with this property. The Legendre Polynomials are one example! We have seen that L 2 is complete with a countable basis. It is a Hilbert space. PHYS Southern Illinois University Function Space and Convergence Types November 1, / 7