Explanation Given: { ϕ } n = 1 ∞ be an orthonormal system with respect to a positive weight function w ( x ) on [ a , b ] and let f be a continuous function on [ a , b ] . Calculation: Consider the function. f ( x ) ∼ ∑ n = 1 ∞ c n ϕ n ( x ) ... ( 1 ) Since, f ( x ) is square-integral function on [ a , b ] . ∫ a b [ f ( x ) ] 2 d x < ∞ … ( 2 ) The Fourier series associated is, f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos ( n x ) + b n sin ( n x ) … ( 3 ) Consider another function S N , S N ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos ( n x ) + b n sin ( n x ) … ( 4 ) Subtract Equation (3) from Equation (4) and square, [ f ( x ) − S N ( x ) ] 2 = f 2 ( x ) − 2 f ( x ) S N ( x ) + S N 2 ( x ) … ( 5 ) Integrate Equation (5) with respect to x . ∫ a b [ f ( x ) − S N ( x ) ] 2 d x = ∫ a b [ f 2 ( x ) − 2 f ( x ) S N ( x ) + S N 2 ( x ) ] d x = ∫ a b [ f 2 ( x ) ] d x − 2 ∫ a b [ f ( x ) S N ( x ) ] d x + ∫ a b [ S N 2 ( x ) ] d x … ( 6 ) Consider the integral, ∫ a b [ S N 2 ( x ) ] d x = ∫ a b [ a 0 2 + ∑ n = 1 ∞ a n cos ( n x ) + b n sin ( n x ) ] 2 d x = a 0 2 4 ∫ a b d x + ∑ n = 1 ∞ a n 2 ∫ a b cos 2 ( n x ) d x + b n ∫ a b sin 2 ( n x ) d x = a 0 2 4 ∫ a b d x + ∑ n = 1 ∞ 1 2 a n 2 [ 1 + cos ( 2 n x ) ] a b + 1 2 b n [ 1 − cos ( 2 n x ) ] a b = ( b − a ) 2 [ a 0 2 4 + ∑ n = 1 ∞ a n 2 + b n ] Consider the integral,

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

See similar textbooks

Videos

Question

Chapter 11.3, Problem 25E

To determine

The equation ∑n=1∞cn2≤∫f2(x)w(x)dx is true if f(x)∼∑n=1∞cnϕn(x).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.3, Problem 24E

Chapter 11.3, Problem 26E

Students have asked these similar questions

Is it possible to show me how to come up with an exponential equation by showing all the steps work and including at least one mistake that me as a person can make. Like a calculation mistake and high light what the mistake is. Thanks so much.

iid 1. The CLT provides an approximate sampling distribution for the arithmetic average Ỹ of a random sample Y₁, . . ., Yn f(y). The parameters of the approximate sampling distribution depend on the mean and variance of the underlying random variables (i.e., the population mean and variance). The approximation can be written to emphasize this, using the expec- tation and variance of one of the random variables in the sample instead of the parameters μ, 02: YNEY, · (1 (EY,, varyi n For the following population distributions f, write the approximate distribution of the sample mean. (a) Exponential with rate ẞ: f(y) = ß exp{−ßy} 1 (b) Chi-square with degrees of freedom: f(y) = ( 4 ) 2 y = exp { — ½/ } г( (c) Poisson with rate λ: P(Y = y) = exp(-\} > y! y²

2. Let Y₁,……., Y be a random sample with common mean μ and common variance σ². Use the CLT to write an expression approximating the CDF P(Ỹ ≤ x) in terms of µ, σ² and n, and the standard normal CDF Fz(·).

Chapter 11 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 3E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 7E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 9E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The equation ∑ n = 1 ∞ c n 2 ≤ ∫ f 2 ( x ) w ( x ) d x is true if f ( x ) ∼ ∑ n = 1 ∞ c n ϕ n ( x ) .

Solution Summary: The author analyzes the Fourier series associated with f(x)=a_02+

Videos

The equation ∑n=1∞cn2≤∫f2(x)w(x)dx is true if f(x)∼∑n=1∞cnϕn(x).

Want to see the full answer?

Chapter 11 Solutions