If f: R R belongs to C(R) and f(j) is periodic for all j > 0, prove

Question:

If f: R †’ R belongs to Cˆž(R) and f(j) is periodic for all j > 0, prove that Sf is term-by-term differentiable on R. In fact, show that

uniformly for all j ˆˆ N.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

An Introduction to Analysis

ISBN: 978-0132296380

4th edition

Authors: William R. Wade

See More Books

Question Posted: Mar 29, 2016 04:30 AM

See More Questions

Let f : R R be such that f(x + y) = f(x) + f(y) for all x, y in R. Assume that exists. Prove that L = 0, and then prove that f has a limit at every point c R. [First note that F(2x) = f(x) + f(x) =...
Define L : (0, ) R by a) Prove that L is differentiable and strictly increasing on (0, ), with L'(x) = 1/x and L(l) = 0. b) Prove that L(x) as x and L(x) - as x 0+. (You may wish to prove for...
Let ÏN: R R be a sequence of continuous, periodic functions on R which satisfy for all N N, and for each 0 uniformly for x R. 2n 2 s 00 0 2m lim a (t)d f(x)
1. The following are selected accounts taken from the adjusted trial balances of the Purell Merchandise company on December 31, 2021: Inventory, January 1, 2021 Selling expenses Loss on sale of...
Assume that your best friend has a full scholarship offer for playing football at the University of Colorado at Boulder. The scholarship covers tuition, dorm room, and meal costs at the university....
Match each description below with the following microscopic pictures. More than one picture may fit each description. A picture may be used more than once or not used at all. a. A gaseous compound b....
1 To what extent do you experience the external pressures from the financial markets for high performance and for frequent and positive financial statements? To what extent do you feel the financial...
Use the fraud triangle to analyze the red flags that existed in the case and the role and responsibilities of the auditors at Deloitte & Touche in The North Face accounting fraud.
Company X , Company Y Data relevant to the December 3 1 , 2 0 2 5 , audit of accounts receivable in two of your clients is presented in the tabulation below. ( a ) Your answer is incorrect. Determine...
Scenario: You are an accountant at Rock Castle Construction. Your boss has asked that you create general purpose financial statements from a trial balance that he has created for you. Additionally,...
Prove that if f: R R belongs to C2(R) and f, f' are both periodic, then Sf converges to f uniformly and absolutely on R. (See also Exercise 14.4.5.)
Suppose that f: R R is periodic on R, integrable on [-, ], and that ak(f) > 0 for k = 0, 1,.... a) Prove that (Skf)(0) > (Sjf)(0) for all k > j > 0. b) Prove that SNf(0) < 22Nf(0) for N N. c) Prove...
Indicate whether each of the following statements is true or false by writing T or F in t he a nswer c olumn. A contract is considered executed as soon as all parties have performed all parts of it.
Given the following differential equation, dydx = sin ( x + y ) Find the following: ( a ) The substitution u = ( b ) The transformed differential equation dudx = ( c ) The implicit solution, given...
Consider the following type declarations TYPE Alinteger; A2 pointer to float; A3 pointer to integer; T1 structure (x: integer; } T2 structure (x: A1; next pointer to integer; } b float; } a :...
https://www.viddler.com/embed/82b62f65 Questions: How do companies decide where to locate their facilities? Why has just-in-time inventory control become a dominant production process used in the...
Adjusting Entries for Interest At December 31 of Year 1, Portland Corporation had two notes payable outstanding (notes 1 and 2). At December 31 of Year 2, Portland also had two notes payable...
We want to get an idea of the actual mass of 235U involved in powering a nuclear power plant. Assume that a single fission event releases 200 MeV of thermal energy. A 1,000 MWe electric power plant...
Define attitudes and work-related attitudes, and describe the basic components of attitudes.
Imagine you are the HR manager at a company, and a female employee came to you upset because she felt a male coworker was creating a hostile work environment by repeatedly asking her out on dates...
(a) Sketch the graph of f(x) = 6 x by starting with the graph of y = x and using the transformations of Section 1.3. (b) Use the graph from part (a) to sketch the graph of f. (c) Use the definition...
(a)If f(x) = x (2/x), find f(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f.
(a)If f(t) = 6/ (1 + t2), find f(t). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f.
Which of the following journal entries will record the payment of a $1,500 salaries payable originally incurred for Salaries Expense? Select one: A. Debit Salaries Expense; credit Salaries Payable B....
What is the definition of substantially appreciated inventory? A. Inventory with a FMV greater than its basis B. Inventory and unrealized receivables with a FMV greater than their basis C. Inventory...
Case Products manufactures two models of DVD storage cases: regular and deluxe. Presented is standard cost information for each model: Cost Components Regular Deluxe Direct materials Lumber 2 board...

If f: R R belongs to C(R) and f(j) is periodic for all j > 0, prove

Question:

di f dxi (x) =-dx) (ak(f) cos kx + bk(f) sin kx) に!

Step by Step Answer:

Fix j N Since It follows from Theorem 1423 that d j dx j a k f cos kx 1k 2 for k ...View the full answer

An Introduction to Analysis

Students also viewed these Numerical Analysis questions