Question: Suppose that f: R R is periodic, piecewise continuous, and of bounded variation on R. Prove that if S is a trigonometric series which

Suppose that f: R → R is periodic, piecewise continuous, and of bounded variation on R. Prove that if S is a trigonometric series which converges to (f(x+) + f(x-))/2 for all x ∈ R, then S is the Fourier series of f.

Step by Step Solution

★★★★★

3.32 Rating (158 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Let gx fx fx2 for each x R By Theorem 1429 ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

741-M-N-A-D-I (824).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

Determine the moments acting at the ends of each member of the frame. EI is the constant. 1.5 k/ft 15 k- 20 ft - 24 ft-

Suppose that f : R R. If f" exists and is bounded on R, and there is an 0 > 0 such that |f'(x)| > 0 for all x R, prove that there exists a > 0 such that if |f(x0)| 0 for some x0 R, then f...

Suppose that f R* = [a, c] for all c (a, b) and that there exists (a, b) and L[a, b] such that |f(x)|

Suppose that f : R R is continuous and periodic. The modulus of continuity of f is defined by a) Show that for k N. b) Prove that for k N. c) Use part b) to give a different proof the...

4. Suppose that f : R --+ R is periodic, piecewise continuous, and of bounded variation on R. Prove that if S is a trigonometric series that converges to (f(x+) + f(x-) )/2 for all x E R, then S is...

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)

Suppose that f : R R satisfies f(x + y) = f(x) + f(y) for each x, y R. a) Show that f(nx) = nf(x) for all x R and n Z. b) Prove that f(qx) = qf(x) for all x e R and q Q. c) Prove that f is...

8. Suppose that f : R ---+ R satisfies f(x + y) = f(x) + f(y) for each x, y E R. (a) Show that f(nx) = nf(x) for all x E Rand n E Z. (b) Prove that f(qx) = qf(x) for all x E Rand q E Q. (c) Prove...

Consider the space C(R, R) of continuous real-valued functions. i) (5 points) Given any f E C(R, R), show that there exists a sequence of polyno- mials (Pn) that converges pointwise to f, i.e. limn ....

Could someone please check my work and let me know if my circled wording is OK Please state all definitions and theorems that you will need: Definition 5.4.1 Let f : D - R . We say that f is...

3. Let 00 ao ~ . S = 2 + 6(ak cos kx + bk sm kx) k=l be a trigonometric series and set for x E Rand N EN. Prove that S is the Fourier series of some continuous periodic function f : R ----; R if and...

Tradable emission permits? a) Force polluters to spend million of dollars on new equipment to meet pollution abetment regulation. b) are granted by the government and allow an increase in pollution....

7. According to the law of there is a positive relationship between price and supply; the quantity supplied supply; the change in supply O demand; change in demand O demand; quantity demanded

Scenar o You are a tax associate with a large tax accounting firm and are compling a comprehensive reporfabout a new outreach initiative for SageGreen Computer Corporation. SageGreen has hired your...

A firms production function is Q = 5L^2/3 x K^1/3. You need to find the marginal product of labor and capital. a) Does this production function exhibit constant, increasing, or decreasing returns to...

Mary operates a convenience store, "Mary's Shoppe". It has become a hangout for teenagers. She hired Frank to act as a security guard between the hours of 3 pm and 5 pm, when the store is at its...

On January 1, 2007, Dole Co. sold land that cost $210,000 for $280,000, receiving a note bearing interest at 10%. The note will be paid in three annual installments of $112,595 starting on December...

Lillian Fok is president of Lakefront Manufacturing, a producer of bicycle tires. Fok makes 1,000 tires per day with the following resources: Labor: 400 hours per day @ $12.50 per hour Raw material:...

The term promotional inducement has been suggested as an alternative to sales promotion. Explain why this term is more descriptive than the established one.

Suppose the cumulative distribution function of the random variable X is Determine the following: (a) P(X (b) P(X > 1.5) (c) P(X M 2) (d) P(X > 6) 0.2x 0sx

Suppose the cumulative distribution function of the random variable X is Determine the following: (a) P(X (b) P(X > - 1.5) (c) P(X (d) P(- 1 x

Determine that cumulative distribution function for the distribution. Exercise 4-1,

finish the code: Given integer spoonsRequested, output: "Full box", if the number of spoons is greater than or equal to 2 9 and less than or equal to 6 2 . "Jumbo box", if the number of spoons is...

Step by step solutions. Problem 30: Integer solutions of cubics ( vVV ) (i) Show that, if in is an integer such that (m - 3)' + m = (m + 3) then m' is a factor of 54. Deduce that there is no integer...

How many cups of frozen yogurt would have to be sold in order for the firm to reach its projected revenues of $1.2 million? NOTE: Each small cup of yogurt is sold for $3 and the cost of producing the...