Problem 1. Suppose {Kδ} is a family of kernels on Rd that satisfy
(1) |Kδ(x)| ≤ Aδ−d for all δ > 0,
(2) |Kδ(x)| ≤ Aδ/|x|d+1 for all δ > 0,
(3) ∫ Rd Kδ(x)dx = 0 for all δ > 0.
Show that if f ∈ L1(Rd ), then
(f ∗ Kδ)(x) → 0 for a.e. x, as δ → 0.
Problem 1. Suppose {Kδ} is a family of kernels on Rd that satisfy (1) |Kδ(x)| ≤...
<C. Problem 1. For all x E R prove that r = 0 if V(e> 0) : Problem 2. For each of the below properties, name a function f: IRR that does not satisfy the property and prove your answer. (d) 3(e>0)
0) : Problem 2. For each of the below properties, name a function f: IRR that does not satisfy the property and prove your answer. (d) 3(e>0)
Problem 29. Suppose 1 < p < 00 and f, g E IP(X, M,). Suppos lfllp and |gl|p are nonzero, and lfgllp llfllp+ llgllp Prove that a.e 19llp
Problem 29. Suppose 1
Problem 5 (7 point) Suppose that f'(x) is continuous and that F(x) is an antiderivative of f(x). You are given the following table of values: r=0 2 = 2 * = 4 x = 6 -2 6 f(x) 6 F(x) 7 2 -4 -3 2 -4 5 3 (a) Evaluate | ((z) – 3)s -3)?f'(x)dx. (b) Evaluate (* 25 r* f" ()dx
Question 49 Solve the problem. Suppose that s* r«x) dx = 3. Find f(x) dx = 3. Find S* fix) dx and sfx) dx . 2 0; -3 4; 3 0; 3 3; -3 Question 50 Evaluate the integral. filt-far 0 등 O 626 0%
Suppose that f is integrable on (a, b) and define (f(x) if f(x) > 0 f+(x) = 3 and f (2)= if f(x) < 0, Show that f+ and f- are integrable on (a, b), and If(x) if f(x) > 0, if f(x) < 0. cb Sisleyde = [* p*(e) ds + [°r(a)di. | f(x) dx = | f+(x) dx + 1 f (x) dx.
Find all the values of x* in the interval [ - 5,0] that satisfy the equation Sx)dx = f(x*)(-a) a in the Mean-Value Theorem for Integrals, if f(x) = x2 + x. Enter your answers, rounded to two decimal places, in increasing order. If there are less than two values, enter any values first and then enter NA in the remaining answer area(s). Show Work is REQUIRED for this question: Open Show Work
Using the properties of the δ-function a) Evaluate6(az - b) f(x) dx for a 0; consider both a > 0 and a <0. b) Evaluate eiaz δ(x) c) Show for any continuous function f(x) that f(ξ) δ(z_ξμέ f(S) δ(S-x) dE and oO use this to deduce that the Dirac-delta operates as an even function, i.e., δ(x-ξ) δίξ_x). La(n-cme-b)dE-6(-b) d) Show that
Using the properties of the δ-function a) Evaluate6(az - b) f(x) dx for a 0; consider both a >...
3) Suppose F(x) is an antiderivative of f(x). Use the graph of the functionf(x) below to answer the following: flx) a) Approximate f'(6), and explain/show how you arrived at your answer 6 4 3 2 b) Explain/show why F'(6) 2 1 2 3 4 5 6 7 c) Approximate o f(x)dx, and explain/show how you arrived at your answer. d) Explain/show why f'(x)dx-3.
Suppose f(x) is an even function on the symmetric interval x 6 [-A, A] and g(x) is an odd function defined on the same interval. Which of the following must be true? A/3 A/3 84(3x) + 1 dx = 2 84(3x) + 1 dx -A/3 0 f(x) is not an odd function. A/2 A/2 ✓ f(x) dx = 2 ✓ f(x) dx -A/2 A | f(x)g?(x) dx = 0 -A
Problem: 15 Consider the function f(x)-x" on 0sxs 1, where α>0. Suppose we want to approximate f best in Lp norm by a constant c, 0< c<1, that is minimize the Lp error 1/p x -cl dx) as a function of c. Find the optimal for p 2 and p 1 and determine Ep(Cp) for each of these p- values.