3. Let (p) be a sequence of orthogonal functions on [a, b] having the property that...
4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose the sequence {f) converges uniformly on every compact subset of (a, b). Prove thatf is differen- tiable on (a, b) and that f'(x) = lim f(x) for all x E (a, b). 4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose...
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(x)g(x) dx. Given a < b, let q(x) be the line mapping a to -1 and b to 1. Prove {p;(q(x))|i = 0,... , n} is a set of orthogonal polynomials with respect to the inner product f(x)g(x) dz, satisfying deg p;(q(x))= i - 6. Let p;(xi = 0,... , n}, with degp;(x) = i, be...
3. (25 pts) Let fe C2[a, b], for a < b, and let {p,}0 be Newton's method, where p,n E [a, b] for all n 2 0. Suppose pn Converges top E [a, b], where f(p) 0, f'(p) 0, and p #p for all n 2 0. Find an expression for X 2 0, where sequence generated by a. Pn+1 -p lim = Pn-pl2 3. (25 pts) Let fe C2[a, b], for a
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists M> 0 such that n(x) M for all r E [0,1] and all n N. (b) Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? Prove or give a counterexample (2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists...
Definition. Let fi, f2.83.... be a sequence of functions defined on an interval I. The series fn(x) is said to have property 6 on I if there erists a convergent series of positive constants, Mn, satisfying \fu(x) S M for all values of n and for every or in the interval I. n=1 Theorem. If the series (1) has property C on the interval (a, b), and if the terms f(x) are continuous functions on (a, b), then nel 1...
Problem (3) A function f(z) is analytic in the disk -1 where the modulus satisfies the bound Here b 2 a > 0 Find an optimal bound on |f'(0) in terms of a and b. Complete arguments required By optimal it is meant that (1) the bound holds for all functions with the stated property and (2) there actually is a function with the stated property such that the bound holds as an equality. The second part of this problem...
are defined on the interval a <t< b #14. Assume that the real functions f(t) and p(t) that f is positive and continuous and p is integrable. Prove that f(t)e'dt< f(t)dt, a. a and that equality holds if and only if the function p assumes the same value mod 27r in all its points of continuity are defined on the interval a
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
Let H be a separable Hilbert space with basis en]nen and define P as the orthogonal projection onto span(e,... ,en) (a) A sequence of operators T, E B(H) is said to converge strongly to T if |Th-Tnhl converges to 0 for all h EH (note that strong convergence is actually weaker than operator norm convergence-think of this as the difference between pointwise and uniform convergence). Show that, for any T E B(H), the sequence P,T Pn converges strongly to T....