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are defined on the interval a <t< b #14. Assume that the real functions f(t) and...
(proof) 14, we assume that the real functions /) and φ(t) are defined on the interval a t b, that/() is positive and co...
(proof)
14, we assume that the real functions /) and φ(t) are defined on the interval a t b, that/() is positive and continuous and ф(1) properly integrable. Then Equality holds if and only if the function ф(t) assumes the same value mod 2π at all its points of continuity.
14, we assume that the real functions /) and φ(t) are defined on the interval a t b, that/() is positive and continuous and ф(1) properly integrable. Then Equality holds...
real analysis hint 13 Suppose fis a continuous function on R', with period 1. Prove that...
real analysis
hint
13 Suppose fis a continuous function on R', with period 1. Prove that lim Σ f(a)-| f(t) dt 0 for every irrational real number α. Hint: Do it first for f(t)= exp (2nikt), k = 0,±1, ±2, 4.13 Let 2 be the set of functions of form P(t)-Σ_NQC2nikt. The equality holds for functions in . For given ε > 0, there is a P E 2 such that llf-Plloo < ε. Then
real analysis 1,3,8,11,12 please 4.4.3 4.4.11a Limits and Continuity 4 Chapter Remark: In the statement of T...
real analysis
1,3,8,11,12 please
4.4.3
4.4.11a
Limits and Continuity 4 Chapter Remark: In the statement of Theorem 4.4.12 we assumed that f was tone and continuous on the interval I. The fact that f is either stric tric. strictly decreasing on / implies that f is one-to-one on t one-to-one and continuous on an interval 1, then as a consequence of the value theorem the function f is strictly monotone on I (Exercise 15). This false if either f is...
3. Let (p) be a sequence of orthogonal functions on [a, b] having the property that...
3. Let (p) be a sequence of orthogonal functions on [a, b] having the property that the zero function is the only con- tinuous real-valued function f satisfying fo, dA ofor all nE N. Prove that the system (p.) is complete. (Hint: First use the hypothesis to prove that if fE P((a, b) satisfies fo, dA -0 for all n E N. then f = 0 a.e. Next use complteness of to prove that Parseval's equality holds for every fE...
Advanced linear algebra, need full proof thanks Let V be an inner product space (real or comple...
advanced linear algebra, need full proof thanks
Let V be an inner product space (real or complex, possibly
infinite-dimensional). Let
{v1, . . . , vn} be an orthonormal set of vectors.
4. Let V be an inner product space (real or complex, possibly infinite-dimensional. Let [vi,..., Vn) be an orthonormal set of vectors. a) Show that 1 (b) Show that for every x e V, with equality holding if and only if x spanfvi,..., vn) (c) Consider the space...
- Let V be the vector space of continuous functions defined f : [0,1] → R...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
2. Consider the set of functions {f(x)} of the real variable x, defined on the interval...
2. Consider the set of functions {f(x)} of the real variable x, defined on the interval -00 < x < oo that approach zero at least as quickly as xas +00 (a) (4 points) Show that the operator B=x+in is Hermitian when acting upon {f(x)}. (b) (4 points) Show that A = x + (4) is not Hermitian and determine the operator At. Determine C = AtA and show that it is Hermitian. (c) (2 points) What well-known problem in...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to f...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typica...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which...
4. Let XC((0. 1) be the space of contimuous real valued functions on interval 0, 1...
4. Let XC((0. 1) be the space of contimuous real valued functions on interval 0, 1 with metric di(f.g) S()-9(t0ldt. R defined by Show that the function p X PS)=max(f(t)|:t€ (0.1]} is not continuous at fo E X which is the identically zero function, folt) Hint: take e= 0 for all t e0, 1. 1 and for any d>0 find a function g EX with p(g)-1 and di(fo- 9) < 6.
(proof)
14, we assume that the real functions /) and φ(t) are defined on the interval a t b, that/() is positive and continuous and ф(1) properly integrable. Then Equality holds if and only if the function ф(t) assumes the same value mod 2π at all its points of continuity.
14, we assume that the real functions /) and φ(t) are defined on the interval a t b, that/() is positive and continuous and ф(1) properly integrable. Then Equality holds...
real analysis
hint
13 Suppose fis a continuous function on R', with period 1. Prove that lim Σ f(a)-| f(t) dt 0 for every irrational real number α. Hint: Do it first for f(t)= exp (2nikt), k = 0,±1, ±2, 4.13 Let 2 be the set of functions of form P(t)-Σ_NQC2nikt. The equality holds for functions in . For given ε > 0, there is a P E 2 such that llf-Plloo < ε. Then
real analysis
1,3,8,11,12 please
4.4.3
4.4.11a
Limits and Continuity 4 Chapter Remark: In the statement of Theorem 4.4.12 we assumed that f was tone and continuous on the interval I. The fact that f is either stric tric. strictly decreasing on / implies that f is one-to-one on t one-to-one and continuous on an interval 1, then as a consequence of the value theorem the function f is strictly monotone on I (Exercise 15). This false if either f is...
3. Let (p) be a sequence of orthogonal functions on [a, b] having the property that the zero function is the only con- tinuous real-valued function f satisfying fo, dA ofor all nE N. Prove that the system (p.) is complete. (Hint: First use the hypothesis to prove that if fE P((a, b) satisfies fo, dA -0 for all n E N. then f = 0 a.e. Next use complteness of to prove that Parseval's equality holds for every fE...
advanced linear algebra, need full proof thanks
Let V be an inner product space (real or complex, possibly
infinite-dimensional). Let
{v1, . . . , vn} be an orthonormal set of vectors.
4. Let V be an inner product space (real or complex, possibly infinite-dimensional. Let [vi,..., Vn) be an orthonormal set of vectors. a) Show that 1 (b) Show that for every x e V, with equality holding if and only if x spanfvi,..., vn) (c) Consider the space...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
2. Consider the set of functions {f(x)} of the real variable x, defined on the interval -00 < x < oo that approach zero at least as quickly as xas +00 (a) (4 points) Show that the operator B=x+in is Hermitian when acting upon {f(x)}. (b) (4 points) Show that A = x + (4) is not Hermitian and determine the operator At. Determine C = AtA and show that it is Hermitian. (c) (2 points) What well-known problem in...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which...
4. Let XC((0. 1) be the space of contimuous real valued functions on interval 0, 1 with metric di(f.g) S()-9(t0ldt. R defined by Show that the function p X PS)=max(f(t)|:t€ (0.1]} is not continuous at fo E X which is the identically zero function, folt) Hint: take e= 0 for all t e0, 1. 1 and for any d>0 find a function g EX with p(g)-1 and di(fo- 9) < 6.
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