f o r i = 1 : nR[ i , i ] = norm(A[ : , i ] , 2)Q[ : , i ] = A[ : , i ] / R[ i , i ]f o r j = i +1:n
R[ i , j ] = Q[ : , i ] ’ ∗ A[ : , j ]A[ : , j ] = A[ : , j ] − R[ i , j ] ∗ Q[ : , i ]
Householder QR:
% Note that the output i s R = A, and W matr i ce sf o r i = 1 : n
x = A[ i : , i ]x [ 1 ] = s i gn (x [ 1 ] ) ∗ norm(x ) + x [ 1 ]W[ i : , i ] = x / norm(x , 2)A[ i : , i : ] = A[ i : , i : ] − 2 ∗ W[ i : , i ] ∗ (W[ i : , i ] ’ ∗ A[ i : , i : ] )
% To so lve , note that% Rx = Q∗b ,f o r i = 1 : n
b [ i : ] = b [ i : ] − 2 ∗ W[ i : , i ] ∗ (W[ i : , i ] ’ ∗ b [ i : ] )
% Use back s ub s t i t u t i o n to s o l v e f o r xx = R\b
2.a.vi) Least Squares
1. Normal Equations: A∗Ax = A∗b
2. Pseudoinverse: A+ = (A∗A)−1A∗ = R−1Q∗ = V Σ−1U
3. Solution methods:
(a) Cholesky: A∗A = R∗R, where R is upper triangular (requires full rank A). Best for speed, bad for error.
(b) QR: Reduces to Rx = Q∗b, use back substitution. Good method unless A is close to rank deficient.
(c) SVD: ΣV ∗x = U∗b, then solve. Stable even if A close to rank deficient, but more time/memory consuming.
2.a.vii) Conditioning
1. Absolute condition number: κ = supδx
||δf ||||δx||
= ||J(x)||
2. Relative condition number: κ = supδx
(||δf ||||f(x)||
/||δx||||x||
)=
||J(x)||||f(x)|| / ||x||
3. Matrix-vector condition: κ = ||A||||x||||Ax||
4. Matrix condition number: κ(A) = ||A||∣∣∣∣A−1
∣∣∣∣ or ||A|| ||A+|| in rank deficient case.
8
2.a.viii) Stability
Given problem f : X → Y , and algorithm f : X → Y , for all x ∈ X,
1. Accuracy:
∣∣∣∣∣∣f(x)− f(x)∣∣∣∣∣∣
||f(x)||= O(εm)
2. Stability: ∃x :||x− x||||x||
= O(εm) and
∣∣∣∣∣∣f(x)− f(x)∣∣∣∣∣∣
||f(x)||= O(εm)
3. Backward stability: f(x) = f(x) for some x as above.
4. Both Householder triangularization is backward stable, and so is MGS provided Q∗b is formed implicitly.
2.a.ix) Gaussian Elimination
1. No pivoting: Use for hand calcs, not stable. A = LU where L = L−11 L−1
2 · · ·L−1m−1 is lower triangular, U is
upper triangular. We have `jk =xjk
xkkfor k < j ≤ m, and
Lk =
1. . .
1−`k+1,k 1
.... . .
−`mk 1
⇒ L =
1 0 · · · 0
`21 1. . .
......
. . .. . . 0
`m1 · · · `m,m−1 1
2. Partial pivoting: Do row interchanges to maximize |xkk|.
3. Full pivoting: (L′m−1 · · ·L′2L′1)(Pm−1 · · ·P2P1)A(Q1Q2 · · ·Qm−1) = L−1PAQ = U . Here, L and P are asbefore, and Q = Q1Q2 · · ·Qm is another permutation matrix. Solve for PAQ = LU .
4. Algorithms:
GE No Pivot:
U = A; L = If o r k = 1 :m−1
f o r j = k+1:mL [ j , k ] = U[ j , k ] / U[ k , k ]U[ j , k : ] = U[ j , k : ] − L [ j , k ] ∗ U[ k , k : ]
GE Partial Pivot:
U = A; L = I ; P = If o r k = 1 :m−1
f o r j = k+1:mSe l e c t i >= k : abs (U[ i , k ] ) i s maximizedU[ k , k : ] , U[ i , k : ] = U[ i , k : ] , U[ k , k : ]P [ k , : ] , P [ i , : ] = P[ i , : ] , P [ k , : ]f o r j = k+1:m
L [ j , k ] = U[ j , k ] / U[ k , k ]U[ j , k : ] = U[ j , k : ] − L [ j , k ] ∗ U[ k , k : ]
2.a.x) Cholesky Factorization
1. Given Hermitian positive definite A, solve for A = R∗R, where R is upper triangular.
A =
[a11 w∗
w K
]=
[α 0∗
w/α I
]︸ ︷︷ ︸
R∗1
[1 0∗
0 K − ww∗
a11
]︸ ︷︷ ︸
A1
[α w∗/α0 I
]︸ ︷︷ ︸
R1
= R∗1 · · ·R∗n︸ ︷︷ ︸R∗
I Rn · · ·R1︸ ︷︷ ︸R
where α =√a11.
9
2. Algorithm:
Cholesky:
R = Afo r k = 1 : n
f o r j = k+1:nR[ j , j : ] = R[ j , j : ] − R[ k , j : ] ∗ R[ k , j : ] ’ / R[ k , k ]
R[ k , k : ] = R[ k , k : ] / s q r t (R[ k , k ] )
2.a.xi) Eigenvalues
1. Eigenvalue Decomposition: A = XΛX−1, with X nonsingular, Λ diagonal.
2. Characteristic Polynomial: PA(z) = det(zI −A).
3. Similarity Transform: X ∈ Cm×m nonsingular. A, B similar if A = X−1BX.
4. Eigenvalue Multiplicity:
(a) Geometric: # of lin. indep. eigenvectors for λ.
(b) Algebraic: multiplicity of root of char. poly. for λ
(c) algebraic ≥ geometric
5. Defective/Degenerate: λ if algebraic mult. > geometric mult.
6. Nondefective: iff it has an eigenvalue decomposition.
7. Unitary Diagonalization: A = QΛQ∗, with Q unitary. ⇔ A normal.
8. Schur Factorization: A = QTQ∗, with T upper triangular. Every square matrix has one.
2.a.xii) Iterative Methods
1. Want to solve Ax = b starting with initial guess.
(b) BDF: Derived from interpolating polynomial. With α0 = 1, this is
k∑i=0
αiyn−i = hβ0f(tn, yn)
(c) Order: Order p if 0 = C0 = · · · = Cp 6= Cp+1, where
C0 =
k∑j=0
αj , Ci = (−1)i
1
i!
k∑j=0
jiαj +1
(i− 1)!
k∑j=0
ji−1βj
(d) Char. Poly.s: ρ(ξ) =
∑kj=0 αjξ
k−j , σ(ξ) =∑kj=0 βjξ
k−j .
(e) 0-Stable: All roots of ρ(ξ) satisfy |ξi| ≤ 1, and if |ξi| = 1 it is simple. Strongly stable if all |ξi| < 1(except ξ = 1), weakly stable if 0-stable but not strongly stable.
(f) Stability Region: z =ρ(eiθ)
σ(eiθ)
8. Predictor Corrector Methods: Use explicit multistep method to predict, then implicit method to correct.
2.b.vi) BVPs
{y′ = A(t)y + q(t) 0 < t < b
B0y(0) +Bby(b) = b
13
1. Fundamental solution: Y ′ = A(t)Y , Y (0) = I. Gives general solution:
y(t) = Y (t)
[c +
∫ t
0
Y −1(s)q(s) ds
], Qc = b−BbY (b)
∫ b
0
Y −1(s)q(s) ds, Q = B0 +BbY (b)
unique solution iff Q nonsingular.
2. Green’s Functions:
3. Shooting:
4. Finite Difference:
3) Analysis
1. Pointwise Convergence: ∀ε > 0, x ∈ I, ∃N(ε, x) : |fn(x)− f(x)| < ε, ∀n > N(x, ε).
2. Uniform Convergence: ∀ε > 0, ∃N(ε) : |fn(x)− f(x)| < ε ∀x and n > N(ε). ALT: limn→∞ ||fn − f ||∞ = 0.
3.a) Metric Spaces
1. A pair (M,d), M a set, d : M ×M → [0,∞) a function, satisfying
(i) d(x, y) = 0⇔ x = y
(ii) d(x, y) = d(y, x)
(iii) d(x, z) ≤ d(x, y) + d(y, z)
2. Convergence: Given (M,d), {xn}, x. xn → x if ∀ε > 0 ∃N(ε) : n > N(ε)⇒ d(xn, x) < ε.
3. Equivalence of Metrics: Given (M,d), (M,d′), equivalent if ∀ε > 0, ∃δ : B′(x, δ) ⊂ B(x, ε), and ∀ε′ > 0,∃δ′ : B(x, δ′) ⊂ B′(x, ε′).
3.a.i) `p Spaces
For 1 ≤ p ≤ ∞ the norm is given by
||x||p =
∞∑j=0
|xj |p1/p
, ||x||∞ = sup0≤j<∞
|xj |
`p = {x : ||x||p <∞}. If 1 ≤ p < q ≤ ∞, then `p ⊂ `q (strict).
1. Holder’s Inequality: 1 ≤ p, q ≤ ∞ with 1/p+ 1/q = 1. Then for x ∈ `p, y ∈ `q,
∞∑j=0
|xjyj | ≤
∞∑j=0
|xj |p1/p ∞∑
j=0
|yj |q1/q
2. Jensen’s Inequality: f : [a, b] → R convex (i.e. f(px1 + (1 − p)x2) ≤ pf(x1) + (1 − p)f(x2) for x1 < x2,0 < p < 1), a ≤ x1 ≤ · · · ≤ xn ≤ b, and pi ∈ (0, 1) with
8. Closure: A ∈ X, and F(A) = {F : F closed, A ⊆ F}. Then the closure is A = ∩F∈F(A)F .
ALT: A = { all points of closure }
(a) A ⊆ A
15
(b) A ⊆ B ⇒ A ⊆ B
(c) A = A
(d) A ∪B = A ∪B(e) ∅ = ∅
9. Gδ: Countable intersection of open sets.
10. Fσ: Countable union of closed sets.
3.b.ii) Defining Topologies and Continuity
1. Base: Collection B = {B(x, ε) : x ∈ X, ε > 0} with
(a) ∪B∈B = X
(b) x ∈ B1, B2 ⇒ ∃B3 ⊂ B1 ∩B2 : x ∈ B3
2. Sub-base: B0 ⊂ X with ∪B∈B0B = X. Then B =
{∩nk=1B
0k ∈ B0
}is a base for the topology.
3. Weak topology: X has two topologies T ,S. S weaker than T means S ⊂ T .
4. Local base:
5. Continuity: Two topological spaces (X, T ), (Y,S), function f : X → Y . f is continuous at x0 if ∀V ∈ Yopen, with f(x0) ∈ V , ∃U ⊂ X with x0 ∈ U , and x ∈ U ⇒ f(x) ∈ V .
(a) A function is continuous iff xn → x⇒ f(xn)→ f(x).
(b) If f continuous, then f−1(open) = open.
6. Convergence: xn → x if ∀U 3 x, ∃N : n ≥ N ⇒ xn ∈ U .
7. Connected: (X, T ) connected if no nonempty sets A,B ⊂ X with A ∪B = X, and A ∩B = ∅.(X, T ) connected and f : X → Y continuous ⇒ (Y,S) connected.
8. Hausdorff Space: For every x, y ∈ X, x 6= y, ∃ open U, V , with x ∈ U , y ∈ V , U ∩ V = ∅.
(a) Convergent sequences have unique limits
(b) Complement of {x} is open.
3.c) Distributions
Topological linear space of functions D. Distributions are complex-valued continuous (with respect to D) linearfunctions D′. I.e. if ϕn → ϕ in D, then 〈T, ϕn〉 → 〈T, ϕ〉.
3. CkN : k times continuously differentiable with support in [−N,N ].
4. Schwartz space S : subset of C∞ with limabsx→∞ |x|k |Dαϕ(x)| = 0 for all k, α.
5. C∞0 Convergence: ϕn → ϕ if:
(a) supp(ϕn) ⊆ [−N,N ] independent of n.
(b) Derivatives ϕ(m)n
n→∞−−−−→ ϕ(m) uniformly, i.e. limn→∞
supx∈[−N,N ]
∣∣∣ϕ(m)n (x)− ϕ(m)(x)
∣∣∣→ 0
ALT: T ∈ D′ iff ∀N , ∃B(N), k(N): |〈T, ϕ〉| ≤ B ||ϕ||N,k ∀ϕ ∈ CkN .
6. Order: Smallest k independent of N : above convergence definition holds.
7. Norm: ||ϕ||N,kdef=
k∑j=0
sup[−N,N ]
∣∣∣ϕ(j)(t)∣∣∣
8. Topology: d(ϕ1, ϕ2)def=
∞∑k=0
||ϕ1 − ϕ2||N,k1 + ||ϕ1 − ϕ2||N,k
· 1
2k
9. Convergence: {Tj} ∈ D′. If limj→∞
〈Tj , ϕ〉 = 〈T, ϕ〉 ∀ ϕ ∈ D, then Tj converges to T .
10. Derivatives:⟨T (n), ϕ
⟩= (−1)n
⟨T, ϕ(n)
⟩.
17
3.d) Measure Theory
3.d.i) Basic Definitions
1. Dense: A ⊂ B. A is dense in B if B ⊆ A. A set is dense iff (Ac)◦ = ∅.
2. Nowhere Dense: Interior of closure is empty: (A)◦ = ∅. A nowhere dense iff (Ac)◦ = X.
3. (A◦)c = Ac and (Ac)◦ = (A)c.
4. First Category: Countable union of nowhere dense sets (also meager).
5. Residual: Complement of first category (also comeager). Residual sets are dense.
6. Second Category: Not first category.
7. Measure Zero: ∀ ε > 0, ∃ {Bn} countable collection of open balls Bn = B(xn, δn) such that
A ⊂∞⋃n=1
Bn and
∞∑n=1
volume of Bn ≤ ε
8. Baire Category Theorem in R:
(a) The complement of a 1st category set is dense.
(b) The intersection of countably many open dense sets is dense.
9. Cantor Intersection Theorem: Given nonempty closed and bounded {Cn} with C0 ⊇ C1 ⊇ · · · ⊇ Cn ⊇ · · · ,then ∩Ck 6= ∅.
3.d.ii) Measurable Spaces
1. (X,B), set X, collection of subsets B satisfying
(i) X ∈ B, ∅ ∈ B(ii) A ∈ B ⇒ Ac ∈ B(iii) {Aj} ∈ B ⇒
⋃nj=1 ∈ B
This defines an algebra of sets, and if n =∞ a σ−algebra.
2. Measurable: (X,B), (Y, C) measurable spaces. Then f : X → Y is measurable if A ∈ C ⇒ f−1(A) ∈ B.
3. Borel σ−algebra: Given (X, T ), the σ−algebra generated by T .
4. Additive Measure: on (X,B). µ : B → [0,∞]
(a) µ(∅) = 0
(b) {Ak} ∈ B mutually disjoint ⇒ µ (⋃nk=1Ak) =
∑nk=1 µ(Ak) where n can by ∞.
3.d.iii) Probability Spaces
1. A measurable space + a measure, (X,B, P ), with P : B → [0, 1].
2. Borel-Cantelli:
(a) Given (X,B, P ) (P (X) = 1). Let E1, E2, . . . be events (∈ B). Then if
∞∑m=1
P (Em) <∞, P (Emi.o.) = P (lim supEm) = 0
(b) If E1, E2, . . . independent and∑∞n=1 P (En) =∞, then P (Eni.o.) = 1.
3. Independent: E1, . . . , En are independent if P (Ei1 ∩ . . . Eik) =∏kj=1 P (Eij ).
4. Borel Zero-One Law: En independently often (i.o.) can only have probability 0 or 1.
18
3.e) Convergence and Compactness
3.e.i) Basic Definitions
In a metric space (M,d).
1. Cauchy Sequence: If ∀ ε > 0, ∃N(ε) : m,n > N ⇒ d(xm, xn) < ε, then {xn} is Cauchy. If xn → x, then{xn} is Cauchy.
2. Complete: Every Cauchy sequence converges to a limit x ∈M .
(a) Rn is complete.
(b) `p is complete.
(c) Lp is complete.
(d) Closed subspace of complete space is complete.
3. Contraction Mapping Theorem: (M,d) complete metric space. f : M → M continuous, and ∃ 0 < k < 1such that d(f(x), f(y)) ≤ kd(x, y) ∀ x, y ∈M . Then there exists a unique z ∈M for which f(z) = z.
4. Sequential Compactness: In (X, T ). A ∈ X is compact if every sequence {xn} ∈ A has a convergentsubsequence with limit in A.
5. Compactness:
(a) In Rn, compact = closed and bounded.
(b) In (M,d), compact = complete and totally bounded or = sequentially compact.
(c) In (X, T ) compact if every open cover has a finite subcover.
(d) Compact sets are closed and bounded.
(e) Closed subsets of compact sets are compact.
(f) Compact sets are complete.
(g) f : M → R continuous and M compact, then f assumes its max and min values.
6. ε−Net: ε > 0, finite collection x1, . . . , xn such that ∪nj=1B(xj , ε) ⊇M .
7. Totally Bounded: M is totally bounded if there is an ε−net for every ε > 0.
8. Open Cover: (X, T ), A ⊆ X, O = {Uα}α∈I (open sets), and A ⊆ ∪α∈IUα.
9. Finite Subcover: Finite subset of O that covers A.
3.f) Uniformity
3.f.i) Basic Definitions
1. Uniformly Continuous: f : (M,d)→ (M ′, d′). If ∀ ε > 0, ∃ δ > 0 : d(x, y) < δ ⇒ d′(f(x), f(y)) < ε.If A ⊂M is compact and f : M →M ′ is continuous, then f is uniformly continuous on A.
2. Uniform Convergence: fn : (M,d)→ (M ′, d′).
(a) fn converges to f if ∀ ε, ∀ x, ∃N(x, ε) : n > N(x, ε)⇒ d′(f(x), fn(x)) < ε.
(b) fn converges uniformly to f if ∀ ε, ∃ N(ε) : n > N(ε)⇒ d′(f(x), fn(x)) < ε ∀x.
(c) {fn} is Cauchy if ∀ ε, ∀ x, ∃N(ε) : n,m > N(x, ε)⇒ d′(fn(x), fm(x)) < ε.
(d) {fn} is uniformly Cauchy if ∀ ε, ∃ N(ε) : n,m > N(ε)⇒ d′(fn(x), fm(x)) < ε ∀x.
3. fn : M →M ′, M ′ complete. fn converges Uniformly iff it is uniformly Cauchy.
4. A ⊂M , fn : A→ R.∑∞n=0 fn(x) converges uniformly on A if the sequence of partial sums converges uniformly.
5. Weirstrass M-test: (for uniform convergence). If ∃Mn ≥ 0 : |fn(x)| ≤ Mn ∀x ∈ A, and∑∞n=0Mn < ∞,
then∑∞n=0 fn converges uniformly.
19
3.f.ii) Interchanging Limits and Integrals
Given fn → f uniformly, fn continuous.
1. limh→0 limn→∞ fn(x0 + h) = limn→∞ limh→0 fn(x0 + h), i.e. f is continuous.
2.∫ ∑
fn =∑∫
fn.
3. Given fn → f (uniform not required) on [a, b], f ′n continuous, and f ′n → g uniformly. Then f ′ = g.
3.f.iii) More on Integrals
1. Riemann integrable: Define Mi = max{f(t) : ti ≤ t ≤ ti+1}, mi = min{f(t) : ti ≤ t ≤ ti+1}, and U(f ; ∆) =n∑i=0
Mi(ti+1 − ti), L(f ; ∆) =
n∑i=0
mi(ti+1 − ti) for some partition ∆. If infall ∆ U(f ; ∆) = supall ∆ L(f ; ∆) then
f is Riemann integrable.
(a) f on [a, b] is R-integrable if it is continuous.
(b) f on [a, b] is R-integrable iff the set of points of discontinuity of f has Lebesgue measure zero.
(c) Riemann integral exists ⇒ Lebesgue integral exists, and they are equal (not converse).
yn < yn+1 = M + 1. Let µi be the length of {t : yi ≤ f(t) < yi+1}. Then the Lebesgue sum is:
n∑i=0
yiµi.
Alternatively, define
f+(x) =
{f(x) if f(x) ≥ 0
0 elsef−(x) =
{−f(x) textiff(x) ≤ 0
0 else
Then f(x) = f+(x)− f−(x), and
∫X
f dµ =
∫X
f+ dµ−∫X
f− dµ (provided at least 1 of the two integrals on
the right is finite).
3. Simple Function: A1, . . . , An ∈ X pairwise disjoint measurable sets. ϕ(x) =
n∑j=1
αjcAj(x) with αj ≥ 0 is a
non-negative simple function, and has integral
∫X
ϕ dµ =
n∑j=1
αjµ(Aj).
4. Measurable Function: (X,B) a measurable space. f : X → [−∞,∞] is measurable if {t : f(t) < α} ∈ B foreach α ∈ R. Limit of measurable functions is measurable.
3.g) Convergence Theorems
3.g.i) Basic Definitions
1. Lim inf: lim infn→∞
xndef= limn→∞
(infm≥n
xm
), alternatively the leftmost limit point (or ±∞).
2. Lim sup: lim supn→∞
xndef= limn→∞
(supm≥n
xm
), alternatively the rightmost limit point (or ±∞).
4. Homoclinic Orbit: connects a fixed point to itself
4.a.iii) Dispersion Relations
4.b) Contour Integration
4.b.i) Complex Basics
1. Cauchy-Riemann: The function f(z) = u(x, y) + iv(x, y) is differentiable at z = x + iy iff ux = vy andvx = −uy, and all partials are continuous in a neighborhood of z.
2. Analytic: At a point if differentiable at that point; in a region if analytic at every point in the region.
3. Entire: Analytic at every point in C except ∞.
4. Singular Point: Where f is not analytic.
4.b.ii) Integration
1. Cauchy’s Integral Formula: f (k)(z0) =k!
2πi
∮C
f(z)
(z − z0)k+1dz.
2. Laurent Series: f(z) =
∞∑n=−∞
cn(z − z0)n where cn =1
2πi
∮C
f(z)
(z − z0)n+1dz.
(a) Strength: c−n for pole of order n.
(b) Residue: c−1 = 12π
∮Cf(z) dz
(c) Essential singularity: ∞ number of c−n terms.
3. Residue: Res (f(z), z0) = limz→z0
1
(k − 1)!
dk−1
dzk−1
[f(z)(z − z0)k
]for a pole of order k.
4. Residue at ∞: Res (f(z),∞) = Res
(− 1
z2f
(1
z
), 0
)
5. Cauchy’s Residue Theorem: For a simple closed contour C,
∮C
f(z) dz = 2πi
N∑j=1
rj , where rj are residues
of poles in the interior of C.
6. Jordan’s Lemma: If f(z)→ 0 uniformly on Reiθ, 0 ≤ θ ≤ π, and∣∣f(Reiθ
∣∣ ≤ G(R), and limR→∞G(R) = 0,
then limR→∞
∫CR
eikzf(z) dz = 0, for k > 0. Take lower arc for k < 0.
4. Has a dense orthonormal basis, for L2[a, b] : ϕn = e2πinx/(b−a)/√b− a
4.c.ii) Theorems
1. Riemann-Lebesgue: Given f ∈ L1[a, b], limn→∞ |cn| = 0. More specifically, for f ∈ Cn[a, b] (actually only
need f (n) ∈ L1[a, b]), |ck| ≤c
|k|nfor |k| ≥ 1, and constant c.
2. Parseval’s Identity: For f ∈ L2[a, b], ||f ||2L2 = 2π
∞∑k=−∞
|ck|2.
3. Carlson’s Theorem: f ∈ L2 ⇒ Sn(f)→ f pointwise almost everywhere, where Sn is the partial sum of theFourier series.
4. For f(x) periodic, piecewise smooth on [a, b], and integrable, then in (a, b), limn→∞ Sn(f) = f(x) where f iscontinuous, and limn→∞ Sn(f) = 1
2 (f(x+) + f(x−)) where f is discontinuous (convergence may break down atendpoints).
5. For f(x) periodic, continuous, and piecwise smooth on [a, b], then Sn(f)→ f uniformly. The convergence is atthe rate ||Sn(f)− f ||∞ = O(n1−p), where f (p) ∈ L1[a, b] (or O(n−p) for f ∈ Cp[a, b]).
6. Gibb’s Phenomenon: Apporximately 10% error near discontinuities in function.
7. Can integrate term by term to get Fourier series of F (x) (but need to make it periodic), and differentiate termby term to get Fourier series of f ′(x) on the condition that f ∈ C, f ′ ∈ L1.
4.d) Distributions
Also see distributions.
4.d.i) Basics
1. Null sequence: In C∞0 (R), {ϕm(x)} such that limm→∞
supx∈[−K,K]
∣∣∣∣ dndxnϕm(x)
∣∣∣∣ = 0 for all n ∈ Z+. In S(R),
{ϕm(x)} such that limm→∞
supx∈R
∣∣∣∣xk dndxnϕm(x)
∣∣∣∣ = 0 for all k, n ∈ Z+.
2. Seminorms: Examples are pm,k(ϕ) = supx∈R
∣∣∣∣xk dmϕdxm
∣∣∣∣, and qm,k(ϕ) = supx∈R
∣∣∣∣(1 + x2)kdmϕ
dxm
∣∣∣∣.23
3. Continuity: Given {ϕm} → ϕ, then T continuous if T [ϕm]→ T [ϕ].
4. To prove T is a distribution, show that T [αϕ + βψ] = αT [ϕ] + βT [ψ] (linearity) and T [ϕm] → 0 for all nullsequences (continuity). Break up integral and use seminorms to prove.
5. If f ∈ Lloc1 , then Tf is a distribution in C∞0 and S.
6. For ψ smooth, then ψT [ϕ] = T [ψϕ].
7. Change of Variables: T [ϕ] ≡∫T (y(x))ϕ(x) dx ≡
∫T (z)ϕ(y−1(z))
|y′(y−1(z))|dy, where y(x) is the change of
variables.
8. Convolution: Tf ? Tg = Tf?g. If Tfn → Tf , Tgn → Tg, then Tfn ? Tgn → S where S depends only on Tf , Tg(not n).
4.e) Fourier Transforms
F [f(x)] = f(k) =
∫ ∞−∞
f(x)e−ikx dx F−1[f(k)
]= f(x) =
1
2π
∫ ∞−∞
f(k)eikx dk
4.e.i) Common Functions/Distributions
1. Sinc and box: F [sinc(x)] = πχ[−1,1], and F[χ[−1,1]
]= 2 sinc(k)
2. Gaussian: F[e−x
2/a]
=√aπ exp
(−ak2
4
)3. Sin and cos: F [sin(ax)] = iπ [δ(k + a)− δ(k − a)] and F [cos(ax)] = π [δ(k + a) + δ(k − a)]
4. Delta: F [δ(x)] = 1, and F [1] = 2πδ(k)
5. Heaviside: F [H(x)] = πδ(k)− iPV(
1k
)6. Principal Value and sign: F
[PV
(1x
)]= −iπsign(k), and F [sign(x)] = −2iPV
(1k
)4.e.ii) Other stuff
1. Poisson Sum:
∞∑n=−∞
δ(x− 2nπ) =1
2π
∞∑k=−∞
e−ikx ⇔∞∑
n=−∞ϕ(nT ) =
1
T
∞∑k=−∞
ϕ
(2πk
T
)
2. f(k) = f(−k), f ∈ R ⇒ f Hermitian, i.e. f(k) = f(−k);f(x) = 2πf(−x)
3. F [f(x− b)] = e−ikbf(k); F [f(ax)] = 1|a| f
(ka
)4. F
[f (n)
]= (ik)nf(k); F
[∫ x−∞ f(t) dt
]= 1
ik f(k) + πf(0)δ(k)
5. F [xnf(x)] = inf (n)(k)
6. f ∈ L1 ⇒ f is bounded.
7. F : L1(R)→ C(R), F : L2(R)→ L2(R)
8. Parseval’s/Plancherel’s Theorem: If f, g ∈ L1(R), then⟨f , g⟩
= 2π 〈f, g〉, so ||f ||22 = 12π
∣∣∣∣∣∣f ∣∣∣∣∣∣22
(for
f ∈ L2(R)).
9. Convolution: F [f ? g] = f · g, and F−1[f ? g
]= 2πf · g.
10. Multi dimensions: F [f(x)] = f(k) =
∫Rn
f(x)e−ik·x dx; F−1[f(k)
]= f(x) =
1
(2π)n
∫Rn
f(k)eikx dk
11. FT of Distributions: Tf [ϕ] = Tf [ϕ] = Tf [ϕ]
24
4.e.iii) Sampling
1. Nyquist Rate: ω0/π, where f(ω) = 0 for |ω| > ω0
1. Hilbert Transform: H[f ] = 1πf ? PV (1/x), so F [H[f ]] = 1
π f(k) · F [PV (1/x)] = −isign(k)f(k).
2. Laplace Transform: L [f ] = F (s) =
∫ ∞0
f(t)e−st dt; f(t) =1
2πi
∫ c+i∞
c−i∞F (s)est ds
4.f) Spectral Theory
4.f.i) Basics
1. Operator: L : X → X. Defined on the domain D(L).
2. Bounded: If ||L|| ≤ ∞, where ||L|| = supu∈X
||Lu||||u||
. This implies ||Lu|| ≤ ||L|| ||u||. Bounded ⇔ continuous.
Find bound by bounding ||Lu||2 = 〈Lu,Lu〉 ≤ c ||u||2, or show unbounded via sequence of uns.
3. Adjoint: An operator L∗ such that 〈Lu, v〉 = 〈u, L∗v〉. If L is bounded and linear, then L∗ exists and isbounded and linear. Also, L∗∗ = L, and ||L|| = ||L∗||.
4. Normal: L∗L = LL∗.
5. Self Adjoint: L∗ = L. formally if they have different domains. All eigenvalues are real. No residualspectrum.
6. Resolvent: Complete normed space X 6= {0} and linear operator L : D → X, D ⊂ X. The resolvent isRλ(L) = (L− λI)−1.
7. Regular value: λ such that Rλ(L) exists, is bounded, and is defined on a dense set in X.
8. Point Spectrum: σp(L) = {λ : (L− λI)−1 d.n.e.}. Eigenvalues of L, Lu = λu. λ ∈ σp(L)⇒ λ ∈ σp(L∗).
9. Continuous Spectrum: σc(L) = {λ : (L− λI)−1 exists and is unbounded}.
10. Residual Spectrum: σc(L) = {λ : (L−λI)−1 exists, possibly bounded, but not defined on a dense set of X}.
11. Rayleigh Quotient: λ =〈Lu, u〉〈u, u〉
.
4.f.ii) Sturm-Liouville
1. Lu =−1
σ(x)((p(x)ux)x + q(x)u(x)) on some domain [a, b]
2. Always self adjoint.
3. Eigenvalues λ1 < λ2 < · · · , and eigenfunctions are orthogonal, complete basis.
25
4.g) Green’s Functions
4.g.i) Variation of Parameters
1. Wronskian: W (x) = det
∣∣∣∣u1 u2
u′1 u′2
∣∣∣∣ = u1u′2 − u2u
′1, for solutions Lu1 = Lu2 = 0 of the S-L operator.
2. v′1(x) = −u2(x)f(x)
p(x)W (x); v′2(x) =
u1(x)f(x)
p(x)W (x).
3. up(x) = u1v1 + u2v2 = −u1(x)
∫ x
0
u2(t)f(t)
p(t)W (t)dt+ u2(x)
∫ x
0
u1(t)f(t)
p(t)W (t)dt
4. u(x) = uh(x) + up(x).
4.g.ii) Green’s Functions
Assuming S-L operator L = (pu′)′ + qu.
1. Satisfies LG(x, t) = δ(x− t) as well as boundary conditions of L, and G is continuous at x = t.
2. Jump condition: Gx(t+, t)−Gx(t1, t) = 1p(t)
3. General solution: find 2 linearly indep. homogenous solutions u1, u2. Then
G(x, t) =
{a1(t)u1(x) + a2(t)u2(x) x < t
b1(t)u2(x) + b2(t)u2(x) x > t
Use BCs, continuity, and jump to find a1, a2, b1, b2. Solution of Lu = f on [c, d] is
![Science, Tech & Eng Quals Presentation [O][III]](https://static.documents.pub/doc/80x56/55b75fecbb61ebeb0e8b477a/science-tech-eng-quals-presentation-oiii.jpg)
