Term

Definition
A transformation from an input space to an output space. 


Term

Definition
System where both input and output spaces are discretetime domains. 


Term

Definition
A system where both the input and output spaces are continuoustime domains. 


Term

Definition
System where input and output spaces are not both continuous or both discrete time domains. 


Term

Definition
Space upon which addition and scalar multiplication are defined such that:
for every x,y,z in X: (x+y) + z = x + (y+z)
there exists a 0_{x} s.t. for every x there exists x s.t. x+ (x) = 0
for every k in R or C, k(x+y) = kx+ky 


Term

Definition
 . 
i)  x  ≥ 0 ,  x  = 0 <=> x=0
ii) kx = k x
iii) xy ≤ xz + zy
defined only on linear spaces 


Term

Definition
a. d: X x X > R
i) d(x,y) ≥ 0 d(x,y)=0 <=> x=y
ii) d(x,y) = d(y,x)
iii) d(x,y)≤d(x,z) + d(z,y)
b. collection of points on which a metric is defined 


Term
a. Continuous Transformation
b. sequential continuity 

Definition
a. T is continuous at x_{0} if
for every ε>0, there exists a δ>0 such that:
xx_{0} ≤ δ => T(x)T(x_{0}) ≤ ε
b. T is sequentially continuous at x_{0} if
x>x_{0 }=> T(x)>T(x_{0})
seq. cont. <=> cont. 


Term

Definition
a) innerproduct defined <.,.> : H x H > C
i)<x,x> ≥ 0, <x,x>=0 <=> x = 0
ii)<x,y> = conj(<y,x>)
iii)<x,y> ≤ <x,z> + <z,y>
iv)<kx,y> = k<x,y>
b) complete ( all Cauchy sequences converge in H) 


Term
CauchySchwarz Inequality 

Definition
<x,y>^{2} ≤ x^{2}y^{2}



Term

Definition
{x_{n}} Cauchy if
for every e>0, there exists N st
for every n,m≥N
x_{n}x_{m}<e 


Term

Definition
all Cauchy sequences converge 


Term

Definition
complete normed linear space 


Term
T or F: Countable unions of countable sets are countable yet the Cartesian product of a finite # of countable sets is not. 

Definition
false: Both are countable 


Term
Give an example of a countable subspace 

Definition


Term
Dense subset
a. English
b. math 

Definition
a. subset in which all elements of another subspace can be arbirtarily closely approximated
b. D C X is a dense subset of a normed linear space X if for all x in X, e>0:
there is a y in D st
xy≤e



Term
Separable (two definitions) 

Definition
a. X is separable iff it contains a complete orthonormal sequence
b. X is separable iff it contains a dense countable subset 


Term

Definition
Ortho => perpendicular
all elements of the sequence are perpendicular to each other (<x,y> = 0)
normal => . = 1
sequence => bijection between set and N 


Term

Definition
only vector orthogonal to all elements of the sequence is the zero vector 


Term
If {en} is c.o.n. seq. in Hilbert space H, then: 

Definition
x = sum(1,inf) {<x,ek>ek} 


Term
T or F: l2 is not separable 

Definition


Term

Definition


Term
Wieirstrass' Approximation Thm 

Definition
let the norm on C([a,b]) be
f = sup (t in [a,b]) of f(t)
the set of polynomials:
sum(1,n){a_{i}t^{i}}, ai in R, n in Z
is dense in C([a,b])
(i.e. polynomials without constant terms can be used to closely approximate continuous functions)



Term
Name the discussed c.o.n. seq. in L2([a,b];R) 

Definition
fk(t) = 1/sqrt(2pi) * e^(ikt) 


Term
How can a computer approximate a nonseparable space of functions in L2(R+;R) with error less than e? 

Definition
sacrifice e/2 for removal of later terms (L2 f'ns go to zero)
sacrfice e/2 to approximate f with a polynomial (Wieirstrass' approx. thm)
=> less than 'e' error 


Term

Definition


Term

Definition
a) λ(A) = int [ 1_{{x in A}}λ(dx) ] with indicator f'n
b) define simple f'n:
fn(x) = sum(1,n){ak 1_{{x in Ak}}}
int [ fn(x)λ(dx) ] = sum(1,n){ak.λ(Ak)}
c) for every f, there is a seq. of simp. f'ns st
lim fn(x) = f(x) for every x
int [ f(x)λ(dx) ] = lim int [ fn(x)λ(dx) ]
(indicator f'ns are dense in L2([a,b];R)



Term
Lebesgue Monotone Convergence thm 

Definition
0≤fn(x)≤f(n+1)(x) and lim fn(x) = f(x)
=>
lim int [ fn(x)λ(dx) ] =int [ lim fn(x)λ(dx) ]
= int [ f(x)λ(dx) ]



Term
Lebesgue Dominated convergence thm 

Definition
lim fn(x) = f(x) and there is g(x) st
f(x)≤g(x) for every x
and
int [ g(x)λ(dx) ] < inf
=> lim int [ fn(x)λ(dx) ] =int [ f(x)λ(dx) ] 


Term
Name a space in which C([a,b]) is dense 

Definition


Term
Name a dense subset of L2 

Definition


Term

Definition
any indicator f'n can be closely approximated by a continuous f'n 


Term

Definition
ψ 0,0 = 1 for t in [0,1]
φ n,k = {2^{n/2} for t in [k2^{n},(k+1/2)2^{n}]
{2^{n/2} for t in [(k+1)2^{n},(k+1)2^{n}]
{0 o.w. 


Term
Dual Space of Normed linear space X 

Definition
all linear functionals on X, f such that f is bounded (and hence continuous)



Term

Definition
sup for x=1 : {f(x)/x}<inf 


Term

Definition
for x in X*:
there is an n in lq st
f(x) = sum(1,n){xi.ni} = <x,n>
for every x in X = lp
where 1/p + 1/q = 1 


Term

Definition
x in lp, y in lq, 1/p + 1/q = 1
=>sum (0,inf) { xi yi} = <x,y> ≤x_{p}y_{q}
hence:
if n in lq
=> sup for x=1 of (<x,n>/x_{p}) ≤ n_{q} 


Term

Definition
if xn>x in n.l. X
lim xn  x = 0 in a 


Term

Definition
f(xn)>f(x) for every f in X* 


Term

Definition


Term

Definition
lq
st 1/p + 1/q = 1
and p,q < inf 


Term
Which subspace are the Haar Wavelets a c.o.n in? 

Definition


Term
Which subspace are the Haar Wavelets a c.o.n in? 

Definition


Term

Definition
∂ (t) = { inf if t = 0, 0 o.w. 


Term

Definition
smooth, cmpt support
no metric defined 


Term

Definition
Schwarz Space
S = {v in T(R;R) st
sup for t in R (t^{a}(d^{b}/dt^{b})v)<inf)}
smooth
metric defined



Term

Definition
xn > x in S if
lim p_{a,b}(xnx)=0 for every a,b in Z
where p_{a,b} (v) = sup for t in R (t^{a}(d^{b}/dt^{b})v)<inf) 


Term

Definition
a function in the dual space of S* or D (linear, continuous & bounded) 


Term

Definition
∂ (bar) (Φ) = Φ(0) = int (inf, inf){∂(t)Φ(t)dt} 


Term

Definition
there exists an f such that:
ƒ = int(inf, inf){f(t)Φ(t)} < inf for all Φ in S 


Term

Definition
there does not exist an f such that:
ƒ = int(inf, inf){f(t)Φ(t)} < inf for all Φ in S



Term
When does equality exist between two dist'ns f and g? 

Definition
if f(Φ) = g(Φ) for all Φ in S



Term
When does a seq. of dist'ns fn
converge to a dist'n f in S*? 

Definition
when lim fn(Φ) = f(Φ) for every Φ in S 


Term
What is the derivative of a distribution f(Φ)? 

Definition


Term

Definition
(f*g)(t) = int (inf, inf) { f(u)g(tu)du} 


Term
if {Φn} is a seq. of dist'ns:
int(inf,inf){Φn(t)dt} = ? 

Definition


Term
if {Φn} is a seq. of dist'ns:
lim int(t>∂){Φn(t)dt} = ?


Definition


Term
T or F:
if {Φn} is a seq. of dist'ns:
Φn≤0 for all t


Definition


Term
for f in C([a,b]), Φn a seq. of dist'ns
what can we say about (Φn*f)(t)? 

Definition
it converges uniformly to f in t 


Term

Definition
each input has a unique output 


Term

Definition
if:
T is countable => DT system
else: CT system
if: input/output T 


Term

Definition
output only depends on current and future input values 


Term

Definition
output only depends on past and present inputs 


Term

Definition
shift in input (u(ta))
results in:
a shift in output (y(ta))
i.e.
if (u(t),y(t)) is in R (relation)
=>
(u(ta),y(ta)) is in R 


Term

Definition
i) U, Y linear spaces
ii) Relation R is a linear subspace of UxY
s.t. ((u1,y1), (u2,y2)) is in R
=> (a*u1+b*u2, a*y1+b*y2) for every a,b in Reals 


Term

Definition


Term

Definition
DT: x(n) = sum(Z){xm.∂(nm)}
for linear system:
y(n) = sum(Z){h(n,m)u(m)}
if u = ∂(nn0)
=>y(n) = h(n,n0) 


Term

Definition
output @ t = n when input is DT ∂ @ t = n0
h(n,m) 


Term
if system is discrete and linear, then... 

Definition
it can be described as:
y(n) = sum(Z){h(n,m)u(m)}



Term
if a system is continuous and linear, then ... 

Definition
it can be described as:
y(t) = int(u){h(t,u)u(u)du}
with h cont. in u



Term
if lim T(xn) = T(x)
then:
(T(x))(n) = ? 

Definition
sum(inf,inf){x(m)h(n,m)}
h(.,.): ZxZ >C 


Term

Definition
h(n,n0+T) = h(nT,n0)
and the system is a convolution system 


Term

Definition
time invariance
and hence
system is a convolution system 


Term

Definition
u in l(inf) > y in l(inf)
or
u in L(inf) > y in L(inf)
i.e.
BIBO stable <=> h_{1}<inf 


Term
Eigenfunction property of harmonic signals for convolution systems 

Definition
harmonic input => harmonic scaling of Fcc of impulse response 


Term

Definition
x+y^{2}=x^{2}+y^{2} 


Term

Definition
turns x1,x2,...,xn into o.n. seq. e1,e2,...,en
e1 = x1/x1
e2 = (x2<x2,e1>e1)/(x2<x2,e1>e1)
...
en = (xnsum(i=1,n){<xn,ei>ei)})/
(xnsum(i=1,n){<xn,ei>ei)}) 


Term
sum(1,inf){rk^{2}}<inf
iff ... 

Definition
sum (1,n){rk.ek}>x in Hilbert H 


Term
sum (1,n){rk.ek}>x in Hilbert H
iff ....


Definition


Term

Definition
ˆh(f) = int(inf,inf){h(t)e^(i2pift)dt}


