1
Q
f∈L2([−π,π])
⟹
f∈C([−π,π])
A
FALSE
2
Q
f∈C([−π,π])
⟹
f∈L2([−π,π])
A
TRUE
3
Q
∥f−g∥L2([−π,π])=0
⟺
f(t)=g(t)
for every t∈R
A
FALSE (for almost every t)
4
Q
Assume that f,g∈C([−π,π])
are 2π
-periodic functions. Then ∥f−g∥L2([−π,π])=0
⟺
f(t)=g(t)
for every t∈R
A
TRUE
5
Q
The definition of the Fourier series applies to all functions in L1([−π,π])
.
A
TRUE
6
Q
The definition of the Fourier series applies to all functions in L2([−π,π])
.
A
TRUE
7
Q
The definition of the Fourier series applies to all functions in C([−π,π])
.
A
TRUE
8
Q
If the Fourier series converges pointwise everywhere, the obtained function is 2π
-periodic.
A
TRUE
9
Q
A trigonometric polynomial belongs to a subspace of L2([−π,π])
spanned by the functions ej(t)=eijt
, j=−n,…,n
for some n∈N
.
A
TRUE
10
Q
The partial sum Snf
of a Fourier series is an orthogonal projection of the function f
to the subspace of L2([−π,π])
spanned by {ej}nj=−n
.
A
TRUE
11
Q
The partial sum of a Fourier series gives the best approximation of a function in L2([−π,π])
with trigonometric polynomials.
A
TRUE
12
Q
The statement that the partial sum of a Fourier series gives the best approximation of a function in L2([−π,π])
with trigonometric polynomials means that there does not exist a function in L2([−π,π])
, which would be closer to the original function in L2
-norm.
A
FALSE
13
Q
L2([−π,π])
is an infinite dimensional vector space.
A
TRUE
14
Q
The Fourier series of every function f∈L2([−π,π])
converges with respect to the L2
-norm.
A
TRUE
15
Q
The statement that Snf
approximates a function f
in L2([−π,π])
means that, the error term ∥f−Snf∥L2([−π,π])
converges to zero as n→∞
.
A
TRUE
16
Q
Fourier coefficients are coordinates of the function with respect to the basis {ej}j∈Z
, where ej(t)=eijt
.
A
TRUE
17
Q
Every element f∈L2([−π,π])
is uniquely determined by its Fourier coefficients fˆ(j)
, j∈Z
.
A
TRUE
18
Q
The Fourier coefficients fˆ(j)
of a function f∈L2([−π,π])
converge to zero as |j|→∞
.
A
TRUE
19
Q
The Fourier coefficients fˆ(j)
of a function f∈L2([−π,π])
converge faster to zero as |j|→∞
if the function f
is smoother.
A
TRUE
20
Q
The zero function is the only element in L2([−π,π])
, whose all Fourier coefficients are zero.
A
TRUE
21
Q
Separation of variables is based on finding special solutions to the PDE as a product of functions so that each function depends only on one variable.
A
TRUE
22
Q
Separation of variables applies to PDE problems on rectangular domains.
A
TRUE
23
Q
Separation of variables is based on transforming a PDE to a system of ODEs.
A
TRUE
24
Q
Separation of variables is based on finding solutions to the corresponding system of ODEs.
A
TRUE
25
All functions r|j|eijθ
, j∈Z
, are solutions to the Laplace equation.
TRUE
26
All functions r|j|eijθ, j∈Z, have zero boundary values on the unit circle.
FALSE
27
The L1([−π,π])
-norm of the function θ↦Pr(θ)
is equal to one for every r∈(0,1)
.
TRUE
28
consider the Dirichlet problem for the Laplace equation in the unit disc: The solution to the original problem can be represented as a convolution of the Poisson kernel with the boundary function.
TRUE
29
consider the Dirichlet problem for the Laplace equation in the unit disc: The solution of the original problem on the boundary of the unit disc can be obtained by inserting r=1
in the convolution formula.
FALSE
30
consider the Dirichlet problem for the Laplace equation in the unit disc: The solution of the original problem is always zero at the origin.
FALSE
31
, consider the space-time model of the initial value problem for the heat equation on the unit circle: The initial value function corresponds to the boundary values at the time zero.
TRUE
32
consider the space-time model of the initial value problem for the heat equation on the unit circle: The boundary values can also be given on the lateral boundaries {−π}×R+
and {π}×R+
.
FALSE
33
consider the space-time model of the initial value problem for the heat equation on the unit circle: The temperature at a given point θ
at all moments of time t>0
are given by the solution on the line {θ}×R+
.
TRUE
34
consider the space-time model of the initial value problem for the heat equation on the unit circle: The functions e−j2teijθ
, j∈Z
, are solutions to the heat equation in (−π,π)×R+
.
TRUE
35
consider the space-time model of the initial value problem for the heat equation on the unit circle: The functions e−j2teijθ
, j≠0
, decay to zero as t→∞
TRUE
36
consider the space-time model of the initial value problem for the heat equation on the unit circle: The functions e−j2teijθ
, have zero initial values.
FALSE
37
consider the one-dimensional initial value problem for the wave equation: Initial values describe the shape and velocity of the string at time zero.
TRUE
38
consider the one-dimensional initial value problem for the wave equation: The solution has its maximum and minimum values at the endpoints of the interval.
FALSE
39
consider the one-dimensional initial value problem for the wave equation: The solution decays to zero as t→∞
.
FALSE
40
consider the one-dimensional initial value problem for the wave equation: If the profile of the string is horizontal at time zero, then all coefficients aj
are zero.
TRUE
41
consider the one-dimensional initial value problem for the wave equation: If the velocity of the string is zero at time zero, then all coefficients bj
are zero.
TRUE
42
consider the one-dimensional initial value problem for the wave equation: If the profile of the string is horizontal at time zero, the solution of the problem is identically zero.
FALSE
43
consider the one-dimensional initial value problem for the wave equation: If the profile of the string is horizontal and the velocity is zero at time zero, the solution of the problem is identically zero.
TRUE
44
Poisson kernel satisfy the conditions of good kernel
TRUE
45
Heat kernel on the unit circle satisfy the conditions of a good kernel
TRUE
46
Dirichlet kernel satisfy the conditions of a good kernel
FALSE
47
integral average
satisfy the conditions of a good kernel
TRUE
48
Consider good kernels: The parameter ϵ>0
gives the scale of the approximation.
TRUE
49
good kernel family {K(epsilon)}epsilon>0 properties (not true/false):
1. for every epsilon>0, 1/2pi and integral [-pi, pi] good kernel dx=1
2. Exists a positive constant M such that, (same integral but kernel in abs) <=M, for every epsilon>0
3. for every delta>0 we have lim epsilon->0 integral(delta<|x|<=pi) abs(kernel)dx=0
50
good kernel condition (1) means:
the total mass of the kernel is one at all scales
51
good kernel condition (3) means:
the mass concentrates near the origin at small scales
52
even function definition:
f(t)=f(-t)
53
odd function definition:
f(-t)=-f(t)
54
for 2pi periodic odd function these fourier coefficients are zero:
aj
55
for 2pi periodic even function these fourier coefficients are zero
bj
56
fourier derivative:
=ij fourier f
57
fourier(j) -> 0 with speed 1/|j| as
|j| -> ∞
58
fourier coefficients of an L2 function always converge to a zero
TRUE
59
total mass of dirichlet kernel is one
TRUE
60
Approximations of the identity can be used to approximate functions with smoother functions.
TRUE
61
Approximations of the identity can be used to study boundary and initial values in PDE problems.
TRUE
62
The pointwise convergence of the Dirichlet kernel can be concluded directly using approximations of the identity.
FALSE
63
The pointwise value of a continuous function is obtained as a limit of weighted integral averages in approximations of the identity.
TRUE
64
L1(Rn)⊂L2(Rn)
.
FALSE
65
L2(Rn)⊂L1(Rn)
.
FALSE
66
The Fourier transform of a function in L1(Rn)
is a bounded function.
TRUE
67
The Fourier transform of a function in L1(Rn)
is a continuous function.
TRUE
68
The Fourier transform of a real function is a real valued function.
FALSE
69
The fourier transform of a function is real ONLY IF:
f is even
70
The Fourier transform of a compactly supported function is a compactly supported function.
FALSE
71
If f∈L1(Rn)
, then fˆ∈L1(Rn)
.
FALSE
72
The fourier transform of a real valued function may be a complex valued function
TRUE
73
The Fourier transform at zero equals to the integral of the function over Rn
.
TRUE
74
The smoothness of a function is reflected in the decay of its Fourier transform.
TRUE
75
If f∈C∞0(Rn)
, then fˆ∈C∞(Rn)
TRUE
76
Convolution becomes multiplication on the Fourier side.
TRUE
77
The L2(Rn)
norm of the Fourier transform is the same as the L2(Rn)
norm of the function up to a multiplicative constant.
TRUE
78
∥ f^∥ L2=(2π)^-n∥f∥ L2
TRUE
79
The L1(Rn)
norm of the Fourier transform is the same as the L1(Rn)
norm of the function up to a multiplicative constant.
FALSE
80
Fourier inversion theorem applies to all functions in C∞0(Rn)
.
TRUE
81
Fourier inversion theorem applies to all functions in L1(Rn)
FALSE
82
Fourier inversion theorem applies to all functions in C(Rn)
FALSE
83
The Fourier transform of a Gaussian function is a Gaussian function.
TRUE
84
Dirichlet problem for the Laplace equation in the upper half-space formula??
laplace(u(x,y))=0
u(x,0)=g(x)
85
Consider the Dirichlet problem for the Laplace equation in the upper half-space: The Laplace equation is Fourier transformed with respect to all variables in the upper-half space.
FALSE (vain horisontaaliset muuttujat puolitasossa (x1, x2, ...))
86
Consider the Dirichlet problem for the Laplace equation in the upper half-space: The Laplace equation is Fourier transformed with the last coordinate fixed.
TRUE
87
Consider the Dirichlet problem for the Laplace equation in the upper half-space: The Laplace equation becomes an ODE with respect to the last variable on the Fourier side.
TRUE
88
Consider the Dirichlet problem for the Laplace equation in the upper half-space: Usually the Laplace equation is satisfied if and only if it is satisfied on the Fourier side.
TRUE
89
When considering Dirichlet problem: Δuˆ(ξ,y)(fourier above deltau)=Δuˆ(ξ,y)(fourier only above u)
FALSE (why? fourier taking only from horizontal x so Δuˆ(ξ,y)(fourier above delta u)=-|ξ|^2Δuˆ(ξ,y)(fourier above u) + second derivative of fourier u through y) This because y is fixed.
90
Dirichlet: uˆ(ξ,y)=gˆ(ξ)e^(−|ξ|y) is a solution to the original problem on the Fourier side.
TRUE
91
Consider the initial value problem for the heat equation in the upper half-space: The initial condition is used to determine free parameters in the solution of the ODE.
TRUE
92
One advantage of the Fourier transform is that a PDE becomes an ODE
TRUE
93
One advantage of the Fourier transform is that representation formulas for solution are convolutions.
TRUE
94
One advantage of the Fourier transform is that it in many cases it also proves the uniqueness of a solution.
TRUE
95
One advantage of the Fourier transform is that it also applies to nonhomogeneous problems.
TRUE
96
Fundamental solutions are solutions to the corresponding PDEs in the upper half-space.
TRUE
97
Consider the Dirichlet problem: The problem has a unique solution.
TRUE
98
Neumann problem formula:
Δu=0
osittaisderivaatta u(sen melkeen v näkösen suhteen)=h
99
Consider the Neumann problem: The zero function is a solution to the problem with the zero boundary values.
TRUE
100
Consider the Neumann problem: The zero function is the only solution to the problem with the zero boundary values.
FALSE
101
Consider the Neumann problem: The solution to the problem with the zero boundary values is unique up to an additive constant.
TRUE
102
Consider the Neumann problem: The total flow of a solution to the problem through the boundary is zero.
TRUE
103
The total flux of the fundamental solution of the Laplace equation through the boundary of every sphere centered at the origin is equal to one.
TRUE
104
The fundamental solution of the Laplace equation is a positive function when n≥3.
TRUE
105
Poisson equation in the whole space: The convolution of the fundamental solution with the source term is a solution.
TRUE
106
Poisson equation in the whole space: The solution is unique.
TRUE
107
Poisson equation in the whole space: The solution is harmonic outside the support of the source term.
TRUE
108
Poisson equation in the whole space: The solution of the problem in the whole space can be used to construct a solution of the problem in subdomains.
TRUE
109
|∂B(x,r)|=r^(n−1)|∂B(0,1)|
TRUE
110
|B(x,r)|=r^n|B(0,1)|
TRUE
111
|∂B(0,1)|=n|B(0,1)|
TRUE
112
Green's function of a set Ω
depends on Ω
TRUE
113
The definition of Green's function of a set is based on a solution of a Dirichlet problem.
TRUE
114
Green's function is harmonic at point x
.
FALSE
115
Green's function of the upper half-space can be written using the fundamental solution of the Laplace equation.
TRUE
116
Green's function of the upper half-space satisfies G(x,y)=G(y,x)
, x,y∈Rn+1+
, x≠y
TRUE
117
Green's function of the upper half-space is the same function as the Poisson kernel for the upper half-space.
FALSE
118
The derivative with respect to the exterior unit normal of Green's function of the upper half-space is up to a sign the same function as the Poisson kernel of the upper half-space.
TRUE
119
The mean value formula holds also for solutions to other PDEs in addition to the Laplace equation.
FALSE
120
The weak maximum principle implies that a harmonic function attains its maximum value at an interior point.
FALSE
121
The weak maximum principle implies that a harmonic function cannot attain its maximum value at an interior point.
FALSE
122
The weak maximum principle implies that a harmonic function cannot attain a strict maximum value at an interior point.
TRUE
123
The weak maximum principle implies that a harmonic function that is continuous up to the boundary attains its maximum on the boundary.
TRUE
124
Solutions to Laplace equation are always solutions to the heat equation.
TRUE
125
If u=u(x)
is a solution to the Laplace equation in Rn
, then v(x,t)=u(x)
is a solution to the heat equation in Rn×(0,∞).
TRUE
126
If u=u(x,t)
is a solution to the heat equation in Rn×(0,∞)
, then v(x)=u(x,t)
is a solution to the Laplace equation in Rn
for every t∈(0,∞).
FALSE (This claim holds only solutions that have already reached equilibrium i.e. ut=0
at every point.)
127
If u=u(x,t)
is a solution to the heat equation in Rn×(0,∞)
, then −u
is a solution to the heat equation in Rn×(0,∞).
TRUE
128
If u=u(x,t)
is a solution to the heat equation in Rn×(0,∞)
, then v(x,t)=u(x,−t)
is a solution to the heat equation in Rn×(−∞,0).
FALSE
129
If a solution to the heat equation in ΩT
is zero on the parabolic boundary ΓT
, then u
is zero everywhere in ΩT.
TRUE
130
If a solution to the heat equation in ΩT
is zero on the lateral boundary ∂Ω×(0,T)
, then u
is zero everywhere in ΩT
.
FALSE
131
In heat equation the solution depends from the boundary AND the initial time condition
TRUE
132
If a solution to the heat equation in ΩT
is zero on the initial boundary Ω×{0}
, then u
is zero everywhere in ΩT.
FALSE (also the lateral boundary affects)
133
If a solution to the heat equation in ΩT
is zero on a time slice Ω×{t}
for some 0
, then it is zero everywhere in ΩT.
FALSE
134
u∈C2(ΩT)∩C(ΩT¯)
is a solution of the heat equation in a bounded space-time cylinder ΩT: u
attains its maximum at t=T
FALSE
135
u∈C2(ΩT)∩C(ΩT¯)
is a solution of the heat equation in a bounded space-time cylinder ΩT: For a fixed t>0
, the function x↦u(x,t)
attains its maximum on the boundary ∂Ω.
FALSE
136
u∈C2(ΩT)∩C(ΩT¯)
is a solution of the heat equation in a bounded space-time cylinder ΩT: u
attains its maximum either in Ω×{t=0}
or in ∂Ω×[0,T].
TRUE
137
u∈C2(ΩT)∩C(ΩT¯)
is a solution of the heat equation in a bounded space-time cylinder ΩT:
u cannot have a strict maximum point in ΩT.
TRUE
138
d'Alembert's formula for the Cauchy problem for the one-dimensional wave equation gives existence of a solution.
TRUE
139
d'Alembert's formula for the Cauchy problem for the one-dimensional wave equation gives uniqueness of a solution.
TRUE
140
d'Alembert's formula for the Cauchy problem for the one-dimensional wave equation gives stability of a solution on the boundary data.
TRUE
141
d'Alembert's formula for the Cauchy problem for the one-dimensional wave equation gives C∞
-smoothness of a solution.
FALSE
142
The Euler-Poisson-Darboux equations are for integral averages of the function instead of the function itself.
TRUE
143
A solution to the Cauchy problem for the wave equation can be obtained as a limit of the solutions Euler-Poisson-Darboux equations.
TRUE
144
d'Alembert's formula is used in the derivation of the Kirchhoff's formula.
TRUE
145
The solution given by the Kirchhoff formula at a point (x,t), with x∈R3 and t>0, depends on the initial data in the whole space.
FALSE
146
consider the Cauchy problem for the wave equation in the two-dimensional case: The domain of dependence consists of points at the initial moment of time that affect the value of a solution at a given point.
TRUE
147
consider the Cauchy problem for the wave equation in the two-dimensional case: The domain of dependence is a disc in a plane.
TRUE
148
consider the Cauchy problem for the wave equation in the two-dimensional case: The range of influence consists of points at which the value of a solution is affected by the value of a solution at a given point.
TRUE
149
consider the Cauchy problem for the wave equation in the two-dimensional case: The range of influence is a cone in the three-dimensional space.
TRUE
150
The solution of the Cauchy problem for the wave equation in the two-dimensional case at a given point depends on the initial values in the whole space.
TRUE
150
The solution of the Cauchy problem for the wave equation in the two-dimensional case at a given point depends only on the initial values near the origin.
FALSE
151
The solution of the Cauchy problem for the wave equation in the two-dimensional case at a given point depends only on the initial values far away from the origin.
FALSE
152
The solution of the Cauchy problem for the wave equation in the two-dimensional case at a given point does not depend on the initial values.
FALSE
153
consider the Cauchy problem for the wave equation in the two-dimensional case.
A disturbance of the initial data near the origin influences the solution at a point x≠0 forever starting at a certain moment of time.
TRUE
154
consider the Cauchy problem for the wave equation in the two-dimensional case.
A disturbance of the initial data near the origin influences the solution at a point x≠0 from the initial moment until a certain moment of time after which it does not have any influence.
FALSE
155
consider the Cauchy problem for the wave equation in the two-dimensional case.
A disturbance of the initial data near the origin influences the solution at a point x≠0 for a short moment of time starting at a certain moment of time.
FALSE
156
consider the Cauchy problem for the wave equation in the two-dimensional case.
A disturbance of the initial data near the origin influences the solution at a point x≠0 forever starting at the initial moment of time.
FALSE
157
consider the Cauchy problem for the wave equation in the two-dimensional case: If a solution is zero in some point at a given moment of time, then it is zero at every point at the same moment of time.
FALSE
158
consider the Cauchy problem for the wave equation in the two-dimensional case: If the initial values are positive everywhere, then the solution is positive everywhere.
FALSE
159
consider the Cauchy problem for the wave equation in the two-dimensional case: initial values and initial speed affects the solution
TRUE
160
consider the Cauchy problem for the wave equation in the two-dimensional case: If a solution is zero everywhere at a given moment of time, then it is zero everywhere before that moment of time.
FALSE
161
consider the Cauchy problem for the wave equation in the two-dimensional case: The nonhomogeneous Cauchy problem can be solved using Duhamel's principle.
TRUE
162
Duhamel's principle is a method for solving non-homogeneous problems.
TRUE
163
consider a solution u∈C2(ΩT)∩C(ΩT¯)
to the wave equation in a bounded space-time cylinder ΩT: u cannot attain its maximum in ΩT
FALSE
164
There is no maximum principle for the wave equation.
TRUE
165
consider a solution u∈C2(ΩT)∩C(ΩT¯)
to the wave equation in a bounded space-time cylinder ΩT: u attains its maximum in ΩT
FALSE
166
consider a solution u∈C2(ΩT)∩C(ΩT¯)
to the wave equation in a bounded space-time cylinder ΩT: u attains its maximum in ΩT(BOUNDARY)
TRUE
167
168
Fourier sarja aina konvergoi L kakkosessa
TRUE
PDE
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