Convergence Divergence – What can I understand about a function through its Fourier series?

I've come across a question that looks at things from a different perspective, and I'm not sure how to do it. I have three functions $ f, g, h $ defined in the $ (0, 2 pi) $ Interval, the Fourier series of which are: $$ f (x): sum_ {n = 1} ^ infty {1 over sqrt n} sin (nx) $$ $$ g (x): sum_ {n = 1} ^ infty {1 over n ^ 2 + 1} cos (nx) + {1 over n ^ 4} sin (nx) $$ $$ h (x): sum_ {n = 1} ^ infty {1 over 2 ^ n} cos (nx) $$

Without going into detail because I want to understand the concept first, I am asked a number of questions about belonging to different ones $ L ^ p $Spaces, convergence and so on. The question is … what information can I collect in the Fourier series? What should I pay attention to in order to know my functions?

Thanks a lot!

Fourier analysis – scaling a Mexican hat wavelet in a continuous wavelet transform

My general question relates to the concept of scaling a wavelet whose analytical form exists. An example is a Mexican hat.

The Mexican Hut-Wavelet, obtained from the second derivative of a Gaussian, has a functional form of 2 * (1- (t / s). ^ 2). * Exp (-t. ^ 2 * 0.5 / s ^ 2) divided by (pi) ^ (1/4) * sqrt (3 * s), where "s" is the standard deviation of a Gaussian and $ t $ is the independent variable.

If we want to see its compressions and dilations graphically, we say on a scale $ a $ = 1 to 5, the typical form of a wavelet function $ psi $ has a scaling parameter $ a $ and a translation parameter $ b $, The functional form of the Mexican hat specified in MATLAB does not have either of them explicitly.

The standard deviation $ s $ is the only variable that controls with width, but this is not exactly the same as the scale $ a $, How should we generate scaled versions of the Mexican hat?

Thank you very much.

P.S. I stopped this digital signal processing, but it did not get an answer.

Mexican hat

Expression in closed form for this Fourier summation?

Consider the function $ f: mathbb {T} ^ m to mathbb {R} $

$$ f ( boldsymbol {x}) = sum limits _ {{ pmb { eta} in mathbb {Z} ^ m}} frac {1} {1+ lambda | pmb { eta} | _ {2k} ^ {2k}} cos ({2 pi pmb { eta} cdot pmb {x}}) $$

Where $$ | pmb { eta} | _ {2k} ^ {2k} = sum limits_ {i = 1} ^ {m} eta_i ^ {2k} $$

$ k in mathbb {N} $ and $ lambda in mathbb {R} ^ + $

Question: Is there a closed expression for this sum? $ f ( pmb {x}) $?

Properties of Fourier transforms on the positive real axis

Consider $ has {f} ( omega) $, the Fourier transform of a function $ f (t) $, To let $ f (t) $ real, positive and just non-zero for $ t> 0 $, With these assumptions, the function is $ hat {f} ( omega) hat {f} (- omega) $ is real (this part is easy to see and just follows out $ f $ be real) and seems to decrease monotonously forever $ omega> 0 $ (Less easy to see, it only comes from me when I check many different functions and see that they all fulfill them). Is this second statement a general fact? It seems that you can achieve something by noticing that $ hat {f} & # 39; ( omega) = i hat {g} ( omega) $, Where $ has {g} $ is the Fourier transform of a real and positive function (positivity results from the fact that $ f (t) $ is not zero for $ x> 0 $), but I'm not sure where to go.

Frequency multiplier in Fourier series extensions

If we extend the signal with Fourier series, why are the angular frequencies in multiples? That is why $$ x (t) = sum_ {k = – infty} ^ { infty} z_k * e ^ {jk omega_0t} = z_0 + z_1e ^ {j omega_0t} + z_2e ^ {j2 omega_0t} + z_3e ^ {j3 omega_0t} + … $$
(Negative expressions are omitted for reasons of space).

My question is why frequencies are multiples of each other. For example, why don't we expand like this: $$ x (t) = z_0 + z_1e ^ {j omega_0t} + z_2e ^ {j2.4 omega_0t} + … $$ (Note that we have used $ j2.4 omega_0 $ instead of $ j2 omega_0 $)

Fourier Analysis – How Do I Compile the Function Correctly?

My calculations are terribly slow (for a larger number of variables than in the given code). I cannot compile the function properly. Can someone help me?

Alf(a_):=((m Pi)/a)
Bet(b_):=((n Pi)/b)
q(p1_,p2_,a_,b_,x0_,y0_):=
 (
  ((4 p1)/(a b)) Sin(Alf(a) x0) Sin(Bet(b) y0)+
    ((4 p2)/(a b)) Sin(Alf(a) (x0+2.021)) Sin(Bet(b) (y0+0.065))+
        ((4 p1)/(a b)) Sin(Alf(a) x0) Sin(Bet(b) (y0+1.050))+
            ((4 p2)/(a b)) Sin(Alf(a) (x0+2.021)) Sin(Bet(b) (y0+0.985))

  )
Delt(e_,h_,v_,a_,b_,k_):=(De(e,h,v) (Alf(a)^2+Bet(b)^2)^2+k)
w(p1_,p2_,a_,b_,x0_,y0_,e_,h_,v_,k_):=q(p1,p2,a,b,x0,y0)/Delt(e,h,v,a,b,k)
Mx(e_,h_,v_,a_,b_,p1_,p2_,k_,x0_,y0_,x_/;0

mathematics – Calculate Discrete Fourier Transform with C #

There is a definition of discrete Fourier transform:

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I want to calculate the amplitude and phase of a transformed sequence.

Enter image description here

I tried :

 public ArrayList phase = new ArrayList();
 public ArrayList amp = new ArrayList();
 public float phi = 0;

 void Start ()
 {
      DFT();
      foreach (var ampList in amp)
      {
        Debug.Log(ampList);
      }
 }


 void DFT()
 {
  //Given sequence of Numbers

  float()  X = {1,2,3};
  float()  Y = {2,3,5};
  float  N = X.Length;
  float re = 0;
  float im = 0;

  for (int k = 0; k < N; k++)
  {
      //The discrete Fourier transform, lists of Re & Im instead of a+ib
      for (int n = 0; n < N; n++)
       {
           float phi =  (Mathf.PI * k *n)/N;
           re += X(n) * Mathf.Cos(phi) - Y(n) * Mathf.Sin(phi);
           im -= Y(n) * Mathf.Cos(phi) - X(n) * Mathf.Sin(phi);
       }

      re = re/N;
      im = im/N;
      var  ampp = Mathf.Sqrt(re*re + im*im);
      var  phasse  = Mathf.Atan2(re,im);

      phase.Add(phasse);
      amp.Add(ampp);
  }
}

But the answer is not correct. I use Wolfram Mathematica to check the answer:

Enter image description here

Enter image description here

Fourier analysis – standing wave formula

Let us consider $ u_ {tt} = u_ {xx} $ be the wave equation of a vibrating string $ (0, pi) $, Suppose, based on the individual variables $ u (x, t) = phi (x) psi (t) $ gives a solution. After some calculations, the following formulas are determined.

$$ phi (x) = lambda sin mx ~~~~, psi (t) = mu_1 cos mt + mu_2 sin mt $$

Why must $ m $ be an integer?

Fourier analysis – function values ​​at many points

I would like to receive the results of the function for 3 points. I want x and y to replace the coordinates of three points to get 6 results – due to 2 values ​​of the coefficient "k". x and y are independent of x0 and y0, but in this particular case they are the same.

{x=x0
y=y0
{2,2},{2.3},{5,4}}


    De(e_, h_, v_) := ((e h^3)/(12 (1 - v^2)))
Alf(a_) := ((m Pi)/a)
Bet(b_) := ((n Pi)/b)
q(p_, a_, b_, x0_, y0_) := ((4 p)/(a b)) Sin(Alf(a) x0) Sin(Bet(b) y0)
Delt(e_, h_, v_, a_, b_,k_) :=( De(e, h, v) (Alf(a)^2 + Bet(b)^2)^2+k)
w(p_, a_, b_, x0_, y0_, e_, h_, v_,k_) := q(p, a, b, x0, y0)/Delt(e, h, v, a, b,k)


Mx(e_, h_, v_, a_, b_, p_,k_, x0_, y0_, x_/;0<=x<=6, y_/;0<=y<=4) :=
De(e, h, v) *
Sum(
Sum(
((Alf(a)^2 + 
v Bet(b)^2) w(p, a, b, x0, y0, e, h, v,k)) Sin(Alf(a) x) Sin(Bet(b) y), 
{n, 1, 80}), 
{m, 1, 80})

k={5*^6,4*^6}

Mx(27000000000 , 0.2, 0.2,6, 4, 10000,k, x0,y0,x, y)