## 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.

Enter image description here
DFT

## 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``````
``` ```
``` ```
``` Author AdminPosted on December 16, 2019Categories ArticlesTags analysis, compile, correctly, Fourier, function ```
``` mathematics – Calculate Discrete Fourier Transform with C # There is a definition of discrete Fourier transform: I want to calculate the amplitude and phase of a transformed sequence. 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: Author AdminPosted on December 10, 2019Categories ArticlesTags Calculate, Discrete, Fourier, Mathematics, transform 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? Author AdminPosted on December 6, 2019Categories ArticlesTags analysis, Formula, Fourier, Standing, wave 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) Author AdminPosted on December 2, 2019Categories ArticlesTags analysis, Fourier, function, points, values Posts navigation Page 1 Page 2 … Page 9 Next page ```