Does anyone know which function this Fourier series comes from? $ sum_ {q = 1} ^ infty frac { cos (2 pi q x)} {q ^ {1+ alpha}} $ Seems like one $ zeta (1+ alpha) $ but with cosine at the top … is there any connection? The parameter is $ 0 < alpha <1 $. Thanks for any advice!

# Tag: Fourier

## Post-processing – refocusing of light field images using the Fourier Slice Photograph set

I am trying to refocus images from a microlens array light field using Renier's Fourier Slice photo set, which can be found in his thesis, Chapter 5, Equation 5.7. You can find this at https://stanford.edu/class/ee367/reading/Ren%20Ng -thesis% 20Lytro.pdf

In other words, the Fourier slice photography set means that a photograph is the inverse 2D Fourier transform of an extended 2D layer in the 4D Fourier transform of the light field.

Part of my confusion is the notation and applying in Python to a captured light field image. At first I understand what the Fourier transform is, but when Ng says 4D Fourier transform the light field, I'm not sure how to interpret it. I currently have my light field as a NumPy array, so my values for u = i, v = j index on the different images with subaperture, each pixel in the image with subaperture having an x and y coordinate, which gives me a 4D Light field array.

```
lf_img = cv2.imread(lf_img, -cv2.IMREAD_ANYDEPTH)
row_lens = lf_img.shape(0)/14 #14 is pixels under a microlens in this direction
col_lens = lf_img.shape(1)/14 # 14 is pixels under a microlens in this direction
vp_imgs = np.zeros((14,14,int(row_lens),int(col_lens),3))
for i in range(14):
for j in range(14):
vp_imgs(i, j, :, :, :) = lf_img(i::14, j::14, 0:3) ##U,V,X,Y,(RGB)
```

Then I see pictures like this http://graphics.stanford.edu/papers/fourierphoto/ in Ng's thesis, where it looks to me like he just took the 2D Fourier transform of each lenslet and arranged it in a grid . So I'm not entirely sure how to interpret this 4D Fourier transform and how to cut. Thank you so much!

Example of what type of light field image I work with:

## Fourier analysis – Wigner distribution

The Wigner distribution of $ u in L ^ 2 ( mathbb R) $ is defined as a function $ W (u) $ on $ mathbb R ^ 2 $ given by

$$

W (u) (x, xi) = int_ mathbb R u (x + frac z2) overline {u (x- frac z2)} e ^ {- 2π i z xi} dz.

$$

It's easy to see $ W (u) $ heard $ L ^ 2 ( mathbb R ^ 2) $ With

$

Vert W (u) Vert_ {L ^ 2 ( mathbb R ^ 2)} = Vert u Vert_ {L ^ 2 ( mathbb R)} ^ 2

$

since $ W (u) $ is the partial Fourier transform $ z $ from $ (x, z) mapsto u (x + frac z2) overline {u (x- frac z2)} $.

However, I believe that for some $ u in L ^ 2 ( mathbb R) $, the function $ W (u) $ is not one of them

$ L ^ 1 ( mathbb R ^ 2) $. Is there an "explicit" $ u $ like this?

## Probability or Statistics – Numerically inverting characteristic function with inverse Fourier transformation

I'm trying to numerically invert this characteristic function

```
cf = (((-I)*t + n*((Lambda) + (Mu)) - n*Sqrt(-4*(Lambda)*(Mu) + (I*t - n*((Lambda) + (Mu)))^2/n^2))/(n*Sqrt((Lambda)*(Mu)*(Rho))))^n/2^n
```

about a numerical inverse Fourier transform that looks like this …

```
Re((1/(2*Pi))*NIntegrate(cf/. {(Lambda) -> 0.5, (Mu) -> 1, (Rho) -> 0.5, n -> 5}, {t, -Infinity, 0, Infinity}, Method -> DoubleExponential))
```

But I keep getting errors that the integral at the borders cannot be evaluated.

Does anyone have any recommendations on how to calculate this thing so that I can get numeric?

Values?

Thank you for your help.

## Example of a periodic semi-differentiable function without absolutely convergent Fourier series

There is an example of a periodic continuous function that is semi-differentiable (i.e. the left derivative and the right derivative exist at every point), but

with a non-absolutely convergent Fourier series?

## Fourier analysis – InverseLaplaceTransform does not work

I am trying to find the inverse Laplace transform of a function that I previously received from a Laplace transform, but the result obtained does not match the original function. Why is this happening?

```
LaplaceTransform(2 z^3 Coth(z) Csch(z)^2, z, s) // FullSimplify
InverseLaplaceTransform(%, s, z) // FullSimplify
```

```
-(48/(-2 + s)^4) + 3 PolyGamma(1, -1 + s/2) +
1/8 s (12 PolyGamma(2, -1 + s/2) + s PolyGamma(3, -1 + s/2))
-2 E^(2 z) z (3 + z (9 + 4 z) + 3 Coth(z) (1 + 2 z - z Coth(z)))
```

The last line does not match the original function I transformed: 2 z ^ 3 Coth (z) Csch (z) ^ 2.

## Integration – Fourier transform of the convolution and the Plancherel theorem

For some functions $ h in mathcal {S} ( mathbb {R}) $ and $ theta, eta in mathbb {R} $we define the Fourier transform as

begin {equation} label {fourier}

tilde {h} ^ { pm} ( theta): = frac {1} {2 pi} int dp h (p) e ^ { pm i theta p},

end {equation}

the function $ g ( theta): = -i / ( theta + i0 ^ +) $ and with the symbol $ star $ denote the convolution, we can calculate the following

begin {align *}

int d theta d eta h ( theta) frac {-i} {( theta – eta + i0 ^ +)} h ( eta) & = int d theta h ( theta) ( h star g) ( theta) \

& = 4 pi ^ 2 int dp widetilde {h} (p) widetilde {h star g} (p) quad quad Big ( text {Plancherels Theorem} Big) \

& = 6 pi ^ 3 int dp | widetilde {h} (p) | ^ 2 widetilde {g} (p) \

& = 4 pi ^ 2 int dp | widetilde {h} (p) | ^ 2 Theta (p) quad quad Big ( text {Heaviside function} Big) \

& = 4 pi ^ 2 int_0 ^ infty dp | tilde {h} (p) | ^ 2.

end {align *}

I'm a little confused about the pre-factor $ 4 pi $. We would take up a factor of from Plancherel's theorem $ 2 pi $, but that we are actually transforming a function $ h $ and convolution so that 3 Fourier transforms were performed in the third line of the following calculation, and therefore there should be a factor of $ (2 pi) ^ 3 $.

begin {align}

int _ {- infty} ^ infty | tilde {h} (p) | ^ 2 dp = int _ {- infty} ^ infty tilde {h} (p) overline { tilde {h} (p)} dp & = int _ {- infty} ^ infty Big ( frac {1} {2 pi} int _ {- infty} ^ infty h ( theta) e ^ {ip theta} d theta Big) Big ( frac {1} { 2 pi} int _ {- infty} ^ infty overline {h ( theta & # 39;)} e ^ {- ip theta & # 39;} d theta & # 39; Big) dp \

& = frac {1} {4 pi ^ 2} int _ {- infty} ^ infty int _ {- infty} ^ infty int _ {- infty} ^ infty h ( theta) overline {h ( theta & # 39;)} e ^ {i ( theta – theta & # 39;) p} d theta d theta & # 39; dp \

& = frac {1} {4 pi ^ 2} int _ {- infty} ^ infty int _ {- infty} ^ infty int _ {- infty} ^ infty h ( theta) overline {h ( theta & # 39;)} e ^ {i ( theta – theta & # 39;) p} dp d theta d theta & # 39; \

& = frac {2 pi} {4 pi ^ 2} int _ {- infty} ^ infty int _ {- infty} ^ infty delta ( theta – theta & # 39; ) h ( theta) overline {h ( theta & # 39;)} d theta d theta & # 39; \

& = frac {1} {2 pi} int _ {- infty} ^ infty | h ( theta) | ^ 2 d theta

end {align}

begin {equation}

Rightarrow int _ {- infty} ^ infty | h ( theta) | ^ 2 d theta = 2 pi int _ {- infty} ^ infty | tilde {h} (p) | ^ 2 dp.

end {equation}

And we take up another factor $ frac {1} {2 pi} $ from the Fourier transform of the function $ g ( theta) $::

begin {equation}

tilde {g} (p) = frac {1} {2 pi} Theta (p).

end {equation}

But I'm really not convinced of my reasoning. Am I making a critical or stupid mistake here?

## Classical analysis and odes – Limitation of a Fourier coefficient of a non-negative periodic function with respect to its $ L ^ 2 $ norm

This question is motivated by the earlier MO question: Show that $ ( sum_ {k = 1} ^ {n} x_ {k} cos {k}) ^ 2 + ( sum_ {k = 1} ^ {n}) x_ {k} sin {k}) ^ 2 le (2+ frac {n} {4}) sum_ {k = 1} ^ {n} x ^ 2_ {k} $.

It is an adjusted asymptotic version of this question.

To let $ f $ be a non-negative function, periodically with period $ 1 $and square can be integrated $ { Bbb R} / { Bbb Z} $. Is it true that

$$

| { widehat f} (1) | ^ 2 = Big | int_0 ^ 1 f (x) e ^ {- 2 pi ix} dx Big | ^ 2 le frac 14 int_0 ^ 1 f (x) ^ 2 dx ?

$$

For example, equality is achieved when $ f (x) = max (0, cos (2 pi x)) $.

Note that $ | widehat f (1) | = | widehat f (-1) | $ and since $ f $ is not negative $ | widehat f (1) | le widehat f (0) $. Therefore

$$

int_0 ^ 1 f (x) ^ 2 dx = sum_n | widehat f (n) | ^ 2 ge 3 | widehat f (1) | ^ 2,

$$

so that the estimate applies $ 1/3 $ instead of $ 1/4 $. There is a lot of scope to improve this argument, and with a more careful application of Bessel's inequality I could get the constant $ 1/4 + 1/4 pi $. But the claimed inequality looks very clean, and I wonder if (i) it is true! (Ii) is known in a certain context and (iii) (hopefully) has elegant evidence?

## Harmonic Analysis – For certain "strange" Fourier transformations over p-adic fields or finite fields

I recently met the following two weird "Fourier Transform" questions.

**(I)**, Suppose that $ F $ is a $ p $-adic field (the same question can be asked about any local field, including $ mathbb {R} $ and $ mathbb {C} $) and $ psi $ to be a solid non-trivial additive character of $ F $. To let $ W $ be a function on $ GL_2 (F) $ which meets the following 2 conditions:

(1). $ W left ( begin {pmatrix} 1 & x \ & 1 end {pmatrix}

g right) = psi (x) W (g) $, for all $ x in F, g in GL_2 (F) $, and

(2) there is an open subset $ K $ of $ GL_2 (F) $ so that

$ W (gk) = W (g) $ for all $ g in GL_2 (F), k in K $.

You can think $ W $ As a ** Whittaker** Function of a smooth representation of $ GL_2 (F) $. The definition of $ W $ to all $ 2 times 2 $ Matrices through zero expansion, i. H. if $ g in mathrm {Mat} _ {2 times 2} (F) $ and $ det (g) = 0 $, then define $ W (g) = 0 $.

To let $ mathcal {S} (F ^ 2) $ let Bruhat-Schwartz's room work $ F ^ 2 $. For a non-zero Whittaker function $ W $ as above and $ phi in mathcal {S} (F ^ 2) $, define

$$ widehat { phi} (x_1, x_2) = int_ {F ^ 2} W left (I_1 + begin {pmatrix} x_1 \ x_2 end {pmatrix} begin {pmatrix} y_1 & y_2 end {pmatrix} right) phi (y_1, y_2) dy_1 dy_2. $$

**Question (I):** Do we know that? $ widehat phi in mathcal {S} (F ^ 2) $ and $ phi mapsto widehat phi $ defines an isomorphism?

This looks like a Fourier transform. It is also easy to see that the function $ x_1 mapsto widehat { phi} (x_1.0) $ has compact support. But I don't know how to generally prove the above statement. I wonder if anyone knew anything like that.

**(II)** The second question is similar, but not the same as the above. Suppose that $ k $ is a finite field and $ psi $ is a non – trivial additive character of $ k $. To let $ mathscr {B} $ the set of functions $ B $ on $ GL_2 (k) $ so that

$$ B left ( begin {pmatrix} 1 & x \ & 1 end {pmatrix} g begin {pmatrix} 1 & y \ & 1 end {pmatrix} right) = psi (x) psi (-y) B (g), for all x, y in k, g in GL_2 (k). $$

One can imagine such a function $ B $ As a ** Bessel** Function.

To the $ B in mathscr {B} $, define

begin {align *} & mathcal {F} _B left ( begin {pmatrix} x_ {11} & x_ {12} \ x_ {21} & x_ {22} end {pmatrix} right) .

& qquad: = sum_ {s_1, s_2, r_1, r_2 in k} psi (x_ {22} r_1 + x_ {21} r_2 + s_1) B left ( begin {pmatrix} x_ {11} & x_ {12} \ x_ {21} & x_ {22} end {pmatrix} left (I_2 + begin {pmatrix} s_2 \ -s_1 end {pmatrix} begin {pmatrix} -r_1 & r_2 end {pmatrix} right) right),

end {align *}

where if $ det (g) = 0 $we see $ B (g) = 0. $

It is not difficult to show that $ mathcal {F} _B in mathscr {B} $. The negative signs in the definition of $ mathcal {F} _B $ is to be ensured $ mathcal {F} _B in mathscr {B}. $

**Question (II)**: Is it true that $ B mapsto mathcal {F} _B $ is a bijection?

If the answer to (II) is positive, I also wonder if there is a local field analog.

**Question (III)**: Are there generalizations too $ GL_n $?

## Physical interpretation of adding a real constant to a Fourier transform of a function

If we share the Fourier transform of two functions / signals, e.g. $ S_1 ( omega) $ and $ S_2 ( omega) $ (for deconvolution purposes) and then perform an inverse transformation, the noise level is often increased. This is due to the division by very small numbers in the denominator. I remember reading a book (Numerical Recipes by William Press) that you can add a small constant value to the denominator $ S_2 ( omega) $, Now $ S_2 ( omega) $ is a complex number. If we add a constant $ a $ to $ x $+$ i $$ y $, it will ($ a $+ $ x $) +$ i $$ y $, In fact, a certain level of noise is reduced by this "trick".

What is the physical meaning of adding a real number to a Fourier transform of a function in the time domain? How can we also interpret adding a complex constant number to a Fourier transform of a function? Thank you very much