# FourierDST

FourierDST[list]

finds the Fourier discrete sine transform of a list of real numbers.

FourierDST[list,m]

finds the Fourier discrete sine transform of type .

# Details • Possible types of discrete sine transform for a list of length giving a result are:
•  1 DST-I 2 DST-II 3 DST-III 4 DST-IV • FourierDST[list] is equivalent to FourierDST[list,2].
• The inverse discrete sine transforms for types 1, 2, 3, 4 are types 1, 3, 2, 4, respectively.
• The list given in FourierDST[list] can be nested to represent an array of data in any number of dimensions.
• The array of data must be rectangular.
• If the elements of list are exact numbers, FourierDST begins by applying N to them.
• FourierDST can be used on SparseArray objects.

# Examples

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## Basic Examples(2)

Find a discrete sine transform:

Find the inverse discrete sine transform:

Find a discrete sine transform of type 1 (DST-I):

Find the inverse discrete sine transform:

## Scope(2)

Use machine arithmetic to compute the discrete sine transform:

Use 24digit precision arithmetic:

Two-dimensional discrete sine transform:

Four-dimensional discrete sine transform:

## Generalizations & Extensions(2)

The list may have complex values:

You can use "I", "II", "III", or "IV" for the types 1, 2, 3, and 4 respectively:

## Applications(2)

### Sine Series Expansion(1)

Get an expansion for an odd function as a sum of sines:

The function values on a uniformly spaced grid with n points on [-L,L):

Compute the DST-I and renormalize:

The function has, in effect, been periodized with a particular odd symmetry:

Plot the expansion error where the points are defined:

### Pseudospectral PDE Discretization(1)

Approximate the second derivative for a function with zero boundary conditions:

Solve the wave equation for a plucked string:

Plot the solution as a surface:

## Properties & Relations(3)

DST-I and DST-IV are their own inverses:

DST-II and DST-III are inverses of each other:

The DST is equivalent to matrix multiplication:

## Possible Issues(1)

FourierDST always returns normalized results:

To get unnormalized results, you can multiply by the normalization: