Built-in Mathematica Symbol

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

.

MORE INFORMATION

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

CLOSE ALL

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:

Find a discrete sine transform:

In[1]:=

Out[1]=

Find the inverse discrete sine transform:

In[2]:=

Out[2]=

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

In[1]:=

Out[1]=

Find the inverse discrete sine transform:

In[2]:=

Out[2]=

Scope (2)

Use machine arithmetic to compute the discrete sine transform:

Use 24-digit 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)

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:

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: