You are lucky if you do understand the Fourier transform; I still don't (caveat emptor).

Assuming a certain classical sign convention (i.e. a $e^{-ift}$ vs a $e^{ift}$ kernel, a change of convention won't change the properties), for a real signal, the discrete Hartley transform (DHT) is the sum of the real and the imaginary part of the discrete Fourier transform (DFT). In other words, $a+ib \to a+b$.

It is purely real, it is an involution, so it is its own inverse. It seems to still be in use for some coding, encryption or watermarking applications, where the use of real transforms can be simpler, while keeping some of the DFT nice properties (with respect to convolution for instance).

While in general it has been found to be no faster than FFT, it can be embedded in the same parametric family (New Parametric Discrete Fourier and Hartley Transforms, and Algorithms for Fast Computation, 2011). For some specific hardware, data or data size, it may be found beneficial. Looking back, it has been used for the Fast computation of discrete cosine transform through fast Hartley transform by H. Malvar. I suspect the Hartley transform may have shed some insights into his development of the Lapped Orthogonal Transform (LOT).

Although quite forgotten, it may rise again some day.

You are lucky if you do understand the Fourier transform; I still don't (caveat emptor).

Assuming a certain classical sign convention (i.e. a $e^{-ift}$ vs a $e^{ift}$ kernel, a change of convention won't change the properties), for a real signal, the discrete Hartley transform (DHT) is the sum of the real and the imaginary part of the discrete Fourier transform (DFT). In other words, it corresponds to the function: $a+ib \mapsto a+b$.

It is purely real, it is an involution, so it is its own inverse. It seems to still be in use for some coding, encryption or watermarking applications, where the use of real transforms can be simpler, while keeping some of the DFT nice properties (with respect to convolution for instance).

While in general it has been found to be no faster than FFT (see Fast Hartley Transform Implementation in MATLAB), it can be embedded into the same parametric family (New Parametric Discrete Fourier and Hartley Transforms, and Algorithms for Fast Computation, 2011). For some specific hardware, data or data size, it may still be found beneficial. Looking back, it has been used for the Fast computation of discrete cosine transform through fast Hartley transform by Henrique (Rico) Malvar. I suspect that the Hartley transform may have shed some insights into the development of the Lapped Orthogonal Transform (LOT).

Although quite forgotten, it may rise again some day.

You are lucky if you do understand the Fourier transform, I still don't. Assuming a certain classical sign convention (a change of convention won't change the properties), for a real signal, the discrete Hartley transform (DHT) is the sum of the real and the imaginary part of the discrete Fourier transform (DFT).

It is purely real, it is an involution, so it is its own inverse. It seems to still be in use for some coding, encryption or watermarking applications, where the use of real transforms can be simpler, while keeping some of the DFT nice properties.

While in general it has been found to be no faster than FFT, it can be embedded in the same parametric family (New Parametric Discrete Fourier and Hartley Transforms, and Algorithms for Fast Computation, 2011). For some specific hardware, data or data size, it may be found beneficial. Although quite forgotten, it may rise again some day.

You are lucky if you do understand the Fourier transform; I still don't (caveat emptor). 

Assuming a certain classical sign convention (i.e. a $e^{-ift}$ vs a $e^{ift}$ kernel, a change of convention won't change the properties), for a real signal, the discrete Hartley transform (DHT) is the sum of the real and the imaginary part of the discrete Fourier transform (DFT). In other words, $a+ib \to a+b$.

It is purely real, it is an involution, so it is its own inverse. It seems to still be in use for some coding, encryption or watermarking applications, where the use of real transforms can be simpler, while keeping some of the DFT nice properties (with respect to convolution for instance).

While in general it has been found to be no faster than FFT, it can be embedded in the same parametric family (New Parametric Discrete Fourier and Hartley Transforms, and Algorithms for Fast Computation, 2011). For some specific hardware, data or data size, it may be found beneficial. Looking back, it has been used for the Fast computation of discrete cosine transform through fast Hartley transform by H. Malvar. I suspect the Hartley transform may have shed some insights into his development of the Lapped Orthogonal Transform (LOT).

Although quite forgotten, it may rise again some day.