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From planetmath and wolfram, the Fourier-Stieltjes transform of a function $\alpha$ is defined as $\displaystyle \int_{\mathbb{R}} e^{itx} d(\alpha(t)).$ The kernel $\displaystyle e^{itx}$ is unlike the one $\displaystyle e^{-itx}$ used in Fourier transformation.

From Wikipedia, the Laplace-Stieltjes transform of a function $g$ is given as $ \displaystyle \int_{\mathbb{R}} e^{-sx}\,dg(x).$ The kernel ${e}^{-sx}$ is same as the one used in Laplace transformation.

I was wondering why the minus sign in the exponent of kernels for Fourier-Stieltjes transform and Fourier transform are not consistent, while consistent for Laplace-Stieltjes transform and Laplace transform?

Are there mistakes in the quoted sources, or other popular variant definitions?

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I wouldn't say "mistake" but more of "different conventions". You'll also see this for different uses of the discrete Fourier transform (DFT). Some use $\exp(-i\cdot)$ for the forward transform and $\exp(i\cdot)$ for the inverse transform, while others have their convention backwards. Just be careful in noting the convention the author of what you're reading is using. – J. M. May 13 '11 at 23:49
up vote 1 down vote accepted

I guess this is mostly a matter of convention. I think the minus convention in Fourier Series comes from the fact that

$\int_0^{2 \pi} f(t) \, e^{-int} \, dt$

Is the hermitian scalar product of $f$ and $\epsilon^n : x \mapsto e^{inx}$, which makes Fourier inversion formula more natural :

$f = \sum_{n \in \mathbb{Z}} \langle f \,|\, \epsilon^n \rangle \, \epsilon^n$

If you look at planetmath and wolfram, their point of view is to take a function $f$ and write it as an integral of $\epsilon$ with regards to a certain $d \alpha$. On the other hand, Wikipedia starts from $\alpha$ and then defines $f$. I guess the (admittedly subtle) difference in point of view might explain their convention.

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