The easiest way to construct multivarite kernels from univariates kernel is by using product kernels (https://www.thebigdatablog.com/kernel-based-estimators-for-multivariate-densities-and-functions/). In particular, concerning your question this means

# g-dimensional product kernel

# u and b are g-dimensinal vectors

K=1

for i in range(0,g):

K=K*max(0, 1. / b[i] * 3. / 4 * (1 - u[i] ** 2))

where H=diag(b) is a diagonal matrix.

But you can also think of all kind of other multivariate kernels which are no product kernels.

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