1x1 convolutions, equivalence with fully connected layer
The interpretation that the 1d convolution given in the OP can be duplicated with four separate fully-connected layers is correct (see diagram).  Also, in at least some implementations, kernel weights used during a 1x1 convolution can be made trainable the same way weights in a fully-connected layer can be made trainable.  These points made, every fully-connected layer can not be mathematically duplicated by an equivalent 1x1 convolution. This is based on the definition that 1x1 convolution performs a &quot;column-wise dot product&quot; such that every pixel column in a multi-layer feature map is reduced to a single number (pixel).  A fully-connected layer intermixes weights differently from the way weights are intermixed when performing a 1x1 convolution.  In summary, fully connected layers and 1x1 convolutions each have their own use cases -- some overlap among these use cases exists; however, the two are not intended to be mathematically equivalent in a general sense.