Terminology - regression with one output and multiple output variables I am trying to predict the response when the input is represented by Fourier transform. These form the features and are typically represented as a vector, $x_1,x_2,...,x_d$ where $d$ is the length of the fourier transform. Based on my understanding each such $d$ dimensional vector can be an input to a regression model. The output is $y_1$ which is a scalar real-valued number and there is another output $y_2$ denoting another scalar real-valued number. These are the dependent variables. I have $N$ number of $d$ dimensional examples of fourier transform each labelled by $y_1$ and $y_2$.

Question 1) When the task is of predicting only one output response using the input fourier transform then is the problem termed univariate regression? Is univariate associated by the input's dimension(which is d>1) or the output's dimension (which is 1)

Question 2) When the task is of jointly predicting the two response variables - $y_1$ and $y_2$ then is the problem termed multivariate regression?
Question 1. Both. If you think in opposite to multivariate terms, than in univariate regression both input and output variables should be 1-d

Question 2. Multivariate regression where more than one independent variable (predictors) and more than one dependent variable (responses), are linearly related. So input needs to be more than 2 also.