Complex approximation by real approximation

It is known that approximation of the complex functions does not, in general, depend on approximation of the real functions. However, if the complex function is viewed as a Cartesian product of two real functions, a relationship appears between the two types of approximation specified above. This means that if 𝐹 is a complex function where 𝐹 : (𝑥, 𝑦) → (𝑢(𝑥, 𝑦), (𝑣(𝑥, 𝑦)), 𝑥, 𝑦 ∈ 𝑅, and 𝑓 1 , 𝑓 2 are real functions with 𝐹 = 𝑓 1 × 𝑓 2 , then there is a relation between an approximation of 𝐹 and an approximations of 𝑓 1 , 𝑓 2 . This paper studies the relation between approximation of complex functions and approximation of real functions, in terms of existence and degree, in certain special cases. The first case is when 𝑢 is a function only of 𝑥 (i.e. 𝑢(𝑥)) and 𝑣 is a function only of 𝑦 (i.e. 𝑣(𝑦)); this means that the complex function 𝐹 becomes 𝐹 : (𝑥, 𝑦) → (𝑢(𝑥), (𝑣(𝑦)), that is, 𝐹 = 𝑓 1 × 𝑓 2 with 𝑓 1 : 𝑥 → 𝑢(𝑥) and 𝑓 2 : 𝑦 → 𝑣(𝑦). The second case is when there exists a relation between 𝑥 and 𝑦 in the 𝑋-𝑌 plane (i.e. √ 𝑦 = 𝑓 (𝑥)), for example 𝑦 = 𝑥 2 or 𝑦 = 𝑥; this case leads to the first one. In addition, the relationship between the polynomial of best approximation and any polynomial of the complex function 𝐹(𝑥, 𝑦) and its analogous real functions arising from it (i.e. 𝑓 1 (𝑥), 𝑓 2 (𝑦)) is considered in the sense of Kolmogorov’s theorem. Furthermore, the relation between the modulus of continuity and the difference of the function 𝐹(𝑥, 𝑦) and the real functions 𝑓 1 , 𝑓 2 is examined, together with other related results.