(Sparse tensor product approximation 1) Assume the condition ( Sparse tensor product setup ). Then

First, we verify that where stands for integer part. Indeed, according to the definition ( Sparse tensor product ), and

Clearly

Hence, the inclusion is proven.

Next, we derive the statement from the propositions ( Jackson inequality for wavelets ), ( Tensor product of function spaces ) and ( Sobolev spaces in N dim as tensor products ). We use the proposition ( Sobolev spaces in N dim as tensor products ), the has representations for various placements of among .

We continue estimation We use the proposition ( Jackson inequality for wavelets ) for each term: Hence We use the proposition ( Sobolev spaces in N dim as tensor products ).

(Sparse tensor product approximation 2) Assume the condition ( Sparse tensor product setup ). Then for , , we have

The proof is essentially the same as the proof of the proposition ( Sparse tensor product approximation 1 ). We estimate We use the definition ( Sobolev spaces with dominating mixed derivative ), the has representation

We continue estimation We use the proposition ( Jackson inequality for wavelets ) for each term: