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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-340/3/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 340 / 3 / d / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

V_{j}

and its detail spaces

W_{j}

, take the completed tensor product

V_{j} = V_{j} \otimes V_{j}

. Since

V_{j + 1} = V_{j} \oplus W_{j}

V_{j + 1} = V_{j} \oplus (W_{j} \otimes V_{j}) \oplus (V_{j} \otimes W_{j}) \oplus (W_{j} \otimes W_{j}) .

(1)

The three tensor-product wavelets are

Ψ^{1} (x, y) = ψ (x) φ (y), Ψ^{2} (x, y) = φ (x) ψ (y), Ψ^{3} (x, y) = ψ (x) ψ (y) .

(2)

Taking products of members of the two one-dimensional families gives an orthonormal basis of each respective completed tensor product. Density and the coarse-scale trivial intersection yield

{2^{j} Ψ^{a} (2^{j} x - k_{1}, 2^{j} y - k_{2}) : a = 1, 2, 3, j, k_{1}, k_{2} \in Z}

(3)

as an orthonormal basis of

L^{2} (R^{2})

. The normalization is

2^{j}

in two dimensions. On the square, tensorize the interval-adapted wavelet basis and include the finite coarse scaling function part.

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