w2AOOnorm

w2AOOnorm is a norm intended for spaces that admit a fixed orthogonal decomposition into blocks. It combines a weight sequence with the standard L2 measure on each block, producing a block-weighted L2 norm often used in multicomponent data analysis and regularization contexts.

Definition: Let V be a real or complex inner product space that decomposes as V = ⊕_{i=1}^k V_i with orthogonal projections P_i onto V_i. Let w2 = (w_i)_{i=1}^k be a sequence of nonnegative weights. For any v ∈ V, the w2AOOnorm of v is defined by

w2AOOnorm(v) = sqrt( sum_{i=1}^k w_i^2 ||P_i v||^2 ).

If every w_i > 0, this is a norm on V.

Properties: w2AOOnorm satisfies positivity, homogeneity, and the triangle inequality, leveraging the Pythagorean structure of the orthogonal decomposition and the standard Euclidean norms on each block.

Special cases: If k = 1 and w_1 > 0, w2AOOnorm(v) equals w_1 ||v||, a scalar multiple of the usual norm. If the decomposition groups coordinates into blocks and w_i = 1 for all i, w2AOOnorm becomes the block L2 norm, a common choice in grouped data settings.

Applications: This norm is suitable for regularization in optimization problems where different components contribute unequally, such as multi-block regression, block-structured sparse recovery, and multiresolution or multi-scale analyses. It provides a size measure that respects the underlying block structure of the space.

See also: weighted norms, block norm, L2 norm, direct sum, orthogonal decomposition.