sumCFt

sumCFt is a notational shorthand used in mathematics and programming to denote a discrete weighted sum of two sequences or time-dependent values, typically written as sum_{t in T} C_t F_t. Here, C_t represents a coefficient sequence and F_t a sequence or signal evaluated at index t. The index set T is usually a finite interval of integers, such as t = 1, 2, ..., n, though variants exist for different domains.

Notation and variants

In written form, sumCFt is commonly expressed as sum_t C_t F_t or Σ_t C_t F_t. The camel-case version sumCFt appears in software documentation, notebooks, and field-specific literature as a compact, code-friendly notation. When T is finite and aligned, sumCFt equals the dot product of the two sequences (C_t) and (F_t). In continuous settings, the analogous concept is the inner product defined by the integral ∫ C(t) F(t) dt over a chosen interval.

Computation and properties

sumCFt is linear in each argument: for sequences a_t, b_t and scalars α, β, sumCFt(α a_t + β b_t) = α sumCFt(a_t) + β sumCFt(b_t). It is bilinear with respect to the pair (C_t, F_t). Computationally, it requires a single pass over the index set T, with time complexity proportional to the length of T. In streaming contexts, sumCFt can be updated incrementally as new terms arrive.

Example

Let C = [2, 3, 1] and F = [5, 7, 4]. Then sumCFt = 2*5 + 3*7 + 1*4 = 10 + 21 + 4 = 35.

Applications

sumCFt appears in weighted sums, inner product calculations, projections of a function onto a basis, and various signal processing and statistical models where a weighted aggregation of paired values is required.

See also

Dot product

Weighted sum

Inner product

Convolution

Projection in function spaces