konvolúciójával

The Hungarian word "konvolúciójával" translates to "with its convolution" in English. In mathematics, convolution is a mathematical operation on two functions that produces a third function. This third function represents how the shape of one of the original functions is modified by the other. It's a way of combining two signals to produce a third signal that expresses how the shape of one is modified by the other.

The operation is often denoted by an asterisk (*), so the convolution of functions f and g is written as f * g. The formula for the convolution of two continuous functions f and g is given by the integral:

(f * g)(t) = ∫[−∞ to ∞] f(τ)g(t − τ) dτ

where t is the independent variable.

Convolution has wide-ranging applications in various fields. In signal processing, it's used to describe the output of a linear time-invariant (LTI) system given its input signal and impulse response. In image processing, convolution is fundamental to applying filters, such as blurring or sharpening, to an image. For instance, a blur filter is essentially a convolution operation where the image is convolved with a blur kernel. The term "konvolúciójával" would be used when discussing the effect or process of applying a convolution, for example, "az kép elmosódott a konvolúciójával" meaning "the image was blurred with its convolution" (referring to the blur kernel). It also appears in probability theory for the sum of independent random variables and in differential equations.