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Topics referred to by the same term
Square-summable may refer to: Square-integrable functions Square-summable sequences; see Hilbert space § Sequence spaces This disambiguation page lists
Square-summable
Function whose squared absolute value has finite integral
mathematics, a square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function
Square-integrable_function
Type of vector space in math
the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is often in the older literature referred to as the
Hilbert_space
Index of articles associated with the same name
Least squares For the "sum of squared differences", see Mean squared error For the "sum of squared error", see Residual sum of squares For the "sum of squares
Sum_of_squares
Statistical measure of the discrepancy between data and an estimation model
residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of
Residual_sum_of_squares
Characterization by prime factors of sums of two squares
theory, the sum of two squares theorem relates the prime decomposition of any integer n > 1 to whether it can be written as a sum of two squares, such that
Sum_of_two_squares_theorem
Statistical quantity
In statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression (SSR – not to be
Explained_sum_of_squares
Statistical quantity
In statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses
Total_sum_of_squares
Number-theoretical function
In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n {\displaystyle
Sum_of_squares_function
Condition under which an odd prime is a sum of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2}
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Concept that permeates much of inferential statistics and descriptive statistics
sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of
Partition_of_sums_of_squares
Theorem in probability
in analysis bounding the expectation a weighted sum of Rademacher random variables with square-summable weights. It is named after Aleksandr Khinchin and
Khintchine_inequality
Square of numbers with equal row, column and diagonal totals
and recreational mathematics, a magic square is a square array of numbers, usually positive integers, where the sums of the numbers in each row, each column
Magic_square
Decomposition of periodic functions
continuous, and therefore bounded, it is square-integrable and its Fourier coefficients are square-summable. Then, by the Cauchy–Schwarz inequality, (
Fourier_series
Measure of the error of an estimator
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures
Mean_squared_error
Function spaces generalizing finite-dimensional p norm spaces
sequences whose series are absolutely convergent, ℓ2, the space of square-summable sequences, which is a Hilbert space, and ℓ∞, the space of bounded sequences
Lp_space
Mathematical theorem
says that integrating term by term a trigonometric series with given square summable coefficients, he gets a series converging uniformly to a continuous
Riesz–Fischer_theorem
Value in statistics
a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals
Lack-of-fit_sum_of_squares
Type of algebraic integer
the distance from its consecutive powers to the nearest integer is square-summable, or ℓ 2, then α is a Pisot number (and, in particular, algebraic).
Pisot–Vijayaraghavan_number
Problem in computer science
Turing run-time complexity of the square-root sum problem? More unsolved problems in computer science The square-root sum problem (SRS) is a computational
Square-root_sum_problem
Topics referred to by the same term
the O2 Academy Liverpool The L2 space of square-integrable functions L2 norm The ℓ2 space of square-summable sequences L2 cohomology, a cohomology theory
L2
Numerical optimization process
A sum-of-squares optimization program is an optimization problem with a linear cost function and constraints that certain polynomials constructed from
Sum-of-squares_optimization
Product of an integer with itself
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with
Square_number
Sum of inverse squares of natural numbers
analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard
Basel_problem
Correlation of a signal with a time-shifted copy of itself, as a function of shift
R_{yy}(\ell )=\sum _{n\in Z}y(n)\,{\overline {y(n-\ell )}}} The above definitions work for signals that are square integrable, or square summable, that is,
Autocorrelation
Orthogonal polynomials
satisfy the following orthogonality relation in the Hilbert space of square summable sequences associated with the Poisson distribution with parameter μ
Charlier_polynomials
Topological space with a dense countable subset
Hilbert space is isometric to the space l 2 {\displaystyle l^{2}} of square-summable sequences. An example of a separable space that is not second-countable
Separable_space
Signal (re‑)construction algorithm
any stationary process, in which case the sample sequence is not square summable, and is not in any ℓp(ℤ,ℂ) space. If x[n] is an infinite sequence of samples
Whittaker–Shannon interpolation formula
Whittaker–Shannon_interpolation_formula
Every natural number can be represented as the sum of four integer squares
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative
Lagrange's four-square theorem
Lagrange's_four-square_theorem
Normed vector space that is complete
numbers ℕ; among them, the space ℓ1 of absolutely summable sequences and the space ℓ2 of square summable sequences; the space c0 of sequences tending to
Banach_space
Probability distribution and special case of gamma distribution
the distribution of a sum of the squares of k {\displaystyle k} independent standard normal random variables. The chi-squared distribution χ k 2 {\displaystyle
Chi-squared_distribution
List of mathematical contexts in which exponentiated terms are summed
In mathematics and statistics, sums of powers occur in a number of contexts: Sums of squares arise in many contexts. For example, in geometry, the Pythagorean
Sums_of_powers
Evaluates how likely it is that any difference between data sets arose by chance
steps: Calculate the chi-squared test statistic, χ 2 {\displaystyle \chi ^{2}} , which resembles a normalized sum of squared deviations between observed
Pearson's_chi-squared_test
Approximation method in statistics
and regression analysis, least squares is a method to determine the best-fit model by minimizing the sum of the squared residuals—the differences between
Least_squares
Concept in convex optimization mathematics
Constant step length, αk=γ/‖g(k)‖2, which gives ‖x(k+1)−x(k)‖2=γ. Square summable but not summable step size, i.e. any step sizes satisfying αk≥0,∑k=1∞αk2<∞,∑k=1∞αk=∞
Subgradient_method
Topics referred to by the same term
(category theory) (also called an amalgamated sum or a cocartesian square, fibered coproduct, or fibered sum), the colimit of a diagram consisting of two
Sum
corresponds to a square antiprism. Specific methods of distributing the points include, for example, minimizing the sum of all reciprocals of squares of distances
List_of_sums_of_reciprocals
Detail in the 1514 engraving Melencolia I
square is a symmetrical magic square and has 86 sum combinations of the magic number 34. For a better overview, all sum combinations are grouped below
Albrecht_Dürer's_magic_square
Fifteen open problems in mathematical physics
components of the spectral measure for some Sturm–Liouville operators with square summable potential". Journal of Differential Equations. 191 (1): 90–104. Bibcode:2003JDE
Simon_problems
Product of sums of four squares expressed as a sum of four squares
Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. For any pair
Euler's_four-square_identity
Calculations in probability theory
squared deviations aka total sum of squares E ( T − C ) = ( k − 1 ) σ 2 {\displaystyle \operatorname {E} (T-C)=(k-1)\sigma ^{2}} treatment squared deviations
Squared deviations from the mean
Squared_deviations_from_the_mean
Topological complex vector space
can be developed. To be specific, H is isomorphic to the space of square summable sequences l2; we may assume that H = l2. For each natural number n
C*-algebra
Statistical hypothesis test
A chi-squared test (also chi-square or χ2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are
Chi-squared_test
Says when a natural number is the sum of three squares of integers
mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers n=x2+y2+z2 if and
Legendre's three-square theorem
Legendre's_three-square_theorem
Statement in complex analysis; formerly the Bieberbach conjecture
Schwarz lemma De Branges de Bourcia, Louis; Rovnyak, James (2014). Square Summable Power Series. Dover Books on Mathematics. Newburyport: Dover Publications
De_Branges's_theorem
Grimoire of the 16th century
sigils for the use on talismans or amulets. Sigill Lunae, a 9×9 magic square (sum 369), to be inscribed on a silver talisman It was first printed in 1591
Archidoxis_magica
Mathematical-logic system
a r e _ s u m ( x , y ) = x 2 + y 2 {\displaystyle \operatorname {square\_sum} (x,y)=x^{2}+y^{2}} can be rewritten in anonymous form as ( x , y ) ↦
Lambda_calculus
Theorem in topology
Hilbert spaces are not compact. For example, in the Hilbert space ℓ2 of square-summable real (or complex) sequences, consider the map f : ℓ2 → ℓ2 which sends
Brouwer_fixed-point_theorem
Ancient Mesopotamian civilization from 3300 to 1900 BC
Sumer (/ˈsuːmər/ SOO-mər) is the earliest known civilization, located in the historical region of southern Mesopotamia (now south-central Iraq), emerging
Sumer
Mathematical description of quantum state
takes functions in the above space to elements of l2(Z), the space of square summable functions Z → C. The latter space is a Hilbert space and the Fourier
Wave_function
Situation where total gains match total losses
In game theory and economic theory, zero-sum game is a mathematical representation of a situation that involves two competing entities, where the result
Zero-sum_game
Unit-distance-preserving maps are isometries
space, the Hilbert space of square-summable sequences of real numbers. "Square-summable" means that the sum of the squares of the values in a sequence
Beckman–Quarles_theorem
Square root of the mean square
mathematics, the root mean square (abbrev. RMS, rms or rms) of a set of values is the square root of the set's mean square. Given a set x i {\displaystyle
Root_mean_square
Relation between sides of a right triangle
the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two
Pythagorean_theorem
Linear operator
one of self-adjoint Jacobi operators acting on the Hilbert space of square summable sequences over the positive integers ℓ 2 ( N ) {\displaystyle \ell
Jacobi_operator
Topic in mathematical analysis
If the sequence is infinite, assume that it is square-summable: ∑ m | u m | 2 < ∞ {\displaystyle \sum _{m}|u_{m}|^{2}<\infty } Hilbert's inequality (see
Hilbert's_inequality
Number of stacked spheres in a pyramid
for summing consecutive squares to give a cubic polynomial, whose values are the square pyramidal numbers, are given by Archimedes, who used this sum as
Square_pyramidal_number
Class of ordinary differential equations
Because of Fourier analysis, since the Fourier coefficients are "square-summable", the Fourier series converges in L2 which is all we need for this
Sturm–Liouville_theory
Statistical error measure
selected performance measure, in the sense that the least squares approach is related to the mean squared error, the equivalent for mean absolute error is least
Mean_absolute_error
On when a space equals the closed convex hull of its extreme points
space Lp space ℓ 2 ( N ) {\displaystyle \ell ^{2}(\mathbb {N} )} of square-summable sequences with the usual norm ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}}
Krein–Milman_theorem
Theorem of stationary processes
coefficients have these properties: Stable, that is, square summable ∑ j = 1 ∞ | b j | 2 {\displaystyle \sum _{j=1}^{\infty }|b_{j}|^{2}} < ∞ {\displaystyle
Wold's_theorem
Statistical measure of how far values spread from their average
as the expected value of the squared deviation from the mean of a random variable. The standard deviation is the square root of the variance. Technically
Variance
Mathematical concept of arrangement of numbers in a square
magic square is a magic square for which each pair of numbers symmetrically opposite to the center sum up to the same value. For an n × n square, filled
Associative_magic_square
Number without repeated prime factors
In mathematics, a square-free integer (or squarefree integer) is an integer that is divisible by no square number other than 1. That is, its prime factorization
Square-free_integer
Type of non-sinusoidal waveform
Square wave sound sample 5 seconds of square wave at 220 Hz Sine wave sound sample For comparison, five seconds of a 220 Hz sine wave. Problems playing
Square_wave_(waveform)
Figurate number
relations to other figurate numbers. Most simply, the sum of two consecutive triangular numbers is a square number, since: T n − 1 + T n {\displaystyle T_{n-1}+T_{n}}
Triangular_number
Indicator for how well data points fit a line or curve
sums of squares formulas: The sum of squares of residuals, also called the residual sum of squares: SSres=∑i(yi−fi)2=∑iei2 The total sum of squares (proportional
Coefficient_of_determination
Space with topology generated by convex sets
compact. For example, let H be the separable Hilbert space ℓ2(ℕ) of square-summable sequences with the usual norm ‖⋅‖2 and let en=(0,…,0,1,0,…) be the
Locally convex topological vector space
Locally_convex_topological_vector_space
Natural number
divisible by 9 if and only if the sum of its digits is divisible by 9. 9 is the only square number that is the sum of two consecutive, positive cubes:
9
Construct in mathematics
because all separable Hilbert spaces are isomorphic to the space of square-summable sequences ℓ 2 {\displaystyle \ell ^{2}} . The homotopy-theoretic interpretation
Gerbe
Infinite sum
from addition is summable if the series added were summable, and, second, the sum of the resulting series is the addition of the sums of the added series
Series_(mathematics)
Addition of several numbers or other values
of the product) ∑i=0ni2=∑i=1ni2=n(n+1)(2n+1)6=n33+n22+n6 (Sum of the first squares, see square pyramidal number.) ∑i=0ni3=(∑i=0ni)2=(n(n+1)2)2=n44+n32+n24
Summation
Noncommutative geometric structure
infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional
Singular_trace
Product of a number by itself
mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as
Square_(algebra)
Square of a triangular number
In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, 1 3 + 2 3 + 3 3 + ⋯ + n 3 = ( 1 + 2 + 3 + ⋯ + n ) 2
Squared_triangular_number
Statistical test
Begin by centering the data in each group The within-group sum of squares is the sum of squares of all 18 values in this table S W = ( 1 ) 2 + ( 3 ) 2 +
One-way_analysis_of_variance
Unsolved problem in mathematics
A magic square is a square array of integer numbers in which each row, column, and diagonal sums to the same number. The order of the square refers to
Magic_square_of_squares
Process of deriving classes from, and organizing them into, a hierarchy
stream(). .mapToInt(this::transform) .sum(); } } class SquareSumComputer extends SumComputer { public SquareSumComputer(int a, int b) { super(a, b); }
Inheritance (object-oriented programming)
Inheritance_(object-oriented_programming)
Financial technology company
com. TechCrunch. Retrieved 29 March 2022. Lunden, Ingrid. "SumUp, Another European Square, Expands To Italy, Spain And Holland, Now Covering 7 Countries
SumUp
Linear operator on dense subset of its apparent domain
{\displaystyle X=\ell ^{2}(\mathbb {N} )} be the Hilbert space of square-summable sequences, with orthonormal basis ( e n ) n ≥ 1 {\displaystyle (e_{n})_{n\geq
Densely_defined_operator
Decomposition of a number into a product
sieve (SNFS) Algebraic factorization (which includes difference of two squares, sum of two cubes, Aurifeuillean factorization, etc.) A general-purpose factoring
Integer_factorization
Simple model for one-dimensional crystal in solid-state physics
pair, that is, two operators L(t) and P(t) in the Hilbert space of square summable sequences ℓ 2 ( Z ) {\displaystyle \ell ^{2}(\mathbb {Z} )} such that
Toda_lattice
Power in alternating current systems
={P^{+} \over S}} , that is, the root of squared sums of line voltages multiplied by the root of squared sums of line currents. P + {\displaystyle P^{+}}
AC_power
American mathematician (1933–2017)
1090/S0273-0979-96-00634-9. de Branges, Louis and Rovnyak, James. Square summable power series. Holt, Rinehart and Winston, New York-Toronto, Ont.-London
Donald_Sarason
}\sum _{jk=-\infty }^{\infty }c_{jk}\psi _{jk}{\bigg \Vert }_{L^{2}}^{2}\leq B\Vert c_{jk}\Vert _{l^{2}}^{2}\,} for all bi-infinite square summable series
Dual_wavelet
Number whose square is a given number
In mathematics, a square root of a number x is a number y such that y2=x; in other words, a number y whose square (the result of multiplying the number
Square_root
Natural number
that every integer is the sum of at most three triangular numbers. Three is the only prime which is one less than a perfect square. Any other number which
3
\operatorname {span} (\{\mathbf {e} _{1},\mathbf {e} _{2}\})} . On the square summable sequences, the operators R : ℓ 2 ( N ) → ℓ 2 ( N ) : ( x 1 , x 2 ,
Partial_isometry
Matrix used in complex analysis
curve. In fact the infinite matrix A acting on the Hilbert space of square summable sequences satisfies A ∗ A = I , {\displaystyle A^{*}A=I,} But if J
Grunsky_matrix
Generalization of the concept of a direct sum in mathematics
\end{cases}}} In the above, ℓ 2 {\displaystyle \ell ^{2}} is the space of square summable sequences. All infinite-dimensional separable Hilbert spaces are isomorphic
Direct_integral
How many ways a positive integer can be represented as the sum of four squares
Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares (of integers)
Jacobi's_four-square_theorem
Operator on a Hilbert space that shifts basis vectors
a model for more general theories. Let ℓ2 be the Hilbert space of square-summable sequences of complex numbers, i.e., ℓ2={(a0,a1,a2,…):an∈ℂ and ∑n=0∞|an|2<∞}The
Unilateral_shift_operator
Natural number
fifteen) is the natural number following 114 and preceding 116. 115 has a square sum of divisors: σ ( 115 ) = 1 + 5 + 23 + 115 = 144 = 12 2 . {\displaystyle
115_(number)
Most general completion of a commutative square given two morphisms with same domain
pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms
Pushout_(category_theory)
Statistic in regression analysis
In statistics, the predicted residual error sum of squares (PRESS) is a form of cross-validation used in regression analysis to provide a summary measure
PRESS_statistic
Shape with four equal sides and angles
In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles
Square
a discrete group acting on A. We let K be the Hilbert space of all square summable H-valued functions on G. There is an action of A on K given by a(k)(g)
Crossed_product
Swedish mathematician
"The composition operators on the space of Dirichlet series with square summable coefficients". Michigan Mathematical Journal. 46 (2): 313–329. doi:10
Hakan_Hedenmalm
Integer that is both a perfect square and a triangular number
a square triangular number (or triangular square number) is a number which is both a triangular number and a square number, in other words, the sum of
Square_triangular_number
Connected sum of two trefoil knots with opposite chirality
In knot theory, the square knot is a composite knot obtained by taking the connected sum of a trefoil knot with its reflection. It is closely related to
Square_knot_(mathematics)
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